A282465 a(n) = 11*Fibonacci(n+3) + Fibonacci(n-8) with n>=0.

1, 46, 47, 93, 140, 233, 373, 606, 979, 1585, 2564, 4149, 6713, 10862, 17575, 28437, 46012, 74449, 120461, 194910, 315371, 510281, 825652, 1335933, 2161585, 3497518, 5659103, 9156621, 14815724, 23972345, 38788069, 62760414, 101548483, 164308897, 265857380, 430166277

Offset: 0

Author

Bruno Berselli, Feb 20 2017

Keywords

• nonn
• easy

Comments

Similar sequences with the formula h*Fibonacci(n+k) + Fibonacci(n+k-h):

h= 1, k=-1: A000045;

h= 2, k= 1: A013655;

h= 3, k=-2: A118658 = 2*A212804;

h= 4, k= 2: A022379 = 3*A000204;

h= 5, k= 1: A022113;

h= 6, k= 2: A022125;

h= 7, k= 3: A097657;

h= 8, k= 2: A022355 = 21*A000045;

h= 9, k= 3: 10, 32, 42, 74, 116, 190, 306, 496, 802, ... = 2*A022140;

h=10, k= 3: 33, 22, 55, 77, 132, 209, 341, 550, 891, ... = 11*A013655;

h=11, k= 3: this sequence.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..4769
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000045, A013655, A022113, A022125, A022355, A022379, A022379, A097657, A118658.

Cf. sequences with g.f. (1 + r*x)/(1 - x - x^2) for r = 2..31, respectively: A000204, A000285, A022095 - A022110, A022391 - A022402.

Programs

Magma

[11*Fibonacci(n+3)+Fibonacci(n-8): n in [0..40]];

Mathematica

Table[11 Fibonacci[n + 3] + Fibonacci[n - 8], {n, 0, 40}]
LinearRecurrence[{1,1},{1,46},36] (* or *) CoefficientList[Series[(1 + 45*x)/(1 - x - x^2) , {x,0,35}],x] (* Indranil Ghosh, Feb 22 2017 *)

PARI

a(n) = 11*fibonacci(n+3) + fibonacci(n-8) \\ Indranil Ghosh, Feb 23 2017

Formula

G.f.: (1 + 45*x)/(1 - x - x^2).

a(n) = a(n-1) + a(n-2).

a(n) = a(i)*Fibonacci(n-i+1) + a(i-1)*Fibonacci(n-i). Examples:

for i= 3, a(3)=93, a(2)= 47: a(n) = 93*Fibonacci(n-2) + 47*Fibonacci(n-3);

for i=-1, a(-1)=45, a(-2)=-44: a(n) = 45*Fibonacci(n+2) - 44*Fibonacci(n+1).

Other formulae:

a(n) = 44*Fibonacci(n) + Fibonacci(n+2),

a(n) = 45*Fibonacci(n) + Fibonacci(n+1),

a(n) = 46*Fibonacci(n) + Fibonacci(n-1),

a(n) = 47*Fibonacci(n) - Fibonacci(n-2).

a(n) = ((91 + sqrt(5))*((1 + sqrt(5))/2)^n - (91 - sqrt(5))*((1 - sqrt(5))/2)^n)/sqrt(20).