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 A068377 Engel expansion of sinh(1). 5

%I

%S 1,6,20,42,72,110,156,210,272,342,420,506,600,702,812,930,1056,1190,

%T 1332,1482,1640,1806,1980,2162,2352,2550,2756,2970,3192,3422,3660,

%U 3906,4160,4422,4692,4970,5256,5550,5852,6162,6480,6806,7140,7482,7832,8190

%N Engel expansion of sinh(1).

%C This sequence is also the Pierce expansion of sin(1). - _G. C. Greubel_, Nov 14 2016

%H Simon Plouffe, <a href="/A068377/b068377.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicSine.html">Hyperbolic Sine</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F 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]

%F From _Colin Barker_, Apr 13 2012: (Start)

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

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

%F E.g.f.: -2 + x + 2*(1 - x + 2*x^2)*exp(x). - _G. C. Greubel_, Oct 27 2016

%t Join[{1}, Table[(2 n - 2) (2 n - 1), {n, 2, 50}]] (* _Bruno Berselli_, Aug 04 2015 *)

%t LinearRecurrence[{3,-3,1}, {1,6,20,42}, 25] (* _G. C. Greubel_, Oct 27 2016; a(1)=1 by _Georg Fischer_, Apr 02 2019*)

%t 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 *)

%t 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 *)

%o (PARI) A068377(n)=(n+n--)*n*2+!n \\ _M. F. Hasler_, Jul 19 2015

%o (Sage)

%o A068377 = lambda n: rising_factorial(n*2,2) if n>0 else 1

%o print [A068377(n) for n in (0..45)] # _Peter Luschny_, Aug 04 2015

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

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, Mar 03 2002

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Last modified August 26 05:42 EDT 2019. Contains 326329 sequences. (Running on oeis4.)