A136201 a(0)=a(1)=a(2)=0, a(3)=1, and a(n) = 2*a(n-1) + 4*a(n-2) - 6*a(n-3) - 3*a(n-4) for n > 3.

0, 0, 0, 1, 2, 8, 18, 53, 124, 328, 780, 1969, 4718, 11648, 28014, 68405, 164824, 400240, 965304, 2337409, 5640122, 13637336, 32914794, 79525973, 191966740, 463636600, 1119239940, 2702647921, 6524535782, 15753313808, 38031163398, 91820423573, 221672010544

Offset

0,5

Comments

Based on a Pell recurrence.

A137255(n+1) - 2*A137255(n) has same recurrence.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,4,-6,-3).

Formula

a(n) = (-A108411(n) + A001333(n))/4. - R. J. Mathar, Apr 01 2008

a(n) = (1/8)*(1+sqrt(2))^n + (1/8)*(1-sqrt(2))^n + (1/24)*3^(n/2)*(-3 - sqrt(3) - 3(-1)^n + (-1)^n*sqrt(3)). - Emeric Deutsch, Mar 31 2008

G.f.: x^3/((1 - 2*x - x^2)*(1 - 3*x^2)). - Alexander R. Povolotsky, Mar 31 2008

E.g.f.: (3*exp(x)*cosh(sqrt(2)*x) - 3*cosh(sqrt(3)*x) - sqrt(3)*sinh(sqrt(3)*x))/12. - Stefano Spezia, Mar 26 2026

Maple

a:=proc(n) options operator, arrow: expand((1/8)*(1+sqrt(2))^n+(1/8)*(1-sqrt(2))^n+(1/24)*3^((1/2)*n)*(-3-sqrt(3)-3*(-1)^n+(-1)^n*sqrt(3))) end proc: seq(a(n), n=0..30); # Emeric Deutsch, Mar 31 2008

Mathematica

LinearRecurrence[{2, 4, -6, -3}, {0, 0, 0, 1}, 50] (* G. C. Greubel, Feb 23 2017 *)
CoefficientList[Series[x^3/(1-2 x-4 x^2+6 x^3+3 x^4), {x, 0, 50}], x] (* Harvey P. Dale, Apr 21 2022 *)

Prog

(PARI) my(x='x+O('x^50)); Vec(x^3/(3*x^4 + 6*x^3 - 4*x^2 - 2*x + 1)) \\ G. C. Greubel, Feb 23 2017

Crossrefs

Cf. A001333, A108411, A137255.

Sequence in context: A267638 A153335 A119853 * A358907 A058082 A005675

Adjacent sequences: A136198 A136199 A136200 * A136202 A136203 A136204

Keyword

nonn,easy

Author

Paul Curtz, Mar 16 2008

Status

approved