A155465 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 7, a(1) = 88, a(2) = 555.

7, 88, 555, 3276, 19135, 111568, 650307, 3790308, 22091575, 128759176, 750463515, 4374021948, 25493668207, 148587987328, 866034255795, 5047617547476, 29419671029095, 171470408627128, 999402780733707, 5824946275775148

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

Second trisection of A118120. Cf. A001652.

Cf. A155464, A155466, A156035 (decimal expansion of 3+2*sqrt(2)).

Programs

Magma

I:=[7,88,555]; [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

LinearRecurrence[{7,-7,1},{7,88,555},30] (* Harvey P. Dale, Apr 29 2012 *)
Table[(3*LucasL[2*n+3,2] + 10*LucasL[2*n+1,2] - 34)/4, {n, 0, 50}] (* G. C. Greubel, Aug 21 2018 *)

PARI

{m=20; v=concat([7, 88, 555], 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) = 7, a(1) = 88.

a(n) = ((31+25*sqrt(2))*(3+2*sqrt(2))^n + (31-25*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.

G.f.: (7+39*x-12*x^2)/((1-x)*(1-6*x+x^2)).

a(n) = (3*A002203(2*n+3) + 10*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