A248848 - OEIS

A248848

Norm of coefficients in the expansion of 1/(1 - 3*x - I*x^2), where I^2=-1.

1, 9, 82, 765, 7129, 66420, 618841, 5765805, 53720578, 500519961, 4663394209, 43449307200, 404821512289, 3771762252921, 35141883671458, 327420421852365, 3050608602778201, 28422823459498740, 264818270254044889, 2467338136208552925, 22988434568917776562, 214185529001504000169

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OFFSET

0,2

COMMENTS

Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. It is the case P1 = 9, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Dec 02 2014

LINKS

Table of n, a(n) for n=0..21.

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 (9,2,9,-1).

FORMULA

G.f.: (1-x^2)/(1 - 9*x - 2*x^2 - 9*x^3 + x^4).

a(n) = 9*a(n-1) + 2*a(n-2) + 9*a(n-3) - a(n-4). - Vaclav Kotesovec, Nov 09 2014

a(n) ~ (1 + 9/sqrt(97) + 3*sqrt((18+2*sqrt(97))/97)) * (9 + sqrt(97) + 3*sqrt(18+2*sqrt(97)))^n / 4^(n+1). - Vaclav Kotesovec, Nov 09 2014

From Peter Bala, Dec 02 2014: (Start)

The following remarks assume an offset of 1:

a(n) = ( T(n,a) - T(n,b) )/(a - b), where T(n,x) denotes the Chebyshev polynomial of the first kind and where a = ( 9 + sqrt(97) )/4 and b = ( 9 - sqrt(97) )/4 denote the roots of the quadratic equation x^2 - 9/2*x - 1 = 0.

a(n) = the bottom left entry of the 2 X 2 matrix 2*T(n,1/2*M), where M is the 2 X 2 matrix [0, 4; 1, 9]. See A100047. (End)

EXAMPLE

G.f.: A(x) = 1 + 9*x + 82*x^2 + 765*x^3 + 7129*x^4 + 66420*x^5 +...

If we expand the complex series:

1/(1 - 3*x + I*x^2) = 1 + 3*x + (9 - I)*x^2 + (27 - 6*I)*x^3 + (80 - 27*I)*x^4 + (234 - 108*I)*x^5 + (675 - 404*I)*x^6 + (1917 - 1446*I)*x^7 + (5347 - 5013*I)*x^8 + (14595 - 16956*I)*x^9 +...

then the terms of this sequence equals the norm of the above coefficients:

a(0) = 1^2 = 1;

a(1) = 3^2 = 9;

a(2) = 9^2 + (-1)^2 = 82;

a(3) = 27^2 + (-6)^2 = 765;

a(4) = 80^2 + (-27)^2 = 7129;

a(5) = 234^2 + (-108)^2 = 66420; ...

MATHEMATICA

Abs[CoefficientList[Series[1/(1 - 3*x - I*x^2), {x, 0, 20}], x]]^2 (* Vaclav Kotesovec, Nov 09 2014 *)

PROG

(PARI) {a(n)=(polcoeff(1/(1-3*x+I*x^2 +x*O(x^n)), n))}

for(n=0, 31, print1(norm(a(n)), ", "))