A100184 Structured octagonal anti-prism numbers.

1, 16, 64, 164, 335, 596, 966, 1464, 2109, 2920, 3916, 5116, 6539, 8204, 10130, 12336, 14841, 17664, 20824, 24340, 28231, 32516, 37214, 42344, 47925, 53976, 60516, 67564, 75139, 83260, 91946, 101216, 111089, 121584, 132720, 144516, 156991, 170164, 184054, 198680

Offset: 1

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A100185 (structured anti-prisms), A100145 (for more on structured numbers).

Programs

GAP

List([1..33], n -> (1/6)*(19*n^3-15*n^2+2*n)); # Muniru A Asiru, Feb 14 2018

Magma

[(1/6)*(19*n^3-15*n^2+2*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Maple

a:=n->(1/6)*(19*n^3-15*n^2+2*n): seq(a(n),n=1..33); # Muniru A Asiru, Feb 14 2018

Mathematica

Rest@ CoefficientList[Series[x (1 + 12 x + 6 x^2)/(1 - x)^4, {x, 0, 32}], x] (* Michael De Vlieger, Feb 15 2018 *)

Formula

a(n) = (1/6)*(19*n^3-15*n^2+2*n). [Corrected by Luca Colucci, Mar 01 2011]

G.f.: x*(1 + 12*x + 6*x^2)/(1 - x)^4. - Colin Barker, Jun 08 2012

a(n) = Sum_{i = 0..n-1} (n + i)*(n + 2*i). - Bruno Berselli, Feb 14 2018

E.g.f.: exp(x)*x*(6 + 42*x + 19*x^2)/6. - Stefano Spezia, Oct 11 2023