A298101 Expansion of x*(1 + x)/((1 - x)*(1 - 322*x + x^2)).

0, 1, 324, 104329, 33593616, 10817040025, 3483053294436, 1121532343768369, 361129931640120384, 116282716455774995281, 37442673568827908360100, 12056424606446130716956921, 3882131280602085262951768464, 1250034215929265008539752488489

Offset: 0

Author

Bruno Berselli, Jan 12 2018

Keywords

• nonn
• easy

Comments

16*k*a(n) provides infinitely many x-values solutions (x,y) of x*(5*x + k) = y^2.

This follows from the fact that 5*16*a(n) + 1 is a perfect square: more precisely, 80*a(n) + 1 = A023039(n)^2.

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

Links

• Colin Barker, Table of n, a(n) for n = 0..300
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Crossrefs

Cf. A023039, A049660, A225786, A253368, A298271.

Programs

Maple

P:=proc(n) trunc(evalf(((2+sqrt(5))^(4*n)+(2-sqrt(5))^(4*n)-2)/320,1000));
end: seq(P(i),i=0..13); # Paolo P. Lava, Jan 18 2018

Mathematica

CoefficientList[x (1 + x)/((1 - x) (1 - 322 x + x^2)) + O[x]^20, x]

Maxima

makelist(coeff(taylor(x*(1+x)/((1-x)*(1-322*x+x^2)), x, 0, n), x, n), n, 0, 20);

PARI

first(n) = Vec(x*(1 + x)/((1 - x)*(1 - 322*x + x^2)) + O(x^n), -n) \\ Iain Fox, Jan 12 2018

Sage

gf = x*(1+x)/((1-x)*(1-322*x+x^2))
print(taylor(gf, x, 0, 20).list())

Formula

G.f.: x*(1 + x)/((1 - x)*(1 - 322*x + x^2)).

a(n) = a(-n) = ((2 + sqrt(5))^(4*n) + (2 - sqrt(5))^(4*n) - 2)/320.

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.

a(n) = A049660(n)^2.

a(n)*(80*a(n) + 1) = 81*A253368(n)^2 for n>0.

a(n)*a(n-2) = (a(n-1) - 1)^2.

a(n) = 322*a(n-1) - a(n-2) + 2.

a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3). - Iain Fox, Jan 12 2018

a(n) = A298271(n)+A298271(n-1). - R. J. Mathar, Nov 20 2020