This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179934 #22 Feb 09 2025 10:07:00
%S A179934 4,9,36,85,352,837,3480,8281,34444,81969,340956,811405,3375112,
%T A179934 8032077,33410160,79509361,330726484,787061529,3273854676,7791105925,
%U A179934 32407820272,77123997717,320804348040,763448871241,3175635660124
%N A179934 Expansion of x*(4+5*x-13*x^2-x^3+x^4) / ( (1-x)*(1-10*x^2+x^4) ).
%C A179934 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).
%C A179934 The last digit has the period (4,9,6,5,2,7,0,1).
%H A179934 G. C. Greubel, <a href="/A179934/b179934.txt">Table of n, a(n) for n = 1..1000</a>
%H A179934 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,10,-10,-1,1).
%F A179934 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]
%F A179934 a(n+4) = 10*a(n+2) - a(n) - 4.
%F A179934 a(n+6) = 11*(a(n+4) - a(n+2)) + a(n).
%F A179934 a(2*n+1) = (2 + (7 + 3*r)*(5 + 2*r)^n + (7 - 3*r)*(5 - 2*r)^n)/4, r = sqrt(6).
%F A179934 a(2*n+2) = (2 + (17 + 7*r)*(5 + 2*r)^n + (17 - 7*r)*(5 - 2*r)^n)/4, r = sqrt(6).
%F A179934 From _R. J. Mathar_, Aug 03 2010: (Start)
%F A179934 a(n) = +a(n-1) +10*a(n-2) -10*a(n-3) -a(n-4) +a(n-5).
%F A179934 G.f.: x*(4+5*x-13*x^2-x^3+x^4) / ( (1-x)*(1-10*x^2+x^4) ). (End)
%F A179934 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
%p A179934 r:= sqrt(6);
%p A179934 for n from 0 to 20 do
%p A179934 a(2*n+1):= round((2 +(7+3*r)*(5+2*r)^n)/4);
%p A179934 a(2*n+2):= round((2 +(17+7*r)*(5+2*r)^n)/4);
%p A179934 end do;
%p A179934 seq(a(n), n = 1..40);
%t A179934 LinearRecurrence[{1,10,-10,-1,1},{4,9,36,85,352},30] (* _Harvey P. Dale_, Dec 23 2012 *)
%o A179934 (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
%o A179934 (SageMath)
%o A179934 def b(n): return ((1+(-1)^n)/2)*chebyshev_U(n//2, 5)
%o A179934 def A179934(n): return (b(n) +7*b(n-1) +7*b(n-2) +b(n-3) -2*bool(n==0) +1)/2
%o A179934 [A179934(n) for n in (1..50)] # _G. C. Greubel_, Apr 27 2022
%Y A179934 Cf. A004189, A181442.
%K A179934 nonn
%O A179934 1,1
%A A179934 _Paul Weisenhorn_, Aug 02 2010
%E A179934 Edited by _G. C. Greubel_, Apr 27 2022
%E A179934 New name using g.f. by _R. J. Mathar_ from _Joerg Arndt_, Apr 27 2022