A191697 a(n) = r1^n + r2^n + r3^n where r1, r2, r3 are the three roots of x^3 - 2*x - 2 = 0.

0, 4, 6, 8, 20, 28, 56, 96, 168, 304, 528, 944, 1664, 2944, 5216, 9216, 16320, 28864, 51072, 90368, 159872, 282880, 500480, 885504, 1566720, 2771968, 4904448, 8677376, 15352832, 27163648, 48060416, 85032960, 150448128, 266186752, 470962176, 833269760

Offset: 1

Author

N. J. A. Sloane, Jun 19 2011

Keywords

• nonn
• easy

Comments

Definition 1.1 defines F_3^n as a Boolean function and definition 1.3 defines the Fourier transform of a Boolean function. - Michael Somos, Aug 04 2012

Examples

			G.f. = 4*x^2 + 6*x^3 + 8*x^4 + 20*x^5 + 28*x^6 + 56*x^7 + 96*x^8 + 168*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500
• X. Zhang, H. Guo, R. Feng and Y. Li, Proof of a conjecture about rotation symmetric functions, Discrete Math., 311 (2011), 1281-1289.
• Index entries for linear recurrences with constant coefficients, signature (0,2,2).

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x^2*(2+3*x)/(1-2*x^2-2*x^3))); // G. C. Greubel, Aug 13 2018

Magma

I:=[0,4,6]; [n le 3 select I[n] else 2*Self(n-2)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 14 2018

Mathematica

Rest[CoefficientList[Series[2*x^2*(2+3*x)/(1-2*x^2-2*x^3), {x, 0, 50}], x]] (* G. C. Greubel, Aug 13 2018 *)
LinearRecurrence[{0, 2, 2}, {0, 4, 6}, 40] (* Vincenzo Librandi, Aug 14 2018 *)

PARI

{a(n) = if( n<1, 0, polsym( x^3 - 2*x - 2, n)[n + 1])}; /* Michael Somos, Aug 04 2012 */

PARI

{a(n) = if( n<1, 0, sum( x=0, 2^n-1, (-1)^sum( i=0, n-1, bittest(x, i) * bittest(x, (i+1)%n) * bittest(x, (i+2)%n))))}; /* Michael Somos, Aug 04 2012 */

Formula

a(n) = hat{F_3^n}(0), the Fourier transform evaluated at 0 of the Boolean function F_3^n defined by F_3^n(x_0, ..., x_{n-1}) = Sum_{ 0Michael Somos, Aug 04 2012

From Michael Somos, Aug 04 2012: (Start)

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

a(n + 3) = 2*a(n + 1) + 2*a(n). (End)

a(n) = 4*A052907(n) +6*A052907(n-1). - R. J. Mathar, Aug 10 2012