A023533 -id:A023533 - cp's OEIS frontend

A024326 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 = A023533.

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

Offset: 1

Clark Kimberling

nonn

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

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024327.

Cf. A023531, A023533.

nmax = 120;
A023533:= A023533 = With[{ms= Table[m(m+1)(m+2)/6, {m, 0, nmax+5}]}, Table[If[MemberQ[ms, n], 1, 0], {n, 0, nmax+5}]];
AbsoluteTiming[Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += A023533[[n + 1]]; n -= m++]; t, {n, nmax}]] (* G. C. Greubel, Jan 29 2022  *)

nmax=120
@CachedFunction
def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
@CachedFunction
def B_list(N):
    A = []
    for m in range(ceil((6*N)^(1/3))):
        A.extend([0]*(binomial(m+2, 3) -len(A)) +[1])
    return A
A023533 = B_list(nmax+5)
@CachedFunction
def A023324(n): return sum( A023531(j)*A023533[n-j+1] for j in (1..((n+1)//2)) )
[A023324(n) for n in (1..nmax)] # G. C. Greubel, Jan 29 2022

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

A023670 Convolution of A023533 with itself.

Original entry on oeis.org

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

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 >;
[(&+[A023533(k)*A023533(n-k+1): k in [1..n]]): 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];
A023670[n_]:= Sum[A023533[k]*A023533[n+1-k], {k, n}];
Table[A023670[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(A023533(k)*A023533(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 14 2022

a(n) = Sum_{k=1..n} A023533(k)*A023533(n-k+1). - G. C. Greubel, Jul 14 2022

A024692 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n+1-k), where k = floor((n+1)/2), s = A023533.

Original entry on oeis.org

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

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 >;
[(&+[A023533(k)*A023533(n-k+1): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Jul 14 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A024692[n_]:= A024692[n]= Sum[A023533[k]*A023533[n+1-k], {k, Floor[(n+1)/2]}];
Table[A024692[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(A023533(k)*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)} A023533(k)*A023533(n-k+1).

A023613 Convolution of Fibonacci numbers and A023533.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 16, 26, 42, 69, 111, 180, 291, 471, 762, 1233, 1995, 3228, 5223, 8452, 13675, 22127, 35802, 57929, 93731, 151660, 245391, 397051, 642442, 1039493, 1681935, 2721428, 4403363, 7124791

Offset: 0

Clark Kimberling

nonn

Danny Rorabaugh, Table of n, a(n) for n = 0..4000

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+2-k): k in [1..n+1]]): n in [0..50]]; // G. C. Greubel, Jul 14 2022

Join[{1,1}, Table[Sum[Fibonacci[m+2 -Binomial[j+3,3]], {j,0,n}], {n,6}, {m, Binomial[n+3,3] -2, Binomial[n+4,3] -3}]]//Flatten (* G. C. Greubel, Jul 14 2022 *)

#Assuming A023533 is available as an array
for n in range(34):
    print(n, sum([A023533[k]*fibonacci(n+2-k) for k in range(1,n+2)]))
# Danny Rorabaugh, Mar 14 2015

a(n) = Sum_{k=1..n+1} A000045(k)*A023533(n+2-k). - Danny Rorabaugh, Mar 13 2015

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.

Original entry on oeis.org

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).

A023543 Convolution of natural numbers with A023533.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 216, 222, 228

Offset: 1

Clark Kimberling

nonn

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

Cf. A000027, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Jul 15 2022

Join[{1,2}, Table[(m+2)*(n+1) -Binomial[n+4,4], {n,6}, {m, Binomial[n+3,3] -2, Binomial[n+4,3] -3}]]//Flatten (* G. C. Greubel, Jul 15 2022 *)

[1,2]+flatten([[(m+2)*(n+1) - binomial(n+4,4) for m in (binomial(n+3,3)-2 .. binomial(n+4,3)-3)] for n in (1..6)]) # G. C. Greubel, Jul 15 2022

From G. C. Greubel, Jul 15 2022: (Start)
a(n) = Sum_{j=1..floor((n+1)/2)} (n - j + 1)*A023533(j).
a(n) = (m+2)*(n+1) - binomial(n+4, 4), for binomial(n+3, 3) - 2 <= m <= binomial(n+4, 3) - 3, and n >= 1, with a(1) = 1, a(2) = 2. (End)

Title updated by Sean A. Irvine, Jun 06 2019

A023665 Convolution of A000201 and A023533.

Original entry on oeis.org

1, 3, 4, 7, 11, 13, 17, 20, 23, 28, 32, 37, 43, 47, 52, 57, 61, 67, 71, 77, 84, 90, 97, 103, 109, 117, 122, 129, 136, 141, 149, 155, 161, 169, 175, 184, 192, 199, 209, 215, 224, 232, 240, 249, 256, 264, 274, 280, 289, 297, 304, 314, 321, 329, 337

Offset: 1

Clark Kimberling

nonn

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

Cf. A000201, A001622, 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-k+1): k in [1..n]]): n in [1..80]]; // G. C. Greubel, Jul 18 2022

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

def A023665(n, k): return sum( floor((k+1 + binomial(n+2,3) - binomial(j+2,3))*golden_ratio) for j in (1..n) )
flatten([[A023665(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} A000201(j) * A023533(n-j+1).

T(n, k) = Sum_{j=1..n} A000201(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

A024693 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 = A014306.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 4

Offset: 1

Clark Kimberling

nonn

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

Cf. A014306, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*(1-A023533(n+1-k)): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // G. C. Greubel, Jul 15 2022

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A024693[n_]:= A024693[n]= Sum[(1-A023533[n-k+2])*A023533[k], {k,Floor[(n+1)/2]}];
Table[A024693[n], {n,0,100}] (* G. C. Greubel, Jul 15 2022 *)

def A023533(n):
    if binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n: return 0
    else: return 1
[sum(A023533(k)*(1-A023533(n-k+1)) for k in (1..((n+1)//2))) for n in (1..100)] # G. C. Greubel, Jul 15 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A023533(k)*A014306(n+1-k).

a(n) = Sum_{k=1..floor((n+1)/2)} A023533(k)*(1 - A023533(n-k+1)). - G. C. Greubel, Jul 15 2022

A025075 a(n) = s(1)t(n+1) + s(2)t(n) + ... + s(k)*t(n-k+2), where k = floor((n+1)/2), s = A023532, t = A023533.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 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, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 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 >;
A025075:= func< n | (&+[A023532(k)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
[A025075(n): n in [1..130]]; // G. C. Greubel, Aug 02 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];
A025075[n_]:= A025075[n]= Sum[A023532[j]*A023533[n-j+2], {j, Floor[(n+1)/2]}];
Table[A025075[n], {n,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 A023532(n): return 0 if is_square(8*n+9) else 1
def A025075(n): return sum(A023532(k)*A023533(n-k+2) for k in (1..((n+1)//2)))
[A025075(n) for n in (1..130)] # G. C. Greubel, Aug 02 2022

a(n)= Sum{k=1..floor((n+1)/2)} A023532(k) * A023533(n-k+2).

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

cp's OEIS Frontend

A024326 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 = A023533.

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A023670 Convolution of A023533 with itself.

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A024692 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n+1-k), where k = floor((n+1)/2), s = A023533.

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A023613 Convolution of Fibonacci numbers and A023533.

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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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Magma

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A023543 Convolution of natural numbers with A023533.

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A023665 Convolution of A000201 and A023533.

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A024693 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 = A014306.

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A024326 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 = A023533.

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.

A024693 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 = A014306.

A025075 a(n) = s(1)t(n+1) + s(2)t(n) + ... + s(k)*t(n-k+2), where k = floor((n+1)/2), s = A023532, 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.