Search: a001906 -id:a001906
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A130259
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Maximal index k of an even Fibonacci number (A001906) such that A001906(k) = Fib(2k) <= n (the 'lower' even Fibonacci Inverse).
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+20
10
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0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5
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OFFSET
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0,4
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COMMENTS
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Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130260 for another version).
a(n)+1 is the number of even Fibonacci numbers (A001906) <=n.
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LINKS
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FORMULA
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a(n) = floor(arcsinh(sqrt(5)*n/2)/(2*log(phi))), where phi=(1+sqrt(5))/2.
G.f.: g(x) = 1/(1-x)*Sum_{k>=1} x^Fibonacci(2*k).
a(n) = floor(1/2*log_phi(sqrt(5)*n+1)) for n>=0.
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EXAMPLE
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MATHEMATICA
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Table[Floor[1/2*Log[GoldenRatio, (Sqrt[5]*n + 1)]], {n, 0, 100}] (* G. C. Greubel, Sep 12 2018 *)
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PROG
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(PARI) vector(100, n, n--; floor(log((sqrt(5)*n+1))/(2*log((1+sqrt(5))/2) ))) \\ G. C. Greubel, Sep 12 2018
(Magma) [Floor(Log((Sqrt(5)*n+1))/(2*Log((1+Sqrt(5))/2)))): n in [0..100]]; // G. C. Greubel, Sep 12 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A125662
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A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
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+20
9
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1, 3, 1, 8, 6, 1, 21, 25, 9, 1, 55, 90, 51, 12, 1, 144, 300, 234, 86, 15, 1, 377, 954, 951, 480, 130, 18, 1, 987, 2939, 3573, 2305, 855, 183, 21, 1, 2584, 8850, 12707, 10008, 4740, 1386, 245, 24, 1, 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1
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OFFSET
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0,2
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COMMENTS
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Subtriangle of the triangle given by [0,3,-1/3,1/3,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A123965.
Riordan array (1/(1-3*x+x^2), x/(1-3*x+x^2)).
Triangle of coefficients of Chebyshev's S(n,x+3) polynomials (exponents of x in increasing order). (End)
For 1 <= k <= n, T(n,k) equals the number of (n-1)-length words over {0,1,2,3} containing k-1 letters equal 3 and avoiding 01. - Milan Janjic, Dec 20 2016
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LINKS
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FORMULA
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T(n,k) = T(n-1,k-1) + 3*T(n-1,k) - T(n-2,k); T(0,0)=1; T(n,k)=0 if k < 0 or k > n.
Sum_{k=0..n} T(n, k) = A001353(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000244(n+1).
T(n, k) = abs( [x^k]( ChebyshevU(n, (3-x)/2) ) ).
Sum_{k=0..n} (-1)^k*T(n, k) = A000027(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000225(n). (End)
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EXAMPLE
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Triangle begins:
1;
3, 1;
8, 6, 1;
21, 25, 9, 1;
55, 90, 51, 12, 1;
...
Triangle [0,3,-1/3,1/3,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins:
1;
0, 1;
0, 3, 1;
0, 8, 6, 1;
0, 21, 25, 9, 1;
0, 55, 90, 51, 12, 1;
...
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MATHEMATICA
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With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 3 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)
Table[Abs[CoefficientList[ChebyshevU[n, (x-3)/2], x]], {n, 0, 12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)
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PROG
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(Magma)
m:=12;
R<x>:=PowerSeriesRing(Integers(), m+2);
A125662:= func< n, k | Abs( Coefficient(R!( Evaluate(ChebyshevU(n+1), (3-x)/2) ), k) ) >;
(SageMath)
def A125662(n, k): return abs( ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k] )
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CROSSREFS
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Diagonal sums: A000244(powers of 3).
Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824, A207823, A125662, A078812, A101950, A049310, A104562, A053122, A207815, A159764, A123967 for k = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 respectively.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A130260
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Minimal index k of an even Fibonacci number A001906 such that A001906(k) = Fib(2k) >= n (the 'upper' even Fibonacci Inverse).
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+20
8
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0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6
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OFFSET
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0,3
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COMMENTS
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Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130259 for another version).
a(n+1) is the number of even Fibonacci numbers (A001906) <=n.
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LINKS
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FORMULA
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a(n) = ceiling(arcsinh(sqrt(5)*n/2)/(2*log(phi))) for n>=0.
a(n) = ceiling(arccosh(sqrt(5)*n/2)/(2*log(phi))) for n>=1.
a(n) = ceiling(log_phi(sqrt(5)*n)/2)=ceiling(log_phi(sqrt(5)*n-1)/2) for n>=1, where phi=(1+sqrt(5))/2.
