A023533 -id:A023533 - cp's OEIS frontend

A023565 Convolution of A023531 and A023533.

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

Offset: 1

Clark Kimberling

nonn

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

Cf. A023531, A023533.

A023531:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 1 else 0 >;
A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*A023531(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 16 2022

A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023565[n_]:= A023565[n]= Sum[A023533[k]*A023531[n-k+1], {k,n}];
Table[A023565[n], {n,100}] (* G. C. Greubel, Jul 16 2022 *)

@CachedFunction
def A023531(n): return 1 if ((sqrt(8*n+9) -3)/2).is_integer()  else 0
@CachedFunction
def A023533(n): return 0 if binomial( floor((6*n-1)^(1/3)) +2, 3)!=n else 1
[sum(A023533(k)*A023531(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 16 2022

a(n) = Sum_{j=1..n} A023533(j) * A023531(n-j+1). - G. C. Greubel, Jul 16 2022

A023604 Convolution of A023532 and A023533.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

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

Cf. A023531, A023532, A023533.

A023532:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 0 else 1 >;
A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*A023532(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 16 2022

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
A023604[n_]:= A023604[n]= Sum[A023533[k]*(1-A023531[n-k+1]), {k,n}];
Table[A023604[n], {n,100}] (* G. C. Greubel, Jul 16 2022 *)

def A023532(n): return 0 if ((sqrt(8*n+9) -3)/2).is_integer() else 1
@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
[sum(A023533(k)*A023532(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 16 2022

From G. C. Greubel, Jul 16 2022: (Start)
a(n) = Sum_{j=1..n} A023532(n-j+1) * A023533(j).
a(n) = Sum_{j=1..n} (1 - A023531(n-j+1)) * A023533(j). (End)

A023623 Convolution of Lucas numbers and A023533.

Original entry on oeis.org

1, 3, 4, 8, 14, 22, 36, 58, 94, 153, 249, 402, 651, 1053, 1704, 2757, 4461, 7218, 11679, 18898, 30579, 49477, 80056, 129533, 209589, 339122, 548711, 887833, 1436544, 2324377, 3760921, 6085298, 9846219

Offset: 1

Clark Kimberling

nonn

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

Cf. A000032, A023533.

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

Table[Sum[LucasL[m+2-Binomial[j+3,3]], {j,0,n}], {n,0,5}, {m, Binomial[n+3,3] -1, Binomial[n+4,3] -2}]//Flatten (* G. C. Greubel, Jul 16 2022 *)

def A023623(n, k): return sum(lucas_number2(k-binomial(j+3,3),1,-1) for j in (0..n))
flatten([[A023623(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_{j=1..n+1} LucasL(j) * A023533(n-j+1). - G. C. Greubel, Jul 16 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

A023660 Convolution of odd numbers and A023533.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 24, 28, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 215, 225, 235, 245, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 405

Offset: 1

Clark Kimberling

nonn

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

Cf. A005408, A023533, A023543, A056556.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[(2*k+1)*A023533(n-k): k in [0..n-1]]): n in [1..80]]; // G. C. Greubel, Jul 17 2022

Table[(2*k+1)*n + 6*Binomial[n+2,4], {n, 7}, {k,0,n*(n+3)/2}]//Flatten (* G. C. Greubel, Jul 17 2022 *)

def A023660(n, k): return (2*k+1)*n + 6*binomial(n+2, 4)
flatten([[A023660(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # G. C. Greubel, Jul 17 2022

From G. C. Greubel, Jul 17 2022: (Start)
a(n) = Sum_{j=0..n-1} (2*j+1)*A023533(n-j).
a(n) = 2*A023543(n-1) + A056556(n).
T(n, k) = (2*k+1)*n + 6*binomial(n+2, 4), for 0 <= k <= n*(n+3)/2 and n >= 1 (as an irregular triangle). (End)

A023668 Convolution of A001950 and A023533.

