A268344 a(n) = 11*a(n - 1) - 3*a(n - 2) for n>1, a(0)=0, a(1)=1.

0, 1, 11, 118, 1265, 13561, 145376, 1558453, 16706855, 179100046, 1919979941, 20582479213, 220647331520, 2365373209081, 25357163305331, 271832676731398, 2914087954129385, 31239469465229041, 334891900255131296, 3590092494410757133

Offset: 0

Author

Ilya Gutkovskiy, Feb 02 2016

Keywords

• nonn
• easy

Comments

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*b(n - 2), with n>1 and b(0)=0, b(1)=1, is x/(1 - k*x + m*x^2). This recurrence gives the closed form b(n) = (2^(-n)*((sqrt(k^2 - 4*m) + k)^n - (k - sqrt(k^2 - 4*m))^n))/sqrt(k^2 - 4*m).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..965
• Index entries for linear recurrences with constant coefficients, signature (11,-3).

Crossrefs

Cf. A190872, A190980.

Programs

Magma

I:=[0,1]; [n le 2 select I[n] else 11*Self(n-1) - 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

Mathematica

LinearRecurrence[{11, -3}, {0, 1}, 20] (* or *) Table[(((11 + Sqrt[109])/2)^n - ((11 - Sqrt[109])/2)^n)/Sqrt[109], {n, 0, 20}]

PARI

x='x+O('x^30); concat([0], Vec(x/(1-11*x+3*x^2))) \\ G. C. Greubel, Jan 14 2018

Formula

G.f.: x/(1 - 11*x + 3*x^2).

a(n) = ( ((11 + sqrt(109))/2)^n - ((11 - sqrt(109))/2)^n )/sqrt(109).