A118239 Engel expansion of cosh(1).

1, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8010, 8372

Offset

1,2

Comments

Differs from A002939 only in first term.

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Engel Expansion

Eric Weisstein's World of Mathematics, Pierce Expansion

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = A002939(n-1) = 2*(n-1)*(2*n-3) for n>1.

From Colin Barker, Apr 13 2012: (Start)

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

G.f.: x*(1 - x + 9*x^2 - x^3)/(1-x)^3. (End)

E.g.f.: -6 + x + 2*(3 - 3*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 27 2016

Mathematica

Join[{1}, Table[(2 n - 2) (2 n - 3), {n, 2, 50}]] (* Bruno Berselli, Aug 04 2015 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {2, 12, 30}, 25]] (* G. C. Greubel, Oct 27 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[Cos[1] , 7!], 50] (* G. C. Greubel, Nov 14 2016 *)

Prog

(PARI) a(n)=max(4*n^2-10*n+6, 1) \\ Charles R Greathouse IV, Oct 22 2014
(Sage)
A118239 = lambda n: falling_factorial(n*2, 2) if n>0 else 1
print([A118239(n) for n in (0..46)]) # Peter Luschny, Aug 04 2015

Crossrefs

Cf. A006784, A002939, A068377.

Sequence in context: A156021 A067348 A002939 * A249055 A127118 A259127

Adjacent sequences: A118236 A118237 A118238 * A118240 A118241 A118242

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 17 2006

Status

approved