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

Author

Eric W. Weisstein, Apr 17 2006

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A006784, A002939, A068377.

Programs

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 *)

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

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

From Amiram Eldar, May 05 2025: (Start)

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

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