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%I #57 Nov 20 2020 05:02:33
%S 0,1,324,104329,33593616,10817040025,3483053294436,1121532343768369,
%T 361129931640120384,116282716455774995281,37442673568827908360100,
%U 12056424606446130716956921,3882131280602085262951768464,1250034215929265008539752488489
%N Expansion of x*(1 + x)/((1 - x)*(1 - 322*x + x^2)).
%C 16*k*a(n) provides infinitely many x-values solutions (x,y) of x*(5*x + k) = y^2.
%C This follows from the fact that 5*16*a(n) + 1 is a perfect square: more precisely, 80*a(n) + 1 = A023039(n)^2.
%C This is a divisibility sequence, that is a(n) divides a(m) if n divides m. It is the case P1 = 324, P2 = 644, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Jan 19 2018
%H Colin Barker, <a href="/A298101/b298101.txt">Table of n, a(n) for n = 0..300</a>
%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F G.f.: x*(1 + x)/((1 - x)*(1 - 322*x + x^2)).
%F a(n) = a(-n) = ((2 + sqrt(5))^(4*n) + (2 - sqrt(5))^(4*n) - 2)/320.
%F a(n) = A225786(n)/48. This is the case k=3 of the first comment. Example: for n = 2, 16*3*a(2) = A225786(2) = 15552 and 15552*(5*15552+3) = 34776^2.
%F a(n) = A049660(n)^2.
%F a(n)*(80*a(n) + 1) = 81*A253368(n)^2 for n>0.
%F a(n)*a(n-2) = (a(n-1) - 1)^2.
%F a(n) = 322*a(n-1) - a(n-2) + 2.
%F a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3). - _Iain Fox_, Jan 12 2018
%F a(n) = A298271(n)+A298271(n-1). - _R. J. Mathar_, Nov 20 2020
%p P:=proc(n) trunc(evalf(((2+sqrt(5))^(4*n)+(2-sqrt(5))^(4*n)-2)/320,1000));
%p end: seq(P(i),i=0..13); # _Paolo P. Lava_, Jan 18 2018
%t CoefficientList[x (1 + x)/((1 - x) (1 - 322 x + x^2)) + O[x]^20, x]
%o (Sage)
%o gf = x*(1+x)/((1-x)*(1-322*x+x^2))
%o print(taylor(gf, x, 0, 20).list())
%o (Maxima) makelist(coeff(taylor(x*(1+x)/((1-x)*(1-322*x+x^2)), x, 0, n), x, n), n, 0, 20);
%o (PARI) first(n) = Vec(x*(1 + x)/((1 - x)*(1 - 322*x + x^2)) + O(x^n), -n) \\ _Iain Fox_, Jan 12 2018
%Y Cf. A023039, A049660, A225786, A253368, A298271.
%K nonn,easy
%O 0,3
%A _Bruno Berselli_, Jan 12 2018