G.f.: g(x)=x/(1-x)*Sum_{k>=0} x^Fib(2*k).
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EXAMPLE
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MATHEMATICA
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Join[{0}, Table[Ceiling[Log[GoldenRatio, Sqrt[5]*n]/2], {n, 1, 100}]] (* G. C. Greubel, Sep 12 2018 *)
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PROG
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(PARI) for(n=0, 100, print1(if(n==0, 0, ceil(log(sqrt(5)*n)/(2*log((1+ sqrt(5))/2)))), ", ")) \\ G. C. Greubel, Sep 12 2018
(Magma) [0] cat [Ceiling(Log(Sqrt(5)*n)/(2*Log((1+ Sqrt(5))/2))): n in [1..100]]; // G. C. Greubel, Sep 12 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A169690
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Let S be the sequence Fibonacci(2n), n>0 (cf. A001906); sequence lists the differences S(j)-S(i) for i<j.
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+20
7
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2, 5, 7, 13, 18, 20, 34, 47, 52, 54, 89, 123, 136, 141, 143, 233, 322, 356, 369, 374, 376, 610, 843, 932, 966, 979, 984, 986, 1597, 2207, 2440, 2529, 2563, 2576, 2581, 2583, 4181, 5778, 6388, 6621, 6710, 6744, 6757, 6762, 6764, 10946, 15127, 16724, 17334, 17567
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OFFSET
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1,1
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COMMENTS
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The sequences S and T (see A169691) are a pair of sequences with the property that the differences between the terms of S are disjoint from the differences between the terms of T, thus answering a question posed by S. W. Golomb.
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LINKS
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MATHEMATICA
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nn=30; With[{fib2=Fibonacci[Range[2, nn, 2]]}, Union[Flatten[Table[ fib2[[n]]- Take[ fib2, n-1], {n, nn/2}]]]] (* Harvey P. Dale, Jun 18 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A340097
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Odd composite integers m such that A001906(m-J(m,5)) == 0 (mod m) and gcd(m,5)=1, where J(m,5) is the Jacobi symbol.
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+20
6
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21, 323, 329, 377, 451, 861, 1081, 1819, 1891, 2033, 2211, 3653, 3827, 4089, 4181, 5671, 5777, 6601, 6721, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 17119, 17513, 17711, 17941, 18407, 19043, 19951, 20473, 23407, 25369, 25651, 25877, 27323, 27511
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OFFSET
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1,1
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COMMENTS
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The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=3 and b=1, we have D=5 and U(m) recovers A001906(m).
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REFERENCES
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D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
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LINKS
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MATHEMATICA
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Select[Range[3, 30000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 5] - 1, 3/2], #] &]
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CROSSREFS
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Cf. A001906, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340098 (a=5, b=1), A340099 (a=7, b=1).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Coprime condition added to definition by Georg Fischer, Jul 20 2022
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STATUS
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approved
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A338007
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Odd composite integers m such that A001906(m)^2 == 1 (mod m).
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+20
4
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9, 21, 63, 99, 231, 323, 329, 369, 377, 423, 451, 861, 903, 1081, 1189, 1443, 1551, 1819, 1833, 1869, 1891, 2033, 2211, 2737, 2849, 2871, 2961, 3059, 3289, 3653, 3689, 3827, 4059, 4089, 4179, 4181, 4879, 5671, 5777, 6447, 6479, 6601, 6721, 6903, 7743
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OFFSET
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1,1
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COMMENTS
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For a, b integers, the generalized Lucas sequence is defined by the relation U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1.
This sequence satisfies the relation U(p)^2 == 1 for p prime and b=1,-1.
The composite numbers with this property may be called weak generalized Lucas pseudoprimes of parameters a and b.
The current sequence is defined for a=3 and b=1.
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REFERENCES
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D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (2020).
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LINKS
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MATHEMATICA
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Select[Range[3, 8000, 2], CompositeQ[#] && Divisible[ChebyshevU[#-1, 3/2]*ChebyshevU[#-1, 3/2] - 1, #] &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A289803
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p-INVERT of the even bisection (A001906) of the Fibonacci numbers, where p(S) = 1 - S - S^2.
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+20
3
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1, 5, 23, 103, 456, 2009, 8833, 38803, 170399, 748176, 3284833, 14421533, 63314735, 277968871, 1220356440, 5357681369, 23521603225, 103265890987, 453363808127, 1990383615264, 8738295434881, 38363361811637, 168425013526727, 739429075564711, 3246283590352104
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OFFSET
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0,2
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COMMENTS
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Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
See A289780 for a guide to related sequences.
