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A023536 Convolution of natural numbers with A023532.

%I A023536 #31 Feb 28 2025 10:03:10
%S A023536 1,2,4,7,10,14,19,25,31,38,46,55,65,75,86,98,111,125,140,155,171,188,
%T A023536 206,225,245,266,287,309,332,356,381,407,434,462,490,519,549,580,612,
%U A023536 645,679,714,750,786,823,861,900,940,981,1023,1066,1110,1155
%N A023536 Convolution of natural numbers with A023532.
%C A023536 From _Vladimir Letsko_, Dec 18 2016: (Start)
%C A023536 Also, a(n) is the number of possible values for the number of diagonals in a convex polyhedron with n+3 vertices.
%C A023536 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)
%H A023536 Vladimir Letsko, <a href="/A023536/b023536.txt">Table of n, a(n) for n = 1..500</a>
%F A023536 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
%F A023536 From _Paul Barry_, May 24 2004: (Start)
%F A023536 a(n) = (n+1)(n+2)/2 - Sum_{k=1..n+1} floor((sqrt(8k+1)-1)/2);
%F A023536 a(n) = Sum_{k=1..n+1} k-floor((sqrt(8k+1)-1)/2). (End)
%t A023536 A023536[n_] := n*(n + 5)/2 - 2 - Sum[Round[Sqrt[2*k + 4]], {k, 2, n}];
%t A023536 Array[A023536, 60] (* _Paolo Xausa_, Feb 28 2025 *)
%o A023536 (Python)
%o A023536 from math import comb, isqrt
%o A023536 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
%Y A023536 Cf. A005230, A279681.
%K A023536 nonn
%O A023536 1,2
%A A023536 _Clark Kimberling_
%E A023536 Corrected by Jan Hagberg (jan.hagberg(AT)stat.su.se), Oct 16 2002