| Displaying 1-6 of 6 results found.
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5, 11, 21, 33, 53, 77, 117, 165, 245, 341, 501, 693, 1013, 1397, 2037, 2805, 4085, 5621, 8181, 11253, 16373, 22517, 32757, 45045, 65525, 90101, 131061, 180213, 262133, 360437, 524277, 720885, 1048565, 1441781, 2097141, 2883573, 4194293, 5767157
FORMULA
a(n) = 2*a(n-2)+11 for n > 2; a(1) = 5, a(2) = 11.
a(n) = (27-5*(-1)^n)*2^(1/4*(2*n-1+(-1)^n))/2-11.
G.f.: x*(5+6*x)/(1-x-2*x^2+2*x^3).
PROG
(Magma) T:=[ n le 2 select n+4 else 2*Self(n-2): n in [1..38] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];
Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.
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1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 5, 3, 4, 1, 5, 4, 7, 2, 6, 3, 7, 2, 4, 5, 8, 3, 11, 4, 11, 1, 6, 5, 6, 4, 13, 7, 10, 2, 13, 6, 17, 3, 8, 7, 17, 2, 9, 4, 10, 5, 19, 8, 6, 3, 14, 11, 19, 4, 23, 11, 12, 1, 10, 6, 23, 5, 14, 6, 29, 4, 29, 13, 8, 7, 9, 10, 31
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Also the Heinz number of the part-wise half (rounded down) of the partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Each n appears A000005(n) times at odd positions (infinitely many at even). To see this, note that our transformation does not distinguish between A066207 and A066208.
FORMULA
Completely multiplicative with a(prime(2k)) = prime(k) and a(prime(2k+1)) = prime(k). Cf. A297002.
EXAMPLE
The prime indices of n = 1501500 are {1,1,2,3,3,3,4,5,6}, so the prime indices of a(n) are {1,1,1,1,2,2,3}; hence we have a(1501500) = 720.
The 6 odd positions of 2124 are: 63, 99, 105, 165, 175, 275, with prime indices:
63: {2,2,4}
99: {2,2,5}
105: {2,3,4}
165: {2,3,5}
175: {3,3,4}
275: {3,3,5}
MATHEMATICA
Table[Times@@(If[#1<=2, 1, Prime[Floor[PrimePi[#1]/2]]^#2]&@@@FactorInteger[n]), {n, 100}]
CROSSREFS
Positions of 2's are 3 and A164095. A004526 is floor(n/2), with an extra first zero.
A109763 lists primes of index floor(n/2).
Expansion of (5-19*x)/(1-10*x+23*x^2).
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5, 31, 195, 1237, 7885, 50399, 322635, 2067173, 13251125, 84966271, 544886835, 3494644117, 22414043965, 143763624959, 922113238395, 5914569009893, 37937085615845, 243335768930911, 1560804720144675, 10011324516035797
COMMENTS
Binomial transform of A161731 without initial 1. Fifth binomial transform of A164095. Inverse binomial transform of A164110.
FORMULA
a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 31.
G.f.: (5-19*x)/(1-10*x+23*x^2).
a(n) = ((5+3*sqrt(2))*(5+sqrt(2))^n + (5-3*sqrt(2))*(5-sqrt(2))^n)/2.
E.g.f: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(5*x). - G. C. Greubel, Sep 08 2017
MATHEMATICA
LinearRecurrence[{10, -23}, {5, 31}, 50] (* or *) CoefficientList[Series[(5 - 19*x)/(1 - 10*x + 23*x^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 08 2017 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+3*r)*(5+r)^n+(5-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
a(n) = 12*a(n-1) - 34*a(n-2) for n > 1; a(0) = 5, a(1) = 36.
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3
5, 36, 262, 1920, 14132, 104304, 771160, 5707584, 42271568, 313200960, 2321178208, 17205305856, 127543611200, 945542935296, 7010032442752, 51971929512960, 385322051101952, 2856819009782784, 21180878379927040
FORMULA
a(n) = ((5+3*sqrt(2))*(6+sqrt(2))^n+(5-3*sqrt(2))*(6-sqrt(2))^n)/2.
G.f.: (5-24*x)/(1-12*x+34*x^2).
E.g.f.: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(6*x). - G. C. Greubel, Sep 11 2017
MATHEMATICA
LinearRecurrence[{12, -34}, {5, 36}, 50] (* G. C. Greubel, Sep 11 2017 *)
PROG
(Magma) [ n le 2 select 31*n-26 else 12*Self(n-1)-34*Self(n-2): n in [1..19] ];
(PARI) x='x+O('x^50); Vec((5-24*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Sep 11 2017
Expansion of (5-9*x)/(1-6*x+7*x^2).
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2
5, 21, 91, 399, 1757, 7749, 34195, 150927, 666197, 2940693, 12980779, 57299823, 252933485, 1116502149, 4928478499, 21755355951, 96032786213, 423909225621, 1871225850235, 8259990522063, 36461362180733, 160948239429957
COMMENTS
Binomial transform of A161941 without initial 2. Third binomial transform of A164095. Inverse binomial transform of A161731 without initial 1.
FORMULA
a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 5, a(1) = 21.
G.f.: (5-9*x)/(1-6*x+7*x^2).
a(n) = ((5+3*sqrt(2))*(3+sqrt(2))^n+(5-3*sqrt(2))*(3-sqrt(2))^n)/2.
E.g.f.: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 08 2017
MATHEMATICA
CoefficientList[Series[(5-9x)/(1-6x+7x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{6, -7}, {5, 21}, 30] (* Harvey P. Dale, Apr 27 2017 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+3*r)*(3+r)^n+(5-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).
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0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 1, 2, 2, 3, 0, 3, 2, 4, 1, 3, 3, 4, 1, 4, 2, 5, 2, 4, 3, 5, 0, 5, 3, 6, 2, 4, 4, 6, 1, 5, 3, 7, 3, 5, 4, 6, 1, 6, 4, 7, 2, 5, 5, 7, 2, 6, 4, 8, 3, 6, 5, 7, 0, 7, 5, 8, 3, 6, 6, 8, 2, 7, 4, 9, 4, 7, 6, 8, 1, 8, 5, 9, 3, 7, 7, 9
COMMENTS
This sequence has graphical features in common with A286326.
FORMULA
Empirically:
- a(n) = 0 iff n is a power of 2 ( A000079), - a(n) = 1 iff n = 3 or belongs to A164095, - a(2*n) = a(n),
- A181497(n) is the least k such that a(k) = n.
EXAMPLE
For n=1:
- a(1) = 0 is suitable.
For n=2:
- a(2) = 0 is suitable.
For n=3:
- a(1) = 0 >= a(2) = 0, so a(3) <> 0,
- a(3) = 1 is suitable.
For n=4:
- a(2) = 0 < a(3) = 1,
- a(4) = 0 is suitable.
For n=5:
- a(3) = 1 >= a(4) = 0, so a(5) <> 0,
- a(1) = 0 < a(3) = 1,
- a(5) = 1 is suitable.
For n=6:
- a(4) = 0 < a(5) = 1,
- a(2) = 0 >= a(4) = 0, so a(6) <> 0,
- a(6) = 1 is suitable.
For n=7:
- a(5) = 1 >= a(6) = 1, so a(7) <> 1,
- a(3) = 1 >= a(5) = 1, so a(7) <> 1,
- a(1) = 0 >= a(4) = 0, so a(7) <> 0,
- a(7) = 2 is suitable.
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