A023532 -id:A023532 - cp's OEIS frontend

A024317 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 = A023532.

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

Offset: 1

Clark Kimberling

nonn

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

Cf. A024312, A024313, A024314, A024315, A024316, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A023531. A023532.

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*(1 - A023531(n-j+1)): j in [1..Floor((n+1)/2)]]) : n in [1..90]]; // G. C. Greubel, Jan 19 2022

A023531[n_]:= SquaresR[1, 8n+9]/2;
a[n_]:= Sum[A023531[j]*(1 - A023531[n-j+1]), {j, Floor[(n+1)/2]}];
Table[a[n], {n, 90}] (* G. C. Greubel, Jan 19 2022 *)

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*(1-A023531(n-j+1)) for j in (1..floor((n+1)/2)) ) for n in (1..90)] # G. C. Greubel, Jan 19 2022

a(n) = Sum_{k=1..floor((n+1)/2)} A023531(k)*A023532(n-k+1). - G. C. Greubel, Jan 19 2022

A024314 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

Original entry on oeis.org

3, 9, 24, 37, 81, 133, 256, 413, 746, 1208, 2098, 3394, 5753, 9309, 15532, 25131, 41499, 67147, 110122, 178181, 290890, 470670, 766068, 1239524, 2013407, 3257761, 5284656, 8550753

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).

Cf. A024312, A024313, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A000045, A023532.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 17 2022

a[n_]:= With[{F=Fibonacci}, 6*F[n+3] +F[n+1] - (1/2)*((1+(-1)^n)*(((n+2)/2 )*LucasL[(n+4)/2] + 5*F[(n+6)/2]) - (1-(-1)^n)*(((n+3)/2)*LucasL[(n+3)/2] +5*F[(n+5)/2] ))];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 17 2022 *)

def A024314_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
a=A024314_list(41); a[1:] # G. C. Greubel, Jan 17 2022

G.f.: x*(3 + 6*x + 6*x^2 - 8*x^3 - 7*x^4 + x^5 - 4*x^6 + 2*x^7)/((1 - x - x^2)*(1 - x^2 - x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
From G. C. Greubel, Jan 17 2022: (Start)
a(2*n) = 6*F(2*n+3) + F(2*n+1) - (n+6)*F(n+3) - (n+1)*F(n+1).
a(2*n+1) = 6*F(2*n+2) + F(2*n) - (n+6)*F(n+2) - (n+1)*F(n), where F(n) = A000045(n). (End)

A023536 Convolution of natural numbers with A023532.

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 19, 25, 31, 38, 46, 55, 65, 75, 86, 98, 111, 125, 140, 155, 171, 188, 206, 225, 245, 266, 287, 309, 332, 356, 381, 407, 434, 462, 490, 519, 549, 580, 612, 645, 679, 714, 750, 786, 823, 861, 900, 940, 981, 1023, 1066, 1110, 1155

Offset: 1

Clark Kimberling

nonn

From Vladimir Letsko, Dec 18 2016: (Start)
Also, a(n) is the number of possible values for the number of diagonals in a convex polyhedron with n+3 vertices.
Let v>4 denote the number of vertices of convex polyhedra. The set of possible numbers of diagonals is the union of sets {(k-1)(v-k-4), ..., (k-1)(v-(k+6)/2)}, where 1 <= k <= floor((sqrt(8v-15)-5)/2), and the set {(k-1)(v-k-4), ..., (v-3)(v-4)/2}, where k = floor((sqrt(8v-15)-3)/2). Note that cardinalities of all sets of this union excluding the last one are consecutive triangular numbers. (End)

Vladimir Letsko, Table of n, a(n) for n = 1..500

Cf. A005230, A279681.

