A024312 -id:A024312

A024318

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (Fibonacci numbers).

+10
20

0, 0, 1, 2, 3, 5, 8, 13, 26, 42, 68, 110, 178, 288, 466, 754, 1254, 2029, 3283, 5312, 8595, 13907, 22502, 36409, 58911, 95320, 154608, 250161, 404769, 654930, 1059699, 1714629, 2774328, 4488957, 7263285

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*Fibonacci(n-j+1). - G. C. Greubel, Jan 19 2022

MATHEMATICA

Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Fibonacci[n+1]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Jan 19 2022 *)

PROG

(Magma)

b:= func< n, j | IsIntegral((Sqrt(8*j+9) -3)/2) select Fibonacci(n-j+1) else 0 >;

A024318:= func< n | (&+[b(n, j): j in [1..Floor((n+1)/2)]]) >;

[A024318(n) : n in [1..80]]; // G. C. Greubel, Jan 19 2022

(Sage)

def b(n, j): return fibonacci(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0

def A024318(n): return sum( b(n, j) for j in (1..floor((n+1)/2)) )

[A024318(n) for n in (1..120)] # G. C. Greubel, Jan 19 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000045, A023531.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

A024316

a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = A023531.

+10
19

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,28

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A023531(n-j+1). - G. C. Greubel, Jan 17 2022

MATHEMATICA

A023531[n_]:= SquaresR[1, 8n+9]/2;

a[n_]:= a[n]= Sum[A023531[j]*A023531[n-j+1], {j, Floor[(n+1)/2]}];

Table[a[n], {n, 110}] (* G. C. Greubel, Jan 17 2022 *)

PROG

(Haskell)

a024316 n = sum $ take (div (n + 1) 2) $ zipWith (*) zs $ reverse zs

where zs = take n $ tail a023531_list

-- Reinhard Zumkeller, Feb 14 2015

(Magma)

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) - 3)/2 ) select 1 else 0 >;

[ (&+[A023531(j)*A023531(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..110]]; // G. C. Greubel, Jan 17 2022

(Sage)

def A023531(n):

if ((sqrt(8*n+9) -3)/2).is_integer(): return 1

else: return 0

[sum( A023531(j)*A023531(n-j+1) for j in (1..floor((n+1)/2)) ) for n in (1..110)] # G. C. Greubel, Jan 17 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024315, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A023531.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

A024313

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023531.

+10
17

3, 3, 10, 17, 37, 59, 114, 185, 334, 540, 938, 1518, 2573, 4163, 6946, 11239, 18559, 30029, 49248, 79685, 130090, 210490, 342596, 554332, 900423, 1456915, 2363370, 3824013, 6197753

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).

FORMULA

G.f.: x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From G. C. Greubel, Jan 17 2022: (Start)

a(2*n) = Lucas(2*n+3) + F(2*n+2) - Lucas(n+3) - (n+1)*F(n+2).

a(2*n+1) = Lucas(2*n+4) + F(2*n+3) - Lucas(n+3) - (n+2)*F(n+2), where F(n) = A000045(n). (End)

MATHEMATICA

a[n_]:= With[{F=Fibonacci}, If[EvenQ[n], LucasL[n+3] + F[n+2] - LucasL[n/2 +3] - (n/2 +1)*F[n/2 +2], LucasL[n+3] + F[n+2] - LucasL[(n+5)/2]-(n+3)/2*Fibonacci[(n+3)/2]]];

Table[a[n], {n, 40}] (* G. C. Greubel, Jan 17 2022 *)

PROG

(Magma)

R<x>:=PowerSeriesRing(Integers(), 40);

Coefficients(R!( x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 17 2022

(Sage)

def A024313_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( x*(3 -2*x^2 +4*x^3 -x^4 -3*x^5 -2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()

a=A024313_list(41); a[1:] # G. C. Greubel, Jan 17 2022

CROSSREFS

Cf. A024312, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A000045, A023531.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

A024314

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

+10
17

3, 9, 24, 37, 81, 133, 256, 413, 746, 1208, 2098, 3394, 5753, 9309, 15532, 25131, 41499, 67147, 110122, 178181, 290890, 470670, 766068, 1239524, 2013407, 3257761, 5284656, 8550753

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).

FORMULA

G.f.: x*(3 + 6*x + 6*x^2 - 8*x^3 - 7*x^4 + x^5 - 4*x^6 + 2*x^7)/((1 - x - x^2)*(1 - x^2 - x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From G. C. Greubel, Jan 17 2022: (Start)

a(2*n) = 6*F(2*n+3) + F(2*n+1) - (n+6)*F(n+3) - (n+1)*F(n+1).

a(2*n+1) = 6*F(2*n+2) + F(2*n) - (n+6)*F(n+2) - (n+1)*F(n), where F(n) = A000045(n). (End)

MATHEMATICA

a[n_]:= With[{F=Fibonacci}, 6*F[n+3] +F[n+1] - (1/2)*((1+(-1)^n)*(((n+2)/2 )*LucasL[(n+4)/2] + 5*F[(n+6)/2]) - (1-(-1)^n)*(((n+3)/2)*LucasL[(n+3)/2] +5*F[(n+5)/2] ))];

Table[a[n], {n, 40}] (* G. C. Greubel, Jan 17 2022 *)

PROG

(Magma)

R<x>:=PowerSeriesRing(Integers(), 40);

Coefficients(R!( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 17 2022

(Sage)

def A024314_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()

a=A024314_list(41); a[1:] # G. C. Greubel, Jan 17 2022

CROSSREFS

Cf. A024312, A024313, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A000045, A023532.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

A024315

a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = floor(n/2), s = (natural numbers >= 3), t = (Fibonacci numbers).