Original entry on oeis.org

2, 5, 7, 12, 18, 22, 28, 33, 38, 46, 53, 61, 70, 77, 85, 93, 100, 109, 116, 126, 137, 147, 158, 168, 178, 190, 199, 210, 221, 230, 242, 252, 262, 274, 285, 299, 312, 324, 339, 350, 364, 377, 390, 404, 416, 429, 444, 455, 469, 482, 494, 509, 521

Offset: 1

Clark Kimberling

nonn

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

Cf. A001622, A001950, A023533, A023665.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Floor(k*(3+Sqrt(5))/2)*A023533(n-k+1): k in [1..n]]): n in [1..80]]; // G. C. Greubel, Jul 18 2022

A023668[n_, k_]:= A023668[n, k]= Sum[Floor[(k+1 +Binomial[n+2,3] -Binomial[j+2, 3])*GoldenRatio^2], {j, n}];
Table[A023668[n, k], {n, 7}, {k,0,n*(n+3)/2}] (* G. C. Greubel, Jul 18 2022 *)

def A023668(n, k): return sum( floor((k+1 + binomial(n+2,3) - binomial(j+2,3))*golden_ratio^2) for j in (1..n) )
flatten([[A023668(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # G. C. Greubel, Jul 18 2022

a(n) = Sum_{j=1..n} A001950(j) * A023533(n-j+1).

T(n, k) = Sum_{j=1..n} A001950(k+1 +binomial(n+2,3) -binomial(j+2,3)), for 0 <= k <= n*(n+3)/2, n >= 1 (as an irregular triangle). - G. C. Greubel, Jul 18 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)

A023672 Convolution of A023533 and primes.

Original entry on oeis.org

2, 3, 5, 9, 14, 18, 24, 30, 36, 48, 53, 65, 77, 85, 97, 111, 121, 131, 149, 163, 174, 192, 204, 220, 242, 260, 272, 294, 310, 320, 350, 364, 382, 410, 436, 453, 469, 495, 513, 543, 569, 587, 615, 647, 661, 687, 715, 739, 759, 799, 827, 855, 869

Offset: 1

Clark Kimberling

nonn

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

Cf. A000040, A023533.

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

A023672[n_, m_]:= A023672[n, m]= Sum[Prime[(m +Binomial[n+2,3] -Binomial[j+2, 3])], {j, n}];
Table[A023672[n, m], {n,10}, {m,Binomial[n+2,2]}]//Flatten (* G. C. Greubel, Jul 18 2022 *)

def A023672(n,k): return sum(nth_prime(k +binomial(n+2,3) -binomial(j+2,3)) for j in (1..n))
flatten([[A023672(n,k) for k in (1..binomial(n+2,2))] for n in (1..10)]) # G. C. Greubel, Jul 18 2022

a(n) = Sum_{j=1..n} A000040(j) * A023533(n-j+1).

a(n) = Sum_{j=1..n} A000040(k + binomial(n+3, 3) - binomial(j+2, 3)), for 1 <= k <= binomial(n+2, 2), n >= 1 (as an irregular triangle). - G. C. Greubel, Jul 18 2022

A023868 a(n) = 1t(n) + 2t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t is A023533.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 4, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32

Offset: 1

Clark Kimberling

nonn

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

Cf. A023533.

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

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2, 3]!= n,0,1];
A023868[n_]:= A023868[n]= Sum[j*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A023868[n], {n, 100}] (* G. C. Greubel, Jul 21 2022 *)

def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3)!= n) else 1
def A023868(n): return sum(j*A023533(n-j+1) for j in (1..((n+1)//2)))
[A023868(n) for n in (1..100)] # G. C. Greubel, Jul 21 2022

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

Title simplified by Sean A. Irvine, Jun 12 2019

A024375 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023532, t = A023533.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

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

Cf. A023532, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A023532:= func< n |  IsSquare(8*n+9) select 0 else 1 >;
A025375:= func< n | (&+[A023532(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]) >;
[A025375(n): n in [1..130]]; // G. C. Greubel, Sep 07 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023532[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 0, 1];
A025375[n_]:= A025075[n]= Sum[A023532[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A025375[n], {n, 130}] (* G. C. Greubel, Sep 07 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
def A023532(n): return 0 if is_square(8*n+9) else 1
def A025375(n): return sum(A023532(k)*A023533(n-k+1) for k in (1..((n+1)//2)))
[A025375(n) for n in (1..130)] # G. C. Greubel, Sep 07 2022

cp's OEIS Frontend

A023565 Convolution of A023531 and A023533.

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A023604 Convolution of A023532 and A023533.

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A023623 Convolution of Lucas numbers and A023533.

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

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A023660 Convolution of odd numbers and A023533.

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Mathematica

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A023668 Convolution of A001950 and A023533.

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Magma

Mathematica

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A023671 Convolution of A023533 and A014306.

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A023672 Convolution of A023533 and primes.

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A023868 a(n) = 1t(n) + 2t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t is A023533.

A024375 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023532, t = A023533.