A014306 -id:A014306 - cp's OEIS frontend

A024327 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 = A014306.

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

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, A024326, A024328.

Cf. A014306, A023531.

A014306:= With[{ms= Table[m(m+1)(m+2)/6, {m,0,20}]}, Table[If[MemberQ[ms, n], 0, 1], {n,0,150}]];
Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += A014306[[n+1]]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Feb 17 2022 *)

nmax=120
@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)
def A014306(n): return 1 - A023533[n]
def b(n, j): return A014306(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
@CachedFunction
def A024327(n): return sum( b(n, j) for j in (1..floor((n+1)/2)) )
[A024327(n) for n in (1..nmax)] # G. C. Greubel, Feb 17 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A023531(k)*A014306(n-k+1). - G. C. Greubel, Feb 17 2022

Title corrected by Sean A. Irvine, Jun 30 2019

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

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)

A024890 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = A023531, t = A014306.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 4, 4, 4, 4, 3, 4, 4, 3, 4, 4, 3, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, 6, 5, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 7, 8, 7, 8, 8, 8, 7, 8, 8, 8, 8, 9, 8, 8, 9, 9, 9

Offset: 2

Clark Kimberling

nonn

a(102) onward corrected by Sean A. Irvine, Jul 27 2019

A025126 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023533, t = A014306.

Original entry on oeis.org

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, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6

Offset: 1

Clark Kimberling

nonn

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

Cf. A014306, A023533.

Cf. A024693. [From R. J. Mathar, Oct 23 2008]

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A025126:= func< n | (&+[(1-A023533(n+2-k))*A023533(k): k in [1..Floor((n+1)/2)]]) >;
[A025126(n): n in [1..130]]; // G. C. Greubel, Sep 14 2022

b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];
A025126[n_]:= A025126[n]= Sum[(1-b[j+1])*b[n-j+1], {j, Floor[(n+2)/2], n}];
Table[A025126[n], {n,130}] (* G. C. Greubel, Sep 14 2022 *)

@CachedFunction
def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..15))
@CachedFunction
def A025126(n): return sum((1-b(j+1))*b(n-j+1) for j in (((n+2)//2)..n))
[A025126(n) for n in (1..130)] # G. C. Greubel, Sep 14 2022

A023544 Convolution of natural numbers with A014306.

Original entry on oeis.org

0, 1, 3, 5, 8, 12, 17, 23, 30, 37, 45, 54, 64, 75, 87, 100, 114, 129, 145, 161, 178, 196, 215, 235, 256, 278, 301, 325, 350, 376, 403, 431, 460, 490, 520, 551, 583, 616, 650, 685, 721, 758, 796, 835, 875, 916, 958, 1001, 1045, 1090, 1136, 1183

Offset: 1

Clark Kimberling

nonn

A023566 Convolution of A023531 and A014306.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 1, 2, 3, 2, 2, 3, 2, 4, 4, 3, 3, 4, 4, 4, 5, 3, 4, 5, 5, 5, 5, 5, 5, 6, 6, 5, 6, 6, 5, 7, 6, 5, 7, 7, 7, 6, 6, 8, 7, 7, 7, 8, 8, 8, 8, 7, 6, 9, 9, 7, 9, 9, 8, 8, 9, 7, 8, 9, 10, 10, 9, 8, 10, 10, 10, 9, 9, 9, 10, 10, 10, 11, 10, 11, 10, 11, 10, 10, 10, 11, 9, 11, 10

Offset: 1

Clark Kimberling

nonn

A023605 Convolution of A023532 and A014306.

Original entry on oeis.org

0, 1, 1, 1, 3, 2, 3, 5, 5, 4, 6, 7, 7, 9, 8, 9, 11, 12, 12, 12, 13, 13, 16, 16, 16, 17, 18, 19, 20, 21, 21, 22, 24, 24, 24, 26, 25, 27, 29, 28, 29, 30, 32, 33, 32, 34, 35, 36, 36, 37, 38, 39, 41, 43, 41, 41, 44, 43, 44, 46, 47, 47, 50, 50, 50, 50, 51, 53, 55, 54, 55

Offset: 1

Clark Kimberling

nonn

A023614 Convolution of Fibonacci numbers and A014306.

Original entry on oeis.org

0, 1, 2, 3, 6, 10, 17, 28, 46, 74, 121, 196, 318, 515, 834, 1350, 2185, 3536, 5722, 9258, 14981, 24240, 39222, 63463, 102686, 166150, 268837, 434988, 703826, 1138815, 1842642, 2981458, 4824101, 7805560

Offset: 0

Clark Kimberling

nonn

A023624 Convolution of Lucas numbers and A014306.

Original entry on oeis.org

0, 1, 4, 7, 12, 22, 37, 62, 102, 166, 269, 438, 710, 1151, 1864, 3018, 4885, 7906, 12794, 20702, 33497, 54202, 87702, 141907, 229612, 371522, 601137, 972662, 1573802, 2546467, 4120272, 6666742, 10787017, 17453762, 28240781, 45694544

Offset: 1

Clark Kimberling

nonn

a(28) corrected and more terms from Sean A. Irvine, Jun 08 2019

cp's OEIS Frontend

A024327 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 = A014306.

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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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Magma

Mathematica

SageMath

Formula

A023671 Convolution of A023533 and A014306.

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Magma

Mathematica

Sage

Formula

A024890 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = A023531, t = A014306.

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Extensions

A025126 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023533, t = A014306.

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

A023544 Convolution of natural numbers with A014306.

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Author

Keywords

A023566 Convolution of A023531 and A014306.

Views

Author

Keywords

A023605 Convolution of A023532 and A014306.

Views

Author

Keywords

A023614 Convolution of Fibonacci numbers and A014306.

Views

Author

Keywords

A023624 Convolution of Lucas numbers and A014306.

Views

Author

Keywords

Extensions

A024327 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 = A014306.

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.

A025126 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A023533, t = A014306.