A068377 Engel expansion of sinh(1).

1, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190

Offset: 1

Author

Benoit Cloitre, Mar 03 2002

Keywords

• nonn
• easy

Comments

This sequence is also the Pierce expansion of sin(1). - G. C. Greubel, Nov 14 2016

Links

• Simon Plouffe, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Engel Expansion.
• Eric Weisstein's World of Mathematics, Hyperbolic Sine.
• Eric Weisstein's World of Mathematics, Pierce Expansion.
• Wikipedia, Engel Expansion.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A002943, A006784, A068379, A068380, A073742 (sinh(1)).

Programs

Mathematica

Join[{1}, Table[(2 n - 2) (2 n - 1), {n, 2, 50}]] (* Bruno Berselli, Aug 04 2015 *)
LinearRecurrence[{3,-3,1}, {1,6,20,42}, 25] (* G. C. Greubel, Oct 27 2016; a(1)=1 by Georg Fischer, Apr 02 2019*)
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 *)
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 *)

PARI

A068377(n)=(n+n--)*n*2+!n \\ M. F. Hasler, Jul 19 2015

Sage

A068377 = lambda n: rising_factorial(n*2,2) if n>0 else 1
print([A068377(n) for n in (0..45)]) # Peter Luschny, Aug 04 2015

Formula

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]

From Colin Barker, Apr 13 2012: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.

G.f.: x*(1 + 3*x + 5*x^2 - x^3)/(1-x)^3. (End)

E.g.f.: -2 + x + 2*(1 - x + 2*x^2)*exp(x). - G. C. Greubel, Oct 27 2016

From Amiram Eldar, May 05 2025: (Start)

Sum_{n>=1} 1/a(n) = 2 - log(2).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - Pi/4 - log(2)/2. (End)