A098586 a(n) = (1/2) * (5*P(n+1) + P(n) - 1), where P(k) are the Pell numbers A000129.

2, 5, 13, 32, 78, 189, 457, 1104, 2666, 6437, 15541, 37520, 90582, 218685, 527953, 1274592, 3077138, 7428869, 17934877, 43298624, 104532126, 252362877, 609257881, 1470878640, 3551015162, 8572908965, 20696833093, 49966575152, 120629983398, 291226541949

Offset: 0

Author

Creighton Dement, Oct 03 2004

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Hermann Stamm-Wilbrandt, 4 interlaced bisections
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Crossrefs

Cf. A006451, A124124.

Programs

Magma

I:=[2,5,13]; [n le 3 select I[n] else 3*Self(n-1) - Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 03 2018

Maple

A:= LREtools[REtoproc](a(n) = 3*a(n-1) - a(n-2) - a(n-3), a(n), {a(0)=2, a(1)=5, a(2)=13}):
seq(A(n),n=0..100); # Robert Israel, Aug 26 2014

Mathematica

LinearRecurrence[{3, -1, -1}, {2, 5, 13}, 28] (* Hermann Stamm-Wilbrandt, Aug 26 2014 *)
CoefficientList[Series[(2-x)/((1-x)*(1-2*x-x^2)), {x,0,50}], x] (* G. C. Greubel, Feb 03 2018 *)

PARI

Vec((2-x)/((1-x)*(1-2*x-x^2)) + O(x^50)) \\ Colin Barker, Mar 16 2016

Formula

a(n) = 3*a(n-1) - a(n-2) - a(n-3) with a(0)=2, a(1)=5, a(2)=13. - Hermann Stamm-Wilbrandt, Aug 26 2014

G.f.: (2-x)/((1-x)*(1-2*x-x^2)). - Robert Israel, Aug 26 2014

a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>5. - Hermann Stamm-Wilbrandt, Aug 27 2014

a(2*n-1) = A006451(2*n), for n>0. - Hermann Stamm-Wilbrandt, Aug 27 2014

a(2*n) = A124124(2*n+2). - Hermann Stamm-Wilbrandt, Aug 27 2014

a(n) = (-2+(5-3*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(5+3*sqrt(2)))/4. - Colin Barker, Mar 16 2016

Extensions

Formula supplied by Thomas Baruchel, Oct 03 2004

More terms from Emeric Deutsch, Nov 17 2004