A105063 a(1)=0, a(2)=0, a(3)=8, a(4)=24, a(n) = 32 + 66*a(n-2) - a(n-4) for n > 4.

0, 0, 8, 24, 560, 1616, 36984, 106664, 2440416, 7038240, 161030504, 464417208, 10625572880, 30644497520, 701126779608, 2022072419144, 46263741881280, 133426135166016, 3052705837384904, 8804102848537944, 201432321525522416

Offset: 1

Author

Pierre CAMI, Apr 05 2005

Keywords

• nonn
• easy

Comments

This sequence has the property 17*a(n)*(a(n) + 1) + 1 is a square.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,66,-66,-1,1).

Crossrefs

Cf. A001090, A103200.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( 8*x^3*(1+x)^2/((1-x)*(1-66*x^2+x^4)) )); // G. C. Greubel, Mar 13 2023

Mathematica

LinearRecurrence[{1,66,-66,-1,1}, {0,0,8,24,560}, 40] (* G. C. Greubel, Mar 13 2023 *)

SageMath

@CachedFunction
def a(n):
    if (n<6): return (0,0,0,8,24,560)[n]
    else: return a(n-1) +66*a(n-2) -66*a(n-3) -a(n-4) +a(n-5)
[a(n) for n in range(1,41)] # G. C. Greubel, Mar 13 2023

Formula

From R. J. Mathar, Nov 13 2009: (Start)

a(n) = a(n-1) +66*a(n-2) -66*a(n-3) -a(n-4) +a(n-5).

G.f.: 8*x^3*(1+x)^2/((1-x)*(1+8*x-x^2)*(1-8*x-x^2)). (End)

a(n) = (1/4)*(-32*[n=0] - 2 + i^n*((23 + 11*(-1)^n)*ChebyshevU(n, 4*I) - i*(187 + 89*(-1)^n)*ChebyshevU(n-1, 4*I))). - G. C. Greubel, Mar 13 2023

Extensions

More terms from R. J. Mathar, Nov 13 2009