OFFSET
0,2
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).
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) = (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2. - Gary Detlefs Dec 20 2010
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 *)
PROG
(PARI) vector(30, n, 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
(SageMath) [(fibonacci(n-2)^2 + fibonacci(n+2)^2 + fibonacci(2*n))/2 for n in (0..30)] # G. C. Greubel, Jan 07 2019
(GAP) List([0..30], n -> (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2); # G. C. Greubel, Jan 07 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Creighton Dement, Jun 04 2007
STATUS
approved