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 A253514 #16 May 13 2025 00:59:37 %S A253514 1,841,755161,678133681,608963290321,546848356574521, %T A253514 491069215240629481,440979608437728699361,395999197307865131396641, %U A253514 355606838202854450265484201,319334544706965988473273415801,286762065540017254794549261905041 %N A253514 Centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754). %H A253514 Colin Barker, <a href="/A253514/b253514.txt">Table of n, a(n) for n = 1..339</a> %H A253514 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (899,-899,1). %F A253514 a(n) = 899*a(n-1)-899*a(n-2)+a(n-3). %F A253514 G.f.: -x*(x^2-58*x+1) / ((x-1)*(x^2-898*x+1)). %F A253514 From _Peter Bala_, Apr 15 2025; (Start) %F A253514 a(n) = (1/64)*(-4 + sqrt(14))^2*(15 + 4*sqrt(14) + (449 + 120*sqrt(14))^n)^2 *(449 + 120*sqrt(14))^(-n). %F A253514 a(-n) = a(n+1). %F A253514 a(n) = (1/16) * (1 - T(2*n+1, -15)), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. Cf. A001110. %F A253514 a(n) = A157877(n)^2 = 1 + 7*A157879(n). %F A253514 a(2) divides a(3*n+2); a(3) divides a(5*n+3); a(4) divides a(7*n+4); a(5) divides a(9*n+5). In general, a(k) divides a((2*k-1)*n + k). (End) %e A253514 841 is in the sequence because it is the 16th centered heptagonal number and the 15th centered octagonal number. %o A253514 (PARI) Vec(-x*(x^2-58*x+1)/((x-1)*(x^2-898*x+1)) + O(x^100)) %Y A253514 Cf. A001110, A016754, A069099, A157877, A157879, A253446, A253447. %K A253514 nonn,easy %O A253514 1,2 %A A253514 _Colin Barker_, Jan 03 2015