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%I #69 Jan 13 2024 13:07:07
%S 1,11,33,67,113,171,241,323,417,523,641,771,913,1067,1233,1411,1601,
%T 1803,2017,2243,2481,2731,2993,3267,3553,3851,4161,4483,4817,5163,
%U 5521,5891,6273,6667,7073,7491,7921,8363,8817,9283,9761,10251,10753,11267
%N a(n) = 6*n^2 + 4*n + 1.
%C The old definition of this sequence was "Generalized polygonal numbers".
%C Column T(n,4) of A080853.
%C Sequence found by reading the line from 1, in the direction 1, 11, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Sep 08 2011
%H G. C. Greubel, <a href="/A080859/b080859.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (C(3,0) + (C(5,2) - 2)*x + C(3,2)*x^2)/(1-x)^3 = (1 + 8*x + 3*x^2)/(1-x)^3.
%F E.g.f.: (1 + 10*x + 6*x^2)*exp(x). - _Vincenzo Librandi_, Apr 29 2016
%F a(n) = C(4,0) + C(4,1)n + C(4,2)n^2.
%F a(n) = A186424(2*n).
%F a(n) = 12*n + a(n-1) - 2 with n > 0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = (n+1)*A000384(n+1) - n*A000384(n). - _Bruno Berselli_, Dec 10 2012
%F a(n) = (n+1)^4 mod n^3 for n >= 7. - _J. M. Bergot_, Aug 14 2017
%F a(n) = (2*n+1)^2 + 2*n^2. - _Robert FERREOL_, Jan 13 2024
%t Table[6 n^2 + 4 n + 1, {n, 0, 50}] (* _Vincenzo Librandi_, Apr 29 2016 *)
%o (PARI) a(n)=6*n^2+4*n+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [6*n^2+4*n+1: n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2016
%Y Subsequence of A186424.
%Y Cf. A000384, A001318, A033579, A033581.
%Y Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 23 2003
%E Definition replaced with the closed form by _Bruno Berselli_, Dec 10 2012
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