A090590 (1,1) entry of powers of the orthogonal design shown below.

1, -6, -20, 8, 176, 288, -832, -3968, -1280, 29184, 68608, -96256, -741376, -712704, 4505600, 14712832, -6619136, -130940928, -208928768, 629669888, 2930769920, 824180736, -21797797888, -50189041664, 74004299776, 549520932864

Offset: 1

Author

Simone Severini, Dec 04 2003

Keywords

• sign
• easy

Comments

+1 +1 +1 +1 +1 +1 +1 +1

-1 +1 +1 -1 +1 -1 -1 +1

-1 -1 +1 +1 +1 +1 -1 -1

-1 +1 -1 +1 +1 -1 +1 -1

-1 -1 -1 -1 +1 +1 +1 +1

-1 +1 -1 +1 -1 +1 -1 +1

-1 +1 +1 -1 -1 +1 +1 -1

-1 -1 +1 +1 -1 -1 +1 +1

Also real part of (1 +- i*sqrt(7))^n. - Bruno Berselli, Jun 24-25 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..202
• Index entries for linear recurrences with constant coefficients, signature (2,-8).

Crossrefs

Cf. A087621, A090591.

Programs

Magma

m:=27; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-8*x)/(1-2*x+8*x^2))); // Bruno Berselli, Jun 24-25 2011

Maple

a := proc(n) option remember: if(n=1)then return 1:elif(n=2)then return -6:fi: return 2*a(n-1)-8*a(n-2): end: seq(a(n),n=1..26); # Nathaniel Johnston, Jun 25 2011

Mathematica

LinearRecurrence[{2,-8},{1,-6},30] (* Harvey P. Dale, Mar 30 2019 *)

Maxima

makelist(expand(((1+sqrt(-1)*sqrt(7))^n+(1-sqrt(-1)*sqrt(7))^n)/2),n,1,26); /* Bruno Berselli, Jun 24-25 2011 */

PARI

a=vector(26); a[1]=1; a[2]=-6; for(i=3, #a, a[i]=2*a[i-1]-8*a[i-2]); a \\ Bruno Berselli, Jun 24-25 2011

Formula

G.f.: x*(1-8*x)/(1-2*x+8*x^2). - T. D. Noe, Dec 11 2006

From Bruno Berselli, Jun 24-25 2011: (Start)

a(n) = (1/2)*((1+i*sqrt(7))^n + (1-i*sqrt(7))^n), where i=sqrt(-1).

a(n) = cos(n*arctan(sqrt(7)))*sqrt(8)^n.

a(n) = 2*a(n-1) - 8*a(n-2) (n > 2). (End)

Extensions

Corrected by T. D. Noe, Dec 11 2006

More terms from Bruno Berselli, Jun 24 2011