A217233 Expansion of (1-2*x+x^2)/(1-3*x-3*x^2+x^3).

1, 1, 7, 23, 89, 329, 1231, 4591, 17137, 63953, 238679, 890759, 3324361, 12406681, 46302367, 172802783, 644908769, 2406832289, 8982420391, 33522849271, 125108976697, 466913057513, 1742543253359, 6503259955919, 24270496570321, 90578726325361

Offset: 0

Author

Bruno Berselli, Sep 28 2012

Keywords

• nonn
• easy

Comments

Numbers with the property a(n)^2+a(n-1)^2 = 2*(a(n)-a(n-1)-(-1)^n)^2.

Examples

			a(3)=23, a(2)=7: 23^2+7^2 = 2*(23-7-(-1)^3)^2 = 578;
a(6)=1231, a(5)=329: 1231^2+329^2 = 2*(1231-329-(-1)^6)^2 = 1623602.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
• R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
• Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Crossrefs

Cf. A109437 (1/(1-3*x-3*x^2+x^3)), A006253 ((1-x)/(1-3*x-3*x^2+x^3)).

Programs

Magma

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+x^2)/(1-3*x-3*x^2+x^3)));

Mathematica

CoefficientList[Series[(1 - 2 x + x^2)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 25}], x]

Maxima

makelist(coeff(taylor((1-2*x+x^2)/(1-3*x-3*x^2+x^3), x, 0, n), x, n), n, 0, 25);

PARI

Vec((1-2*x+x^2)/(1-3*x-3*x^2+x^3)+O(x^26))

Formula

G.f.: (1-x)^2/((1+x)*(1-4*x+x^2)).

a(n) = (4*(-2)^n+(1-sqrt(3))^(2*n+1)+(1+sqrt(3))^(2*n+1))/(6*2^n).

a(n) = -a(-n-1) = 3*a(n-1)+3*a(n-2)-a(n-3) = 4*a(n-1)-a(n-2)+4*(-1)^n.

a(n)+a(n-1) = A052530(n) with a(-1)=-1.

a(n)-a(n-2) = A003699(n) with n>1.

Sum(a(i), i=0..n) = A006253(n).