+10
17

3, 6, 17, 27, 59, 96, 185, 299, 540, 874, 1518, 2456, 4163, 6736, 11239, 18185, 30029, 48588, 79685, 128933, 210490, 340580, 554332, 896928, 1456915, 2357338, 3824013, 6187383

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).

FORMULA

G.f.: x*(3 +3*x +2*x^2 -2*x^3 -4*x^4 -x^5 -2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From G. C. Greubel, Jan 16 2022: (Start)

a(2*n) = L(2*n+4) + F(2*n+3) - F(n+5) - (n+2)*F(n+3), n >= 1.

a(2*n-1) = L(2*n+3) + F(2*n+2) - F(n+3) - (n+3)*F(n+2), n >= 1, where L(n) = A000032(n) and F(n) = A000045(n). (End)

MATHEMATICA

a[n_]:= With[{F=Fibonacci}, If[EvenQ[n], LucasL[n+4] +F[n+3] -F[(n+10)/2] -((n+ 4)/2)*F[(n+6)/2], LucasL[n+4] +F[n+3] -F[(n+7)/2] -((n+7)/2)*F[(n+5)/2]]];

Table[a[n], {n, 40}] (* G. C. Greubel, Jan 16 2022 *)

PROG

(Magma)

R<x>:=PowerSeriesRing(Integers(), 40);

Coefficients(R!( x*(3+3*x+2*x^2-2*x^3-4*x^4-x^5-2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 16 2022

(Sage)

def A024315_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( x*(3+3*x+2*x^2-2*x^3-4*x^4-x^5-2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()

a=A024315_list(41); a[1:] # G. C. Greubel, Jan 16 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A000045.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

A024317

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A023532.

+10
17

0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 4, 4, 3, 4, 3, 4, 3, 3, 4, 4, 3, 4, 5, 5, 4, 5, 4, 4, 5, 3, 5, 5, 5, 4, 5, 5, 5, 6, 5, 5, 6, 6, 5, 5, 5, 6, 6, 5, 5, 6, 5, 6, 7, 7, 5, 7, 7, 7, 7, 4, 7, 6, 6, 7, 7, 6, 6, 7, 7, 7, 8, 7

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,11

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = Sum_{k=1..floor((n+1)/2)} A023531(k)*A023532(n-k+1). - G. C. Greubel, Jan 19 2022

MATHEMATICA

A023531[n_]:= SquaresR[1, 8n+9]/2;

a[n_]:= Sum[A023531[j]*(1 - A023531[n-j+1]), {j, Floor[(n+1)/2]}];

Table[a[n], {n, 90}] (* G. C. Greubel, Jan 19 2022 *)

PROG

(Magma)

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;

[ (&+[A023531(j)*(1 - A023531(n-j+1)): j in [1..Floor((n+1)/2)]]) : n in [1..90]]; // G. C. Greubel, Jan 19 2022

(Sage)

def A023531(n):

if ((sqrt(8*n+9) -3)/2).is_integer(): return 1

else: return 0

[sum( A023531(j)*(1-A023531(n-j+1)) for j in (1..floor((n+1)/2)) ) for n in (1..90)] # G. C. Greubel, Jan 19 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024315, A024316, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A023531. A023532.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

A024319

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (Lucas numbers).

+10
17

0, 0, 3, 4, 7, 11, 18, 29, 58, 94, 152, 246, 398, 644, 1042, 1686, 2804, 4537, 7341, 11878, 19219, 31097, 50316, 81413, 131729, 213142, 345714, 559377, 905091, 1464468, 2369559, 3834027, 6203586

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*Lucas(n-j+1). - G. C. Greubel, Jan 19 2022

MATHEMATICA

A023531[n_]:= SquaresR[1, 8n+9]/2;

a[n_]:= Sum[A023531[j]*LucasL[n-j+1], {j, Floor[(n+1)/2]}];

Table[a[n], {n, 40}] (* G. C. Greubel, Jan 19 2022 *)

PROG

(Magma)

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;

[ (&+[A023531(j)*Lucas(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..40]]; // G. C. Greubel, Jan 19 2022

(Sage)

def A023531(n):

if ((sqrt(8*n+9) -3)/2).is_integer(): return 1

else: return 0

[sum( A023531(j)*lucas_number2(n-j+1, 1, -1) for j in (1..floor((n+1)/2)) ) for n in (1..40)] # G. C. Greubel, Jan 19 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A023531.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

A024320

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (1, p(1), p(2), ... ).

