A179934 Expansion of x*(4+5*x-13*x^2-x^3+x^4) / ( (1-x)*(1-10*x^2+x^4) ).

4, 9, 36, 85, 352, 837, 3480, 8281, 34444, 81969, 340956, 811405, 3375112, 8032077, 33410160, 79509361, 330726484, 787061529, 3273854676, 7791105925, 32407820272, 77123997717, 320804348040, 763448871241, 3175635660124

Offset

1,1

Comments

Previous name was: a(n) red balls and b(n) blue balls in an urn; draw 2 balls without replacement; Probability(2 red balls) = 6*Probability(2 blue balls); b(n) = A181442(n).

The last digit has the period (4,9,6,5,2,7,0,1).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-1,1).

Formula

a(n) = (1 + sqrt(1 + 24*b(n) + (b(n) - 1))/2; this is equivalent to the Pell equation A(n)^2 - 6*B(n)^2 = -5 with the two fundamental solutions (7;3) and (17;7) and the solution (5;2) for the unit form; a(n) = (A(n) + 1)/2; b(n) = (B(n) + 1)/2.

a(n+4) = 10*a(n+2) - a(n) - 4.

a(n+6) = 11*(a(n+4) - a(n+2)) + a(n).

a(2*n+1) = (2 + (7 + 3*r)*(5 + 2*r)^n + (7 - 3*r)*(5 - 2*r)^n)/4, r = sqrt(6).

a(2*n+2) = (2 + (17 + 7*r)*(5 + 2*r)^n + (17 - 7*r)*(5 - 2*r)^n)/4, r = sqrt(6).

From R. J. Mathar, Aug 03 2010: (Start)

a(n) = +a(n-1) +10*a(n-2) -10*a(n-3) -a(n-4) +a(n-5).

G.f.: x*(4+5*x-13*x^2-x^3+x^4) / ( (1-x)*(1-10*x^2+x^4) ). (End)

a(n) = (b(n) +7*b(n-1) +7*b(n-2) +b(n-3) -2*bool(n==0) +1)/2, where b(n) = ((1 + (-1)^n)/2)*ChebyshevU(n/2, 5). - G. C. Greubel, Apr 27 2022

Maple

r:= sqrt(6);
for n from 0 to 20 do
    a(2*n+1):= round((2 +(7+3*r)*(5+2*r)^n)/4);
    a(2*n+2):= round((2 +(17+7*r)*(5+2*r)^n)/4);
end do;
seq(a(n), n = 1..40);

Mathematica

LinearRecurrence[{1, 10, -10, -1, 1}, {4, 9, 36, 85, 352}, 30] (* Harvey P. Dale, Dec 23 2012 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x*(4+5*x-13*x^2-x^3+x^4)/((1-x)*(1-10*x^2+x^4)) )); // G. C. Greubel, Apr 27 2022
(SageMath)
def b(n): return ((1+(-1)^n)/2)*chebyshev_U(n//2, 5)
def A179934(n): return (b(n) +7*b(n-1) +7*b(n-2) +b(n-3) -2*bool(n==0) +1)/2
[A179934(n) for n in (1..50)] # G. C. Greubel, Apr 27 2022

Crossrefs

Cf. A004189, A181442.

Sequence in context: A029806 A133125 A126161 * A239213 A346537 A339999

Adjacent sequences: A179931 A179932 A179933 * A179935 A179936 A179937

Keyword

nonn

Author

Paul Weisenhorn, Aug 02 2010

Extensions

Edited by G. C. Greubel, Apr 27 2022

New name using g.f. by R. J. Mathar from Joerg Arndt, Apr 27 2022

Status

approved