A085487 a(n) = p^n + q^n, p = (1 + sqrt(21))/2, q = (1 - sqrt(21))/2.

1, 11, 16, 71, 151, 506, 1261, 3791, 10096, 29051, 79531, 224786, 622441, 1746371, 4858576, 13590431, 37883311, 105835466, 295252021, 824429351, 2300689456, 6422836211, 17926283491, 50040464546, 139671882001, 389874204731, 1088233614736

Offset: 1

Author

Gary W. Adamson, Jul 02 2003

Keywords

• nonn
• easy

Comments

p + q = 1, p*q = -5, p - q = sqrt(21).

The Lucas sequence V(1,-5) apart from the initial term a(0) = 2. - Peter Bala, Jun 23 2015

Examples

			a(5) = 151 = p^5 + q^5, with p = 2.79128...; q = -1.79128...
		

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Wikipedia, Lucas sequence
• Index entries for linear recurrences with constant coefficients, signature (1,5).

Crossrefs

Cf. A015440.

Programs

Magma

I:=[ 1,11]; [n le 2 select I[n] else Self(n-1)+5*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jul 20 2013

Mathematica

CoefficientList[Series[(10 x + 1) / (1 - x - 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)

Maxima

a(n):=n*sum((binomial(k,n-k)*1^(2*k-n)*(5)^(n-k))/k,k,1,n); /* Dmitry Kruchinin, May 16 2011 */

Sage

[lucas_number2(n,1,-5) for n in range(1, 11)] # Zerinvary Lajos, May 14 2009

Formula

G.f.: (10*x^2+x)/(1-x-5*x^2).

a(n) = n*sum(k=1..n, (C(k,n-k)*1^(2*k-n)*(5)^(n-k))/k). - Dmitry Kruchinin, May 16 2011

a(n) = a(n-1) + 5a(n-2), n>1.

a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 21*x^2))/2 )^n. - Peter Bala, Jun 23 2015