A023536[n_] := n*(n + 5)/2 - 2 - Sum[Round[Sqrt[2*k + 4]], {k, 2, n}];
Array[A023536, 60] (* Paolo Xausa, Feb 28 2025 *)

from math import comb, isqrt
def A023536(n): return comb(n+2,2)-sum(isqrt((k<<3)+1)-1>>1 for k in range(1,n+2)) # Chai Wah Wu, Feb 27 2025

a(n) = (n(n + 5) - 4 )/2 - Sum_{k=2..n} floor(1/2 + sqrt(2(k + 2))). - Jan Hagberg (jan.hagberg(AT)stat.su.se), Oct 16 2002
From Paul Barry, May 24 2004: (Start)
a(n) = (n+1)(n+2)/2 - Sum_{k=1..n+1} floor((sqrt(8k+1)-1)/2);
a(n) = Sum_{k=1..n+1} k-floor((sqrt(8k+1)-1)/2). (End)

Corrected by Jan Hagberg (jan.hagberg(AT)stat.su.se), Oct 16 2002

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).

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)

A023859 a(n) = 1t(n) + 2t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2), and t = A023532.

Original entry on oeis.org

1, 0, 1, 3, 5, 4, 7, 6, 9, 13, 18, 17, 23, 21, 27, 25, 32, 40, 49, 47, 56, 54, 64, 62, 73, 71, 82, 94, 107, 105, 119, 117, 132, 130, 145, 142, 158, 155, 172, 190, 209, 207, 227, 224, 244, 241, 262, 259, 281, 278, 301, 298, 322, 346, 371, 368, 394, 391, 418, 415, 443, 440, 469, 466

Offset: 1

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Array[Sum[k Boole@ Not@ IntegerQ@ Sqrt[8 # + 9] &[# + 1 - k], {k, Floor[(# + 1)/2]}] &, 64] (* Michael De Vlieger, Jun 12 2019 *)

Title simplified by Sean A. Irvine, Jun 12 2019

A024368 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = A023532, t = (Lucas numbers).

Original entry on oeis.org

1, 3, 4, 7, 15, 25, 47, 76, 123, 199, 340, 550, 919, 1487, 2453, 3969, 6422, 10391, 16936, 27403, 44538, 72064, 116924, 189187, 306632, 496141, 802773, 1298914, 2103051, 3402808, 5508066, 8912238

Offset: 1

Clark Kimberling

nonn

A024371 Sum_{ k=1 ... floor(n/2) } A023532(k)*Fib(n-k).

Original entry on oeis.org

0, 1, 2, 3, 5, 11, 18, 34, 55, 89, 144, 246, 398, 665, 1076, 1775, 2872, 4647, 7519, 12255, 19829, 32228, 52146, 84607, 136897, 221881, 359011, 580892, 939903, 1521782, 2462295, 3985674, 6448956, 10437214, 16887767, 27329162, 44219513

Offset: 1

Clark Kimberling

nonn

			a(5) = 1*Fib(5) + 1*Fib(3) = 8 + 3 = 11.

Also equals Sum_{ k = 1 ... floor((n+1)/2) } A023532(k)*Fib(n-k+1)

Edited by Joshua Zucker, May 14 2007

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

A024377 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = A023532, t = (primes).

Original entry on oeis.org

2, 3, 5, 7, 16, 20, 35, 43, 53, 65, 86, 106, 137, 153, 196, 220, 246, 270, 329, 353, 412, 454, 521, 559, 642, 686, 732, 780, 869, 917, 1032, 1080, 1199, 1263, 1388, 1456, 1597, 1673, 1735, 1823, 1974, 2062, 2227, 2301, 2476, 2572, 2761, 2863, 3062, 3166, 3365, 3475, 3591

Offset: 1

Clark Kimberling

nonn

Cf. A000040, A023532.

cp's OEIS Frontend

A024317 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 = A023532.

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A024314 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

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A023536 Convolution of natural numbers with A023532.

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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.

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A023604 Convolution of A023532 and A023533.

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A023859 a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2), and t = A023532.

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A024368 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = A023532, t = (Lucas numbers).

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A024371 Sum_{ k=1 ... floor(n/2) } A023532(k)*Fib(n-k).

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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.

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A024317 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 = A023532.

A024314 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

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.

A023859 a(n) = 1t(n) + 2t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2), and t = A023532.

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.