A023533 -id:A023533 - cp's OEIS frontend

A024466 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (Fibonacci numbers), t = A023533.

1, 0, 0, 1, 1, 2, 3, 0, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 988, 1598, 2586, 4184, 6770, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17712, 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.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Fibonacci(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Jul 25 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2,3]!= n,0,1];
A024466[n_]:= A024466[n]= Sum[Fibonacci[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A024466[n], {n, 100}] (* G. C. Greubel, Jul 25 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3)!= n) else 1
def A024466(n): return sum(fibonacci(j)*A023533(n-j+1) for j in (1..((n+1)//2)))
[A024466(n) for n in (1..100)] # G. C. Greubel, Jul 25 2022

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

A024476 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (Lucas numbers), t = A023533.

Original entry on oeis.org

1, 0, 0, 1, 3, 4, 7, 0, 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2208, 3574, 5782, 9356, 15138, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39604, 64082, 103686, 167768

Offset: 1

Clark Kimberling

nonn

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

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+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Aug 01 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]]+2, 3]!= n,0,1];
A024476[n_]:= A024476[n]= Sum[LucasL[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A024476[n], {n, 100}] (* G. C. Greubel, Aug 01 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3)!= n) else 1
def A024476(n): return sum(lucas_number2(j,1,-1)*A023533(n-j+1) for j in (1..((n+1)//2)))
[A024476(n) for n in (1..100)] # G. C. Greubel, Aug 01 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A000032(k) * A023533(n-k+1).

A024601 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (odd natural numbers), t = A023533.

Original entry on oeis.org

1, 0, 0, 1, 3, 5, 7, 0, 0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 32, 36, 40, 44, 48, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47

Offset: 1

Clark Kimberling

nonn

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

Cf. A005408, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[(2*k-1)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Aug 01 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]]+2, 3]!= n, 0,1];
A024601[n_]:= A024601[n]= Sum[(2*j-1)*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A024601[n], {n, 100}] (* G. C. Greubel, Aug 01 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3)!= n) else 1
def A024601(n): return sum((2*j-1)*A023533(n-j+1) for j in (1..((n+1)//2)))
[A024601(n) for n in (1..100)] # G. C. Greubel, Aug 01 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A005408(k-1) * A023533(n-k+1).

A024687 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A000201 (lower Wythoff sequence), t = A023533.

Original entry on oeis.org

1, 0, 0, 1, 3, 4, 6, 0, 0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 26, 30, 33, 36, 40, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 36, 40, 42, 46, 50, 52, 56, 58, 62, 66, 68, 72, 76, 78, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40

Offset: 1

Clark Kimberling

nonn

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

Cf. A000201, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Floor(k*(1+Sqrt(5))/2)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Aug 01 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2, 3]!= n, 0,1];
A024687[n_]:= A024687[n]= Sum[Floor[j*GoldenRatio]*A023533[n-j+1], {j, Floor[(n+ 1)/2]}];
Table[A024687[n], {n, 100}] (* G. C. Greubel, Aug 01 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3)!= n) else 1
def A024687(n): return sum(floor(j*golden_ratio)*A023533(n-j+1) for j in (1..((n+1)//2)))
[A024687(n) for n in (1..100)] # G. C. Greubel, Aug 01 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A000201(k) * A023533(n-k+1).

A024690 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A001950 (upper Wythoff sequence), t = A023533.

Original entry on oeis.org

2, 0, 0, 2, 5, 7, 10, 0, 0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 43, 49, 54, 59, 65, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 59, 65, 69, 75, 81, 85, 91, 95, 101, 107, 111, 117, 123, 127, 39, 41, 44, 47, 49, 52

Offset: 1

Clark Kimberling

nonn

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

Cf. A001950, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A001950:= func< n |  Floor(n*(3+Sqrt(5))/2) >;
A024690:= func< n | (&+[A001950(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]) >;
[A024690(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];
A001950[n_]:= Floor[n*GoldenRatio^2];
A024690[n_]:= A024690[n]= Sum[A001950[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A024690[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 A001950(n): return floor(n*golden_ratio^2)
def A024690(n): return sum(A001950(k)*A023533(n-k+1) for k in (1..((n+1)//2)))
[A024690(n) for n in (1..130)] # G. C. Greubel, Sep 07 2022

A024694 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023533, t = A000040.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 24, 30, 36, 46, 50, 60, 70, 74, 84, 94, 102, 108, 149, 161, 171, 187, 197, 209, 229, 243, 253, 271, 281, 289, 313, 323, 339, 363, 381, 391, 403, 421, 502, 530, 552, 568, 592, 618, 630, 650, 674, 696, 712, 746, 768, 794, 802, 830, 846, 872, 906, 922

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 >;
A024694:= func< n | (&+[A023533(k)*NthPrime(n+1-k): k in [1..Floor((n+1)/2)]]) >;
[A024694(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];
A024694[n_]:= A024694[n]= Sum[Prime[n-j+1]*A023533[j], {j, Floor[(n+1)/2]}];
Table[A024694[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 A024694(n): return sum(nth_prime(n-k+1)*A023533(k) for k in (1..((n+1)//2)))
[A024694(n) for n in (1..130)] # G. C. Greubel, Sep 07 2022

A024865 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000027, t = A023533.

