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 A237516 #44 Aug 22 2025 17:39:19
%S A237516 1,15,91,325,861,1891,3655,6441,10585,16471,24531,35245,49141,66795,
%T A237516 88831,115921,148785,188191,234955,289941,354061,428275,513591,611065,
%U A237516 721801,846951,987715,1145341,1321125,1516411,1732591,1971105,2233441,2521135,2835771,3178981,3552445,3957891,4397095,4871881
%N A237516 Pyramidal centered square numbers.
%C A237516 a(n) is sum of natural numbers filled in order-n diamond.
%C A237516 First differences give A173962.
%C A237516 The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A002378(n). - _Miguel-Ángel Pérez García-Ortega_, Jun 05 2025
%H A237516 Colin Barker, <a href="/A237516/b237516.txt">Table of n, a(n) for n = 1..1000</a>
%H A237516 José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, <a href="/A383834/a383834.pdf">El Libro de las Ternas Pitagóricas</a>.
%H A237516 Kival Ngaokrajang, <a href="/A237516/a237516.pdf">Illustration for n = 1..6</a>.
%H A237516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Diamond.html">Diamond</a>.
%H A237516 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A237516 a(n) = 2*n^4 - 4*n^3 + 5*n^2 - 3*n + 1.
%F A237516 a(n) = Sum_{i = 1..(2*n*(n + 1) + 1)} i.
%F A237516 From _Colin Barker_, Jan 17 2015: (Start)
%F A237516 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A237516 G.f.: -x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5. (End)
%F A237516 a(n) = A000217(A001844(n-1)). - _Ivan N. Ianakiev_, Jun 14 2015
%F A237516 a(n) = A002061(n)*A001844(n-1). - _Bruce J. Nicholson_, May 14 2017
%F A237516 a(n) = (A002378(n)+1)*(2*A002378(n)+1). - _Miguel-Ángel Pérez García-Ortega_, Jun 05 2025
%F A237516 E.g.f.: -1 + exp(x)*(1 + 7*x^2 + 8*x^3 + 2*x^4). - _Elmo R. Oliveira_, Aug 22 2025
%t A237516 Table[Sum[i, {i, 2n(n + 1) + 1}], {n, 0, 29}] (* _Alonso del Arte_, Feb 09 2014 *)
%t A237516 LinearRecurrence[{5,-10,10,-5,1},{1,15,91,325,861},60] (* _Harvey P. Dale_, Apr 21 2018 *)
%t A237516 a=Table[(n(n+1)),{n,0,29}];Apply[Join,Map[{(#+1)(2#+1)}&,a]] (* _Miguel-Ángel Pérez García-Ortega_, Jun 05 2025 *)
%o A237516 (Small Basic)
%o A237516 For n = 0 to 50
%o A237516 a = 0
%o A237516 For n1 = 1 to 2*n*(n+1)+1
%o A237516 a = a + n1
%o A237516 EndFor
%o A237516 TextWindow.Write(a+", ")
%o A237516 EndFor
%o A237516 (PARI) Vec(-x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5 + O(x^100)) \\ _Colin Barker_, Jan 17 2015
%Y A237516 Cf. A000217, A001844, A002061, A173962.
%Y A237516 Cf. A002378, A008514, A384288, A384566.
%K A237516 nonn,easy,changed
%O A237516 1,2
%A A237516 _Kival Ngaokrajang_, Feb 08 2014