login
A248848
Norm of coefficients in the expansion of 1/(1 - 3*x - I*x^2), where I^2=-1.
0
1, 9, 82, 765, 7129, 66420, 618841, 5765805, 53720578, 500519961, 4663394209, 43449307200, 404821512289, 3771762252921, 35141883671458, 327420421852365, 3050608602778201, 28422823459498740, 264818270254044889, 2467338136208552925, 22988434568917776562, 214185529001504000169
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
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
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)), ", "))
(Magma) I:=[1, 9, 82, 765]; [n le 4 select I[n] else 9*Self(n-1)+2*Self(n-2)+9*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015
CROSSREFS
Cf. A100047.
Sequence in context: A163460 A081191 A060531 * A045741 A283498 A294956
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Nov 02 2014
STATUS
approved