A100184 - OEIS

%I #32 Oct 11 2023 15:52:54

%S 1,16,64,164,335,596,966,1464,2109,2920,3916,5116,6539,8204,10130,

%T 12336,14841,17664,20824,24340,28231,32516,37214,42344,47925,53976,

%U 60516,67564,75139,83260,91946,101216,111089,121584,132720,144516,156991,170164,184054,198680

%N Structured octagonal anti-prism numbers.

%H Vincenzo Librandi, <a href="/A100184/b100184.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F G.f.: x*(1 + 12*x + 6*x^2)/(1 - x)^4. - _Colin Barker_, Jun 08 2012

%F a(n) = Sum_{i = 0..n-1} (n + i)*(n + 2*i). - _Bruno Berselli_, Feb 14 2018

%F E.g.f.: exp(x)*x*(6 + 42*x + 19*x^2)/6. - _Stefano Spezia_, Oct 11 2023

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

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

%o (Magma) [(1/6)*(19*n^3-15*n^2+2*n): n in [1..40]]; // _Vincenzo Librandi_, Aug 18 2011

%o (GAP) List([1..33], n -> (1/6)*(19*n^3-15*n^2+2*n)); # _Muniru A Asiru_, Feb 14 2018

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

%K nonn,easy

%O 1,2

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