Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #46 Sep 08 2022 08:46:15
%S 1,34,189,616,1525,3186,5929,10144,16281,24850,36421,51624,71149,
%T 95746,126225,163456,208369,261954,325261,399400,485541,584914,698809,
%U 828576,975625,1141426,1327509,1535464,1766941,2023650,2307361,2619904,2963169,3339106,3749725,4197096
%N a(n) = (n + 1)^2*(5*n^2 + 10*n + 2)/2.
%C Partial sums of centered dodecahedral numbers (A005904).
%H OEIS Wiki, <a href="https://oeis.org/wiki/Centered_Platonic_numbers">Centered Platonic numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlatonicSolid.html">Platonic Solid</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)
%F G.f.: (1 + 29*x + 29*x^2 + x^3)/(1 - x)^5.
%F E.g.f.: exp(x)*(2 + 66*x + 122*x^2 + 50*x^3 + 5*x^4)/2.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F Sum_{n>=0} 1/a(n) = (5 - Pi^2 - sqrt(15)*Pi*cot(sqrt(3/5)*Pi))/9 = 1.0377796966... . - _Vaclav Kotesovec_, Apr 10 2016
%p A269237:=n->(n + 1)^2*(5*n^2 + 10*n + 2)/2: seq(A269237(n), n=0..50); # _Wesley Ivan Hurt_, Oct 15 2017
%t Table[(n + 1)^2 ((5 n^2 + 10 n + 2)/2), {n, 0, 35}]
%t LinearRecurrence[{5, -10, 10, -5, 1}, {1, 34, 189, 616, 1525}, 36]
%o (PARI) x='x+O('x^99); Vec((1+29*x+29*x^2+x^3)/(1-x)^5) \\ _Altug Alkan_, Apr 10 2016
%o (Magma) [(n + 1)^2*(5*n^2 + 10*n + 2)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Oct 15 2017
%Y Cf. A005904, A006566, A008354, A014820, A037270, A132366.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Apr 09 2016