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A344410 a(n) = (3*n^2 - 1) * (3*n^2 - 2) * (3*n^3 - 3*n + 1)/2.

%I A344410 #36 Dec 10 2024 09:56:23
%S A344410 1,1,1045,23725,195661,975061,3578401,10680265,27453385,63016921,
%T A344410 132361021,258815701,477132085,837244045,1408778281,2286380881,
%U A344410 3595928401,5501691505,8214519205,12001111741,17194450141,24205450501,33535911025,45792819865,61704091801
%N A344410 a(n) = (3*n^2 - 1) * (3*n^2 - 2) * (3*n^3 - 3*n + 1)/2.
%C A344410 a(n) is both (3*n+2)-gonal number and (3*n+2)-gonal pyramidal number.
%H A344410 Seiichi Manyama, <a href="/A344410/b344410.txt">Table of n, a(n) for n = 0..10000</a>
%H A344410 Numberphile, <a href="https://www.numberphile.com/cannon-ball-numbers">Cannon Ball Numbers</a>.
%H A344410 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A344410 Let p(k,m) = A057145(k,m) denote m-th k-gonal number. Then
%F A344410   a(n) = p(3*n+2, 3*n^3-3*n+1);
%F A344410   a(n) = Sum_{j=1..3*n^2-2} p(3*n+2, j) for n > 0.
%F A344410 G.f.: (1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8.
%F A344410 a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - _Wesley Ivan Hurt_, Sep 05 2022
%t A344410 Table[PolygonalNumber[3*n + 2, 3*n^3 - 3*n + 1], {n, 0, 24}] (* _Amiram Eldar_, May 17 2021 *)
%t A344410 LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,1,1045,23725,195661,975061,3578401,10680265},30] (* _Harvey P. Dale_, Aug 10 2021 *)
%o A344410 (PARI) a(n) = (3*n^2-1)*(3*n^2-2)*(3*n^3-3*n+1)/2;
%o A344410 (PARI) p(k, n) = n*((k-2)*n-k+4)/2;
%o A344410 a(n) = p(3*n+2, 3*n^3-3*n+1);
%o A344410 (PARI) my(N=40, x='x+O('x^N)); Vec((1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8)
%Y A344410 Cf. A016789, A027669, A057145, A344376, A344377.
%K A344410 nonn,easy
%O A344410 0,3
%A A344410 _Seiichi Manyama_, May 17 2021