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%I #35 Mar 09 2024 16:26:51
%S 1,3,6,17,43,114,297,779,2038,5337,13971,36578,95761,250707,656358,
%T 1718369,4498747,11777874,30834873,80726747,211345366,553309353,
%U 1448582691,3792438722,9928733473,25993761699,68052551622,178163893169
%N Expansion of g.f.: (1-x)*(1+2*x)/((1+x)*(1-3*x+x^2)).
%C 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)
%C From _Andrew Rupinski_, Jan 31 2011: (Start)
%C 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):
%C 1 1 2 3 5 8 13 ...
%C 1 3 4 7 11 18 29 ...
%C 2 4 6 10 16 26 42 ...
%C 3 7 10 17 27 44 71 ...
%C 5 11 16 27 43 70 113 ...
%C 8 18 26 44 70 114 184 ...
%C 13 29 42 71 113 184 297 ...
%C ...
%C (End)
%H G. C. Greubel, <a href="/A129905/b129905.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F a(n+2) - a(n) = A054486(n+1).
%F a(n) = ( (4-sqrt(5))*((1+sqrt(5))/2)^(2*n) + (4 + sqrt(5))*((1-sqrt(5))/2 )^(2*n) + 2*(-1)^n)/5.
%F a(n) = -2*(-1)^n/5-8*A001906(n)/5+7*A001906(n+1)/5. - _R. J. Mathar_, Nov