This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A069126 #58 Feb 16 2025 08:32:45
%S A069126 1,14,40,79,131,196,274,365,469,586,716,859,1015,1184,1366,1561,1769,
%T A069126 1990,2224,2471,2731,3004,3290,3589,3901,4226,4564,4915,5279,5656,
%U A069126 6046,6449,6865,7294,7736,8191,8659,9140,9634,10141,10661,11194
%N A069126 Centered 13-gonal numbers.
%C A069126 Centered tridecagonal numbers or centered triskaidecagonal numbers. - _Omar E. Pol_, Oct 03 2011
%H A069126 Ivan Panchenko, <a href="/A069126/b069126.txt">Table of n, a(n) for n = 1..1000</a>
%H A069126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>
%H A069126 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H A069126 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1)
%F A069126 a(n) = (13n^2 - 13n + 2)/2.
%F A069126 Binomial transform of [1, 13, 13, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 13, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 29 2007
%F A069126 a(n) = 13*n+a(n-1)-13 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F A069126 G.f.: -x*(1+11*x+x^2) / (x-1)^3. - _R. J. Mathar_, Feb 04 2011
%F A069126 a(n) = A152741(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F A069126 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A069126 Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(5/13)*Pi/2)/sqrt(65).
%F A069126 Sum_{n>=1} a(n)/n! = 15*e/2 - 1.
%F A069126 Sum_{n>=1} (-1)^n * a(n)/n! = 15/(2*e) - 1. (End)
%F A069126 E.g.f.: exp(x)*(1 + 13*x^2/2) - 1. - _Stefano Spezia_, May 15 2022
%e A069126 a(5) = 131 because 131 = (13*5^2 - 13*5 + 2)/2 = (325 - 65 + 2)/2 = 262/2 = 131.
%t A069126 FoldList[#1 + #2 &, 1, 13 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t A069126 LinearRecurrence[{3,-3,1},{1,14,40},60] (* _Harvey P. Dale_, Jan 20 2014 *)
%t A069126 With[{nn=50},Total/@Thread[{PolygonalNumber[13,Range[nn]],Range[0,nn-1]^2}]] (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Aug 29 2016 *)
%o A069126 (PARI) a(n)=13*n(n-1)/2+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A069126 Cf. A001263, A001844, A003215, A005448, A005891, A069099, A152741.
%K A069126 easy,nice,nonn
%O A069126 1,2
%A A069126 _Terrel Trotter, Jr._, Apr 07 2002