A323221 a(n) = n*(n + 5)*(n + 7)/6 + 1.

1, 9, 22, 41, 67, 101, 144, 197, 261, 337, 426, 529, 647, 781, 932, 1101, 1289, 1497, 1726, 1977, 2251, 2549, 2872, 3221, 3597, 4001, 4434, 4897, 5391, 5917, 6476, 7069, 7697, 8361, 9062, 9801, 10579, 11397, 12256, 13157, 14101, 15089, 16122, 17201, 18327, 19501

Offset: 0

Author

Peter Luschny, Jan 25 2019

Keywords

• nonn
• easy

Comments

a(n) is related to the total angular defect of certain polytopes. See Hilton and Pedersen, Cor. 1; compare A275874.

Examples

			For n = 2 the sum formula gives:
I(2) = {{0,0}, {0,1}, {1,0}, {0,2}, {1,1}, {2,0}, {0,3}, {1,2}, {2,1}, {3,0}};
a(2) = 1 + 1 + 1 + 2 + 1 + 2 + 5 + 2 + 2 + 5 = 22.
		

Links

• P. Hilton and J. Pedersen, Descartes, Euler, Poincaré, Pólya and Polyhedra, L'Enseign. Math., 27 (1981), 327-343.

Crossrefs

Çf. A323224 (column 4), A323233 (row 4), A034856 (first difference), A275874.

Programs

Maple

a := n -> n*(35 + 12*n + n^2)/6 + 1:
seq(a(n), n = 0..45);

Mathematica

a[n_] := n (35 + 12 n + n^2)/6 + 1;
Table[a[n], {n, 0, 45}]

Formula

Let I(n) denote the set of all tuples of length n with elements from {0, 1, 2, 3} with sum <= 3 and C(m) denote the m-th Catalan number. Then for n > 0

a(n) = Sum_{(j1,...,jn) in I(n)} C(j1)*C(j2)*...*C(jn).

a(n) = [x^n] (3*x^3 - 8*x^2 + 5*x + 1)/(x - 1)^4.

a(n) = n! [x^n] exp(x)*(x^3 + 15*x^2 + 48*x + 6)/6.

a(n) = a(n - 1)*(n*(n + 5)*(n + 7) + 6)/(n*(n + 2)*(n + 7) - 18) for n > 0.

a(n) = A323224(n, 4).

a(n) = A275874(n+4) + 1.