A023536 Convolution of natural numbers with A023532.

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

Author

Clark Kimberling

Keywords

• nonn

Comments

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)

Links

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

Crossrefs

Cf. A005230, A279681.

Programs

Mathematica

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

Python

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

Formula

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)

Extensions

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