A129905 Expansion of g.f.: (1-x)*(1+2*x)/((1+x)*(1-3*x+x^2)).

1, 3, 6, 17, 43, 114, 297, 779, 2038, 5337, 13971, 36578, 95761, 250707, 656358, 1718369, 4498747, 11777874, 30834873, 80726747, 211345366, 553309353, 1448582691, 3792438722, 9928733473, 25993761699, 68052551622, 178163893169

Offset: 0

Author

Creighton Dement, Jun 04 2007

Keywords

• easy
• nonn

Comments

Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B] with A = + .5'i + .5'j + .5'k + 'ji' + .5e ; B = + .5i' + .5j' + .5k' + 'ij' + .5e (apart from initial term)

From Andrew Rupinski, Jan 31 2011: (Start)

Form the infinite recursive array R(i,j) as follows: R(1,j) = F(j), R(2,j) = L(j) and for i > 2, R(i,j) = R(i-1,j) + R(i-2,j) where F(j) is the j-th Fibonacci number and L(j) is the j-th Lucas number. Then for i > 0, R(i,i) = a(i-1):

1 1 2 3 5 8 13 ...

1 3 4 7 11 18 29 ...

2 4 6 10 16 26 42 ...

3 7 10 17 27 44 71 ...

5 11 16 27 43 70 113 ...

8 18 26 44 70 114 184 ...

13 29 42 71 113 184 297 ...

...

(End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A001906, A054486.

Programs

GAP

List([0..30], n -> (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2); # G. C. Greubel, Jan 07 2019

Magma

[(Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2: n in [0..30]]; // G. C. Greubel, Jan 07 2019

Mathematica

CoefficientList[ Series[(1+x-2x^2)/(1-2x-2x^2+x^3), {x, 0, 27}], x] (* Or *)
t[1, k_] := Fibonacci@ k; t[2, k_] := LucasL@ k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 2, k]; Table[ t[n, n], {n, 28}] (* Robert G. Wilson v *)

PARI

vector(30, n, n--; (fibonacci(n-2)^2 + fibonacci(n+2)^2 + fibonacci(2*n))/2) \\ G. C. Greubel, Jan 07 2019

SageMath

[(fibonacci(n-2)^2 + fibonacci(n+2)^2 + fibonacci(2*n))/2 for n in (0..30)] # G. C. Greubel, Jan 07 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).

a(n+2) - a(n) = A054486(n+1).

a(n) = ( (4-sqrt(5))*((1+sqrt(5))/2)^(2*n) + (4 + sqrt(5))*((1-sqrt(5))/2 )^(2*n) + 2*(-1)^n)/5.

a(n) = -2*(-1)^n/5-8*A001906(n)/5+7*A001906(n+1)/5. - R. J. Mathar, Nov 10 2009

a(n) = (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2. - Gary Detlefs Dec 20 2010