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 A068377 #62 May 05 2025 01:33:25
%S A068377 1,6,20,42,72,110,156,210,272,342,420,506,600,702,812,930,1056,1190,
%T A068377 1332,1482,1640,1806,1980,2162,2352,2550,2756,2970,3192,3422,3660,
%U A068377 3906,4160,4422,4692,4970,5256,5550,5852,6162,6480,6806,7140,7482,7832,8190
%N A068377 Engel expansion of sinh(1).
%C A068377 This sequence is also the Pierce expansion of sin(1). - _G. C. Greubel_, Nov 14 2016
%H A068377 Simon Plouffe, <a href="/A068377/b068377.txt">Table of n, a(n) for n = 1..1000</a>
%H A068377 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>.
%H A068377 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperbolicSine.html">Hyperbolic Sine</a>.
%H A068377 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.
%H A068377 Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>.
%H A068377 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A068377 a(n) = (2*n-2)*(2*n-1) = A002943(n-1) = 2*A000217(2n-2) for n>1. [Corrected and extended by _M. F. Hasler_, Jul 19 2015]
%F A068377 From _Colin Barker_, Apr 13 2012: (Start)
%F A068377 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
%F A068377 G.f.: x*(1 + 3*x + 5*x^2 - x^3)/(1-x)^3. (End)
%F A068377 E.g.f.: -2 + x + 2*(1 - x + 2*x^2)*exp(x). - _G. C. Greubel_, Oct 27 2016
%F A068377 From _Amiram Eldar_, May 05 2025: (Start)
%F A068377 Sum_{n>=1} 1/a(n) = 2 - log(2).
%F A068377 Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - Pi/4 - log(2)/2. (End)
%t A068377 Join[{1}, Table[(2 n - 2) (2 n - 1), {n, 2, 50}]] (* _Bruno Berselli_, Aug 04 2015 *)
%t A068377 LinearRecurrence[{3,-3,1}, {1,6,20,42}, 25] (* _G. C. Greubel_, Oct 27 2016; a(1)=1 by _Georg Fischer_, Apr 02 2019*)
%t A068377 Rest@ CoefficientList[Series[x (1 + 3 x + 5 x^2 - x^3)/(1 - x)^3, {x, 0, 46}], x] (* _Michael De Vlieger_, Oct 28 2016 *)
%t A068377 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[Sin[1] , 7!], 50] (* _G. C. Greubel_, Nov 14 2016 *)
%o A068377 (PARI) A068377(n)=(n+n--)*n*2+!n \\ _M. F. Hasler_, Jul 19 2015
%o A068377 (Sage)
%o A068377 A068377 = lambda n: rising_factorial(n*2,2) if n>0 else 1
%o A068377 print([A068377(n) for n in (0..45)]) # _Peter Luschny_, Aug 04 2015
%Y A068377 Cf. A000217, A002943, A006784, A068379, A068380, A073742 (sinh(1)).
%K A068377 nonn,easy
%O A068377 1,2
%A A068377 _Benoit Cloitre_, Mar 03 2002