A218134 Norm of coefficients in the expansion of 1/(1 - 2*x - i*x^2), where i is the imaginary unit.

1, 4, 17, 80, 369, 1700, 7841, 36160, 166753, 768996, 3546289, 16354000, 75417809, 347795396, 1603886913, 7396455680, 34109360321, 157298104900, 725393076049, 3345209499600, 15426707209777, 71141522037604, 328074947492321, 1512944453384000, 6977067089461281

Offset: 0

Author

Paul D. Hanna, Oct 21 2012

Keywords

• nonn

Comments

The radius of convergence of g.f. equals 1 + sqrt(2) - sqrt(2)*sqrt(1 + sqrt(2)) = 0.216845335...

The following remarks assume an offset of 1. This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4. It is the case P1 = 4, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

Examples

			G.f.: A(x) = 1 + 4*x + 17*x^2 + 80*x^3 + 369*x^4 + 1700*x^5 + 7841*x^6 +...
The terms equal the norm of the complex coefficients in the expansion:
1/(1 - 2*x - i*x^2) = 1 + 2*x + (4 + i)*x^2 + (8 + 4*i)*x^3 + (15 + 12*i)*x^4 + (26 + 32*i)*x^5 + (40 + 79*i)*x^6 + (48 + 184*i)*x^7 +...
so that
a(1) = 2^2, a(2) = 4^2 + 1, a(3) = 8^2 + 4^2, a(4) = 15^2 + 12^2, a(5) = 26^2 + 32^2, ...
		

Links

• Peter Bala, Linear divisibility sequences and Chebyshev polynomials
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
• Index entries for linear recurrences with constant coefficients, signature (4,2,4,-1).

Crossrefs

Cf. A105309, A218135.

Cf. also A000129, A057087, A100047.

Programs

Mathematica

LinearRecurrence[{4, 2, 4, -1}, {1, 4, 17, 80}, 25] (* Jean-François Alcover, Nov 02 2019 *)

PARI

{a(n)=norm(polcoeff(1/(1-2*x-I*x^2+x*O(x^n)), n))}
for(n=0,31,print1(a(n),", "))

Formula

G.f.: (1 - x^2)/(1 - 4*x - 2*x^2 - 4*x^3 + x^4).

From Peter Bala, Mar 25 2014: (Start)

The following formulas assume an offset of 1.

a(n) = 1/(2*sqrt(2))*(T(n,1 + sqrt(2)) - T(n,1 - sqrt(2))), where T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 2]. Note, the bottom left element of the matrix M^n gives the Lucas sequence A000129.

a(n) = U(n-1,exp(2*i*Pi/8))*U(n-1,exp(-2*i*Pi/8)) = U(n-1,(1 + i)/sqrt(2))*U(n-1,(1 - i)/sqrt(2)), where U(n,x) denotes the Chebyshev polynomial of the second kind.

The o.g.f. is the Chebyshev transform of the rational function x/(1 - 4*x - 4*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)