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

Author

Paul Weisenhorn, Aug 02 2010

Keywords

• nonn

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).

Crossrefs

Cf. A004189, A181442.

Programs

Magma

R:=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

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 *)

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

Formula

a(n) = (1 + sqrt(1 + 24*b(n)*(b(n) - 1)))/2 where b(n) = A181442(n); 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. [corrected by Jason Yuen, Feb 09 2025]

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

Extensions

Edited by G. C. Greubel, Apr 27 2022

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