A023613 -id:A023613 - cp's OEIS frontend

A024595 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (F(2), F(3), ...), t = A023533.

1, 0, 0, 1, 2, 3, 5, 0, 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1598, 2586, 4184, 6770, 10954, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28658, 46370, 75028, 121398, 196426

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000045, A023533, A023613.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Fibonacci(k+1)*A023533(n-k+1): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Jul 14 2022

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A024595[n_]:= A024595[n]= Sum[Fibonacci[k+1]*A023533[n+1-k], {k, Floor[(n+1)/2]}];
Table[A024595[n], {n,100}] (* G. C. Greubel, Jul 14 2022 *)

def A023533(n):
    if binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n: return 0
    else: return 1
[sum(fibonacci(k+1)*A023533(n-k+1) for k in (1..((n+1)//2))) for n in (1..100)] # G. C. Greubel, Jul 14 2022

a(n) = Sum_{k=1..floor((n+1)/2)} Fibonacci(k+1)*A023533(n-k+1).

A025109 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (F(2), F(3), F(4), ...), t = A023533.

Original entry on oeis.org

0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1598, 2586, 4184, 6770, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28658, 46370, 75028, 121398, 196426, 317824, 514250

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A000045, A023533, A023613, A024595.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Fibonacci(k+1)*A023533(n-k+1): k in [1..Floor(n/2)]]): n in [2..100]]; // G. C. Greubel, Jul 14 2022

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] + 2, 3] != n, 0, 1];
A025109[n_]:= A025109[n]= Sum[Fibonacci[k+1]*A023533[n+1-k], {k, Floor[n/2]}];
Table[A025109[n], {n, 2, 100}] (* G. C. Greubel, Jul 14 2022 *)

def A023533(n):
    if binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n: return 0
    else: return 1
[sum(fibonacci(k+1)*A023533(n-k+1) for k in (1..(n//2))) for n in (2..100)] # G. C. Greubel, Jul 14 2022

a(n) = Sum_{k=1..floor(n/2)} Fibonacci(k+1)*A023533(n-k+1).

a(36) corrected by Sean A. Irvine, Aug 07 2019

Offset corrected by G. C. Greubel, Jul 14 2022

A023655 Convolution of (F(2), F(3), F(4), ...) and A023533.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 26, 42, 68, 111, 180, 291, 471, 762, 1233, 1995, 3228, 5223, 8451, 13675, 22127, 35802, 57929, 93731, 151660, 245391, 397051, 642442, 1039493, 1681935, 2721428, 4403363, 7124791

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..4700

Essentially the same as A023613.

Cf. A000045, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Fibonacci(k+2)*A023533(n-k): k in [0..n-1]]): n in [1..50]]; // G. C. Greubel, Jul 16 2022

Table[Sum[Fibonacci[m+1 -Binomial[j+3,3]], {j,0,n}], {n,0,5}, {m, Binomial[n+3,3] +1, Binomial[n+4,3]}]//Flatten (* G. C. Greubel, Jul 16 2022 *)

def A023655(n, k): return sum(fibonacci(k+1-binomial(j+3,3)) for j in (0..n))
flatten([[A023655(n, k) for k in (binomial(n+3,3)+1..binomial(n+4,3))] for n in (0..5)]) # G. C. Greubel, Jul 16 2022

a(n) = Sum_{k=0..n-1} Fibonacci(k+2) * A023533(n-k), n >= 1. - G. C. Greubel, Jul 16 2022

A023671 Convolution of A023533 and A014306.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A014306, A023533, A023670, A056556,

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[(1-A023533(k))*A023533(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 18 2022

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] + 2, 3] != n, 0, 1];
A023671[n_]:= A023613[n]= Sum[(1-A023533[k])*A023533[n-k+1], {k,n}];
Table[A023671[n], {n, 100}] (* G. C. Greubel, Jul 18 2022 *)

@CachedFunction
def A023533(n): return 1 if binomial(floor((6*n-1)^(1/3)) +2, 3)!=n else 0
def A023671(n): return sum((1-A023533(k))*A023533(n-k+1) for k in (1..n))
[A023671(n) for n in (1..100)] # G. C. Greubel, Jul 18 2022

a(n) = Sum_{j=1..n} A023533(n-j+1)*A014306(j).
From G. C. Greubel, Jul 18 2022: (Start)
a(n) = Sum_{j=1..n} A023533(n-j+1)*(1 - A023533(j)).
a(n) = A056556(n) - A023670(n). (End)

cp's OEIS Frontend

A024595 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (F(2), F(3), ...), t = A023533.

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A025109 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (F(2), F(3), F(4), ...), t = A023533.

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Author

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Links

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Programs

Magma

Mathematica

SageMath

Formula

Extensions

A023655 Convolution of (F(2), F(3), F(4), ...) and A023533.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A023671 Convolution of A023533 and A014306.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A024595 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (F(2), F(3), ...), t = A023533.

A025109 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (F(2), F(3), F(4), ...), t = A023533.