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A179934
Expansion of x*(4+5*x-13*x^2-x^3+x^4) / ( (1-x)*(1-10*x^2+x^4) ).
1
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).
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
Sequence in context: A029806 A133125 A126161 * A239213 A346537 A339999
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