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LINKS
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Rigoberto Flórez, Javier González, Mateo Matijasevick, Cristhian Pardo, José Luis Ramírez, Lina Simbaqueba, and Fabio Velandia, Lattice paths in corridors and cyclic corridors, Contrib. Disc. Math. (2024) Vol. 19. No. 2, 36-55. See p. 11.
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FORMULA
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G.f.: (1 - 2 x + x^2)/(1 - 7 x + 13 x^2 - 7 x^3 + x^4).
a(n) = 7*a(n-1) - 13*a(n-2) + 7*a(n-3) - a(n-4).
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MATHEMATICA
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z = 60; s = x/(1 - 3*x + x^2); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A001906 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289803 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A337777
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Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m)=A001906(m) and V(m)=A005248(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=1, respectively.
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+20
3
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4, 44, 836, 1364, 2204, 7676, 7964, 9164, 11476, 12524, 23804, 31124, 32642, 39556, 73124, 80476, 99644, 110564, 128876, 156484, 192676, 199924, 287804, 295196, 315524, 398924, 542242, 715604, 780044, 934876, 987524, 1050524, 1339516, 1390724, 1891124, 1996796
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OFFSET
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1,1
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COMMENTS
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For a, b integers, the following sequences are defined:
generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1;
generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b. The current sequence is defined for a=3 and b=1.
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LINKS
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MATHEMATICA
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Select[Range[2, 20000, 2], CompositeQ[#] && Divisible[LucasL[2#] - 3, #] && Divisible[ChebyshevU[#-1, 3/2]*ChebyshevU[#-1, 3/2] - 1, #] &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A340122
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Odd composite integers m such that A001906(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.
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+20
3
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9, 21, 27, 63, 81, 189, 243, 323, 329, 351, 377, 423, 451, 567, 729, 783, 861, 891, 963, 1081, 1701, 1743, 1819, 1891, 1967, 2033, 2187, 2211, 2871, 2889, 2961, 3321, 3653, 3807, 3827, 4089, 4181, 5103, 5229, 5671, 5777, 5901, 6561, 6601, 6721, 6741, 7587
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OFFSET
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1,1
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COMMENTS
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The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == J(p,D) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
The composite integers m with the property U(k*m-J(m,D)) == J(m,D)*U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a. Here b=1, a=3, D=5 and k=2, while U(m) is A001906(m).
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REFERENCES
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D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
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LINKS
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MATHEMATICA
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Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] &&
Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 5] - 1, 3/2] - JacobiSymbol[#, 5], #] &]
Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#]
&& Divisible[Fibonacci[2*(2*#-JacobiSymbol[#, 5]), 1] - JacobiSymbol[#, 5], #] &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 1, 3, 4, 8, 11, 21, 29, 55, 76, 144, 199, 377, 521, 987, 1364, 2584, 3571, 6765, 9349, 17711, 24476, 46368, 64079, 121393, 167761, 317811, 439204, 832040, 1149851, 2178309, 3010349, 5702887, 7881196, 14930352, 20633239, 39088169
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OFFSET
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0,3
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COMMENTS
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Sequence relates bisections of Lucas and Fibonacci numbers (see also A098149).
Floretion Algebra Multiplication Program, FAMP code: 4jesleftforsumseq[ + .25'i + .25i' + .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' + .25e], vesleftforsumseq = A000045, sumtype: (Y[15], *, inty*sum) (internal program code)
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LINKS
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FORMULA
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G.f.: (1+x+x^3)/((1+x-x^2)*(1-x-x^2)).
a(n) = ((3/20)*sqrt(5) + 3/4)*(1/2 + (1/2)*sqrt(5))^n + (-(3/20)*sqrt(5) + 3/4)*(1/2 - (1/2)*sqrt(5))^n + (-(3/20)*sqrt(5) - 1/4)*(-1/2 + (1/2)*sqrt(5))^n + ((3/20)*sqrt(5) - 1/4) *(-1/2 - (1/2)*sqrt(5))^n.
a(n) = 3*a(n-2) - a(n-4), n >= 4; a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4. - Daniel Forgues, May 07 2011
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MAPLE
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a:= n-> (<<0|1>, <-1|3>>^iquo(n, 2, 'r'). <<1, 3+r>>)[1, 1]:
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MATHEMATICA
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CoefficientList[Series[(1+x+x^3)/((1+x-x^2)(1-x-x^2)), {x, 0, 40}], x] (* Harvey P. Dale, Aug 10 2021 *)
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PROG
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(GAP) a:=[1, 1, 3, 4];; for n in [5..40] do a[n]:=3*a[n-2]-a[n-4]; od; a; # Muniru A Asiru, Aug 09 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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