A394826 The tetrahemihexahedral numbers.

1, 6, 20, 47, 91, 156, 246, 365, 517, 706, 936, 1211, 1535, 1912, 2346, 2841, 3401, 4030, 4732, 5511, 6371, 7316, 8350, 9477, 10701, 12026, 13456, 14995, 16647, 18416, 20306, 22321, 24465, 26742, 29156, 31711, 34411, 37260, 40262, 43421, 46741, 50226, 53880

Offset

1,2

Comments

Submitted on behalf of a third-party: mecejide and ericabq from the Polytope Discord.

Links

Table of n, a(n) for n=1..43.

H. K. Kim, On regular polytope numbers, Proceedings of the American Mathematical Society, 131(1), 65-75 (2002).

Polytope Wiki, Tetrahemihexahedron.

Wikipedia, Tetrahemihexahedron.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = (n + 2)*(4*n^2 - 5*n + 3)/6.

From Stefano Spezia, May 30 2026: (Start)

G.f.: x*(1 + 2*x + 2*x^2 - x^3)/(1 - x)^4.

E.g.f.: exp(x)*(6 + 15*x^2 + 4*x^3)/6 - 1. (End)

Mathematica

a[n_] := (4*n^3 + 3*n^2 - 7*n + 6)/6; Array[a, 45] (* Amiram Eldar, May 30 2026 *)

Prog

(Python)
def A394826(n): return n*(n*(4*n+3)-7)//6+1 # Chai Wah Wu, Jun 05 2026

Crossrefs

Cf. A000292.

Sequence in context: A353692 A005564 A011928 * A055455 A203552 A331754

Adjacent sequences: A394823 A394824 A394825 * A394827 A394828 A394829

Keyword

nonn,easy

Author

Keith J. Bauer, May 29 2026

Status

approved