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
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Mar 03 2002
STATUS
approved