%I #18 Aug 03 2024 17:41:09
%S 1,2,5,22,34,161,494,2365,3685,17654,54281,260074,405262,1941725,
%T 5970362,28605721,44575081,213572042,656685485,3146369182,4902853594,
%U 23490982841,72229432934,346072004245,539269320205,2583794540414,7944580937201,38064774097714,59314722368902,284193908462645,873831673659122,4186779078744241,6524080191258961,31258746136350482,96113539521566165,460507633887768742,717589506316116754
%N Solutions a(n) to (r(n)-5)*(r(n)-6) = 21 *a(n)*(a(n)-1).
%C The associated r(n) are in A180509, which gives a combinatorial interpretation of the pairs (r(n),a(n)).
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,110,-110,0,0,-1,1).
%F G.f. ( -1-x-3*x^2-17*x^3+98*x^4-17*x^5-3*x^6-x^7-x^8 ) / ( (x-1)*(x^8-110*x^4+1) ). - _R. J. Mathar_, Feb 05 2011
%F Explicit formulas: r=sqrt(21), s=55+12*r, t=55-12*r:
%F a(4*n)=(42+(21+r)*s^n+(21-r)*t^n)/84.
%F a(4*n+1)=(42+(63+13*r)*s^n+(63-13*r)*t^n)/84.
%F a(4*n+2)=(42+(189+41*r)*s^n+(189-41*r)*t^n)/84.
%F a(4*n+3)=(42+(903+197*r)*s^n+(903-197*r)*t^n)/84.
%F a(n) = 111*a(n-4) - 111*a(n-8) + a(n-12).
%F a(n) = +a(n-1) +110*a(n-4) -110*a(n-5) -a(n-8) +a(n-9). - _R. J. Mathar_, Jan 05 2011
%e For n=2: a(2)=5; b(2)=26; binomial(26,7)=657800; binomial(26,5)*binomial(5,2)=657800.
%p n:=0: for s from 1 to 100 do r:=(sqrt(84*s^2-84*s+1)+11)/2: if (floor(r)=r) then a[n]:=s: b[n]:=r: n:=n+1: end if: end do:
%t LinearRecurrence[{1,0,0,110,-110,0,0,-1,1},{1,2,5,22,34,161,494,2365,3685},40] (* _Harvey P. Dale_, Aug 03 2024 *)
%K nonn
%O 0,2
%A _Paul Weisenhorn_, Jan 29 2011