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 A118239 #38 May 05 2025 01:32:39
%S A118239 1,2,12,30,56,90,132,182,240,306,380,462,552,650,756,870,992,1122,
%T A118239 1260,1406,1560,1722,1892,2070,2256,2450,2652,2862,3080,3306,3540,
%U A118239 3782,4032,4290,4556,4830,5112,5402,5700,6006,6320,6642,6972,7310,7656,8010,8372
%N A118239 Engel expansion of cosh(1).
%C A118239 Differs from A002939 only in first term.
%C A118239 This sequence is also the Pierce expansion of cos(1). - _G. C. Greubel_, Nov 14 2016
%H A118239 G. C. Greubel, <a href="/A118239/b118239.txt">Table of n, a(n) for n = 1..1000</a>
%H A118239 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>.
%H A118239 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.
%H A118239 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A118239 a(n) = A002939(n-1) = 2*(n-1)*(2*n-3) for n>1.
%F A118239 From _Colin Barker_, Apr 13 2012: (Start)
%F A118239 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A118239 G.f.: x*(1 - x + 9*x^2 - x^3)/(1-x)^3. (End)
%F A118239 E.g.f.: -6 + x + 2*(3 - 3*x + 2*x^2)*exp(x). - _G. C. Greubel_, Oct 27 2016
%F A118239 From _Amiram Eldar_, May 05 2025: (Start)
%F A118239 Sum_{n>=1} 1/a(n) = log(2) + 1.
%F A118239 Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - Pi/4 + log(2)/2. (End)
%t A118239 Join[{1}, Table[(2 n - 2) (2 n - 3), {n, 2, 50}]] (* _Bruno Berselli_, Aug 04 2015 *)
%t A118239 Join[{1}, LinearRecurrence[{3,-3,1},{2,12,30},25]] (* _G. C. Greubel_, Oct 27 2016 *)
%t A118239 PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Cos[1] , 7!], 50] (* _G. C. Greubel_, Nov 14 2016 *)
%o A118239 (PARI) a(n)=max(4*n^2-10*n+6, 1) \\ _Charles R Greathouse IV_, Oct 22 2014
%o A118239 (Sage)
%o A118239 A118239 = lambda n: falling_factorial(n*2,2) if n>0 else 1
%o A118239 print([A118239(n) for n in (0..46)]) # _Peter Luschny_, Aug 04 2015
%Y A118239 Cf. A006784, A002939, A068377.
%K A118239 nonn,easy
%O A118239 1,2
%A A118239 _Eric W. Weisstein_, Apr 17 2006