%I #50 Sep 29 2023 19:35:15
%S 1,2,7,19,41,76,127,197,289,406,551,727,937,1184,1471,1801,2177,2602,
%T 3079,3611,4201,4852,5567,6349,7201,8126,9127,10207,11369,12616,13951,
%U 15377,16897,18514,20231,22051,23977,26012,28159,30421,32801,35302
%N a(n) = n-th centered n-gonal number.
%C a(n) is n times the n-th triangular number plus 1. - _Thomas M. Green_, Nov 16 2009
%C From _Gary W. Adamson_, Jul 31 2010: (Start)
%C Equals (1, 2, 3, 4, ...) convolved with (1, 0, 4, 7, 10, 13, ...).
%C Example: a(5) = 76 = (6, 5, 4, 3, 2, 1) dot (1, 0, 4, 7, 10, 13) = (6 + 0 + 16 + 21 + 20 + 13). (End)
%H Vincenzo Librandi, <a href="/A100119/b100119.txt">Table of n, a(n) for n = 0..1000</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 + n*(n + n^2)/2 = 1 + (1/2)*n^2 + (1/2) * n^3 = 1 + mean(n^2, n^3). - _Joshua Zucker_, May 03 2006
%F Equals A002411(n) + 1. - _Olivier GĂ©rard_, Jun 20 2007
%F G.f.: (1 - 2*x + 5*x^2 - x^3) / (x-1)^4. - _R. J. Mathar_, Apr 04 2012
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 25 2012
%F a(n) = (A098547(n)+1)/2. - _Richard Turk_, Jul 18 2017
%F a(n) = A060354(n+2) - A000290(n+1) = A006003(n+1) - A005563(n) and for n>0 A005920(n) - A068601(n+1). - _Bruce J. Nicholson_, Jun 23 2018
%e a(2) = 2*3 + 1 = 7, a(3) = 3*6 + 1 = 19, a(4) = 4*10 + 1 = 41. - _Thomas M. Green_, Nov 16 2009
%t Table[(n^3+n^2)/2+1,{n,0,6!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 06 2010 *)
%t LinearRecurrence[{4,-6,4,-1},{1,2,7,19},40] (* _Vincenzo Librandi_, Jun 25 2012 *)
%o (Magma) I:=[1, 2, 7, 19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 25 2012
%o (PARI) a(n) = n^2*(n+1)/2+1; \\ _Altug Alkan_, Sep 21 2018
%Y See also A101357 (Cumulative sums of the n-th n-gonal numbers).
%Y A diagonal of A101321.
%Y Cf. A060354, A098547, A101357.
%Y Cf. A006003, A005920.
%K easy,nonn
%O 0,2
%A _Jonathan Vos Post_, Dec 26 2004
%E Corrected and extended by _Joshua Zucker_, May 03 2006