A083580 -id:A083580 - cp's OEIS frontend

A083586 Binomial transform of A083580.

0, 1, 5, 22, 96, 419, 1819, 7824, 33302, 140317, 585873, 2427146, 9989068, 40886295, 166600967, 676368388, 2737758594, 11055063953, 44553867901, 179280877950, 720513441080, 2892804682891, 11605214879475, 46528078138232

Offset: 0

Paul Barry, May 01 2003

Index entries for linear recurrences with constant coefficients, signature (11,-43,69,-36).

Cf. A083580.

LinearRecurrence[{11,-43,69,-36},{0,1,5,22},30] (* Harvey P. Dale, Jul 19 2015 *)

a(n) = 2*4^n/3 -5/12 -n*3^(n-1)/2 -3^n/4.
O.g.f.: x*(1-6*x+10*x^2)/((-1+x)(-1+4*x)(-1+3*x)^2). - R. J. Mathar, Apr 02 2008
E.g.f.: exp(x)*(8*exp(3*x) - 3*exp(2*x)*(1 + 2*x) - 5)/12. - Stefano Spezia, Apr 12 2025

A083579 Generalized Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 4, 8, 19, 39, 82, 166, 337, 677, 1360, 2724, 5455, 10915, 21838, 43682, 87373, 174753, 349516, 699040, 1398091, 2796191, 5592394, 11184798, 22369609, 44739229, 89478472, 178956956, 357913927, 715827867, 1431655750, 2863311514

Offset: 0

Paul Barry, May 01 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A083580.

a:=[0,1,1,4];; for n in [5..40] do a[n]:=3*a[n-1]-a[n-2]-3*a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, May 24 2019

I:=[0,1,1,4]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2) - 3*Self(n-3)+2*Self(n-4): n in [1..40]]; // G. C. Greubel, May 25 2019

LinearRecurrence[{3,-1,-3,2}, {0,1,1,4}, 40] (* G. C. Greubel, May 25 2019 *)

concat(0, Vec(x*(1-2*x+2*x^2)/(1-3*x+x^2+3*x^3-2*x^4) + O(x^40))) \\ G. C. Greubel, May 25 2019

(x*(1-2*x+2*x^2)/(1-3*x+x^2+3*x^3-2*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

a(n) = (2^(n+3) - 5*(-1)^n - 3*(2*n+1))/12.
a(n+2) = a(n+1) + 2*a(n) + n, a(0)=0, a(1)=1.
G.f.: x*(1 - 2*x + 2*x^2)/(1 - 3*x + x^2 + 3*x^3 - 2*x^4). - Colin Barker, Jan 16 2012

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A083586 Binomial transform of A083580.

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