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
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