A155466 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.

28, 207, 1248, 7315, 42676, 248775, 1450008, 8451307, 49257868, 287095935, 1673317776, 9752810755, 56843546788, 331308470007, 1931007273288, 11254735169755, 65597403745276, 382329687301935, 2228380720066368

Offset: 0

Author

Klaus Brockhaus, Jan 30 2009

Keywords

• nonn
• easy

Comments

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Third trisection of A118120. Cf. A001652.

Cf. A155464, A155465, A156035 (decimal expansion of 3+2*sqrt(2)).

Programs

Magma

I:=[28, 207, 1248]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..50]]; // G. C. Greubel, Aug 21 2018

Mathematica

Table[(10*LucasL[2*n+3,2] + 3*LucasL[2*n+1, 2] -34)/4, {n, 0, 50}] (* or *) LinearRecurrence[{7,-7,1}, {28, 207, 1248}, 50] (* G. C. Greubel, Aug 21 2018 *)

PARI

{m=19; v=concat([28, 207, 1248], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

Formula

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 28, a(1) = 207.

a(n) = ((73+53*sqrt(2))*(3+2*sqrt(2))^n + (73-53*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.

G.f.: (28+11*x-5*x^2)/((1-x)*(1-6*x+x^2)).

a(n) = (10*A002203(2*n+3) + 3*A002203(2*n+1) - 34)/4. - G. C. Greubel, Aug 21 2018

Extensions

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009