Original entry on oeis.org

0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 5, 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, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30

Offset: 2

Clark Kimberling

nonn

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

Cf. A000027, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A024865:= func< n | (&+[k*A023533(n+1-k): k in [1..Floor(n/2)]]) >;
[A024865(n): n in [2..130]]; // G. C. Greubel, Sep 07 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A024865[n_]:= A024865[n]= Sum[j*A023533[n-j+1], {j, Floor[n/2]}];
Table[A024865[n], {n, 2, 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 A024865(n): return sum(k*A023533(n-k+1) for k in (1..(n//2)))
[A024865(n) for n in (2..130)] # G. C. Greubel, Sep 07 2022

A024889 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023531, t = A023533.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0

Offset: 2

Clark Kimberling

nonn

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

Cf. A023531, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A023531:= func< n |  IsSquare(8*n+9) select 1 else 0 >;
A024889:= func< n | (&+[A023531(k)*A023533(n+1-k): k in [1..Floor(n/2)]]) >;
[A024889(n): n in [2..130]]; // G. C. Greubel, Aug 02 2022

A023533[n_]:= 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];
A024889[n_]:= A024889[n]= Sum[A023531[j]*A023533[n-j+1], {j, Floor[n/2]}];
Table[A024889[n], {n, 2, 130}] (* G. C. Greubel, Aug 02 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3) != n) else 1
def A023531(n): return 1 if is_square(8*n+9) else 0
def A024889(n): return sum(A023531(k)*A023533(n-k+1) for k in (1..(n//2)))
[A024889(n) for n in (2..130)] # G. C. Greubel, Aug 02 2022

a(n) = Sum_{j=2..floor(n/2)} A023531(k)*A023533(n-k+1).

A025086 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000045, t = A023533.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 988, 1598, 2586, 4184, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17712, 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.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A025086:= func< n | (&+[Fibonacci(k)*A023533(n+1-k): k in [1..Floor(n/2)]]) >;
[A025086(n): n in [2..100]]; // G. C. Greubel, Sep 08 2022

b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];
A025086[n_]:= A025086[n]= Sum[Fibonacci[n-j+1]*b[j], {j, Floor[(n+3)/2], n}];
Table[A025086[n], {n,2,100}] (* G. C. Greubel, Sep 08 2022 *)

@CachedFunction
def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..10))
@CachedFunction
def A025086(n): return sum(fibonacci(n-j+1)*b(j) for j in (((n+3)//2)..n))
[A025086(n) for n in (2..100)] # G. C. Greubel, Sep 08 2022

Offset corrected by G. C. Greubel, Sep 08 2022

A025096 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000032, t = A023533.

Original entry on oeis.org

0, 0, 1, 3, 4, 0, 0, 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2208, 3574, 5782, 9356, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39604, 64082, 103686, 167768, 271454

Offset: 2

Clark Kimberling

nonn

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

Cf. A000032, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A025096:= func< n | (&+[Lucas(k)*A023533(n+1-k): k in [1..Floor(n/2)]]) >;
[A025096(n): n in [2..130]]; // G. C. Greubel, Sep 08 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2,3]!= n,0,1];
A025096[n_]:= A025096[n]= Sum[LucasL[j]*A023533[n-j+1], {j, Floor[n/2]}];
Table[A025096[n], {n,2,100}] (* G. C. Greubel, Sep 08 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
def A025096(n): return sum(lucas_number2(k,1,-1)*A023533(n-k+1) for k in (1..(n//2)))
[A025096(n) for n in (2..100)] # G. C. Greubel, Sep 08 2022

Offset corrected by G. C. Greubel, Sep 08 2022

cp's OEIS Frontend

A024466 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (Fibonacci numbers), t = A023533.

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A024476 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (Lucas numbers), t = A023533.

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A024601 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (odd natural numbers), t = A023533.

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A024687 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A000201 (lower Wythoff sequence), t = A023533.

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A024690 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A001950 (upper Wythoff sequence), t = A023533.

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A024694 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023533, t = A000040.

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A024865 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000027, t = A023533.

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A024889 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023531, t = A023533.

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A024466 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (Fibonacci numbers), t = A023533.

A024476 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (Lucas numbers), t = A023533.

A024601 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (odd natural numbers), t = A023533.

A024687 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A000201 (lower Wythoff sequence), t = A023533.

A024690 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A001950 (upper Wythoff sequence), t = A023533.

A024694 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023533, t = A000040.

A024865 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000027, t = A023533.

A024889 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023531, t = A023533.

A025086 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000045, t = A023533.

A025096 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000032, t = A023533.