+10
17

0, 0, 2, 3, 5, 7, 11, 13, 24, 30, 36, 46, 50, 60, 70, 74, 103, 117, 131, 139, 157, 171, 177, 193, 207, 221, 278, 294, 310, 330, 348, 360, 390, 408, 424, 448, 470, 486, 573, 611, 625, 653, 673, 699, 739, 761, 781, 803, 835, 863, 891, 925, 1054, 1078, 1104, 1136, 1180, 1214

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A023531(1) + Sum_{j=2..floor((n+1)/2)} A023531(j)*Prime(n-j+1). - G. C. Greubel, Jan 19 2022

MATHEMATICA

A023531[n_]:= SquaresR[1, 8*n+9]/2;

p[n_]:= If[n==1, 1, Prime[n-1]];

a[n_]:= Sum[A023531[j]*p[n-j+1], {j, Floor[(n+1)/2]}];

Table[a[n], {n, 60}] (* G. C. Greubel, Jan 19 2022 *)

PROG

(Magma)

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;

p:= func< n | n eq 1 select 1 else NthPrime(n-1) >;

[ (&+[A023531(j)*p(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..60]]; // G. C. Greubel, Jan 19 2022

(Sage)

def A023531(n):

if ((sqrt(8*n+9) -3)/2).is_integer(): return 1

else: return 0

def p(n):

if (n==1): return 1

else: return nth_prime(n-1)

[sum( A023531(j)*p(n-j+1) for j in (1..floor((n+1)/2)) ) for n in (1..60)] # G. C. Greubel, Jan 19 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000040, A023531.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

A024321

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (composite numbers).

+10
17

0, 0, 6, 8, 9, 10, 12, 14, 25, 28, 32, 35, 37, 40, 44, 46, 64, 69, 73, 77, 81, 85, 89, 93, 96, 100, 128, 133, 139, 144, 148, 154, 162, 166, 170, 176, 181, 187, 223, 229, 236, 242, 248, 255, 262, 268, 275, 281, 287, 294, 301, 308, 354, 361, 370, 380, 386, 394, 401, 408, 418, 425

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A002808(n-j+1). - G. C. Greubel, Jan 19 2022

MATHEMATICA

A023531[n_]:= SquaresR[1, 8n+9]/2;

Composite[n_]:= FixedPoint[n +PrimePi[#] +1 &, n];

a[n_]:= Sum[A023531[j]*Composite[n-j+1], {j, Floor[(n+1)/2]}];

Table[a[n], {n, 70}] (* G. C. Greubel, Jan 19 2022 *)

PROG

(Magma)

A002808:= [n : n in [2..100] | not IsPrime(n) ];

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;

[ (&+[A023531(j)*A002808[n-j+1]: j in [1..Floor((n+1)/2)]]) : n in [1..70]]; // G. C. Greubel, Jan 19 2022

(Sage)

A002808 = [n for n in (1..250) if sloane.A001222(n) > 1]

def A023531(n):

if ((sqrt(8*n+9) -3)/2).is_integer(): return 1

else: return 0

[sum( A023531(j)*A002808[n-j] for j in (1..floor((n+1)/2)) ) for n in (1..70)] # G. C. Greubel, Jan 19 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A002808, A023531.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

A024322

a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (F(2), F(3), ...).

+10
17

0, 0, 2, 3, 5, 8, 13, 21, 42, 68, 110, 178, 288, 466, 754, 1220, 2029, 3283, 5312, 8595, 13907, 22502, 36409, 58911, 95320, 154231, 250161, 404769, 654930, 1059699, 1714629, 2774328, 4488957, 7263285

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

From G. C. Greubel, Jan 20 2022: (Start)

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A000045(n-j+1).

a(n) = Sum_{j=1..floor((n+1)/2)} A010054(j+1)*A000045(n-j+2). (End)

MATHEMATICA

A010054[n_]:= SquaresR[1, 8n+1]/2;

a[n_]:= Sum[A010054[j+1]*Fibonacci[n-j+2], {j, Floor[(n+1)/2]}];

Table[a[n], {n, 40}] (* G. C. Greubel, Jan 20 2022 *)

PROG

(Magma)

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) - 3)/2 ) select 1 else 0 >;

[ (&+[A023531(j)*Fibonacci(n-j+2): j in [1..Floor((n+1)/2)]]) : n in [1..40]]; // G. C. Greubel, Jan 20 2022

(Sage)

def A023531(n):

if ((sqrt(8*n+9) -3)/2).is_integer(): return 1

else: return 0

[sum( A023531(j)*fibonacci(n-j+2) for j in (1..floor((n+1)/2)) ) for n in (1..40)] # G. C. Greubel, Jan 20 2022

CROSSREFS

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024323, A024324, A024325, A024326, A024327.

Cf. A000045, A010054, A023531.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved