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

1,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A006784, A073742 (sinh(1)).

Sequence in context: A143711 A338120 A077539 * A002943 A009946 A290154

Adjacent sequences: A068374 A068375 A068376 * A068378 A068379 A068380

Keyword

nonn,easy

Author

Benoit Cloitre, Mar 03 2002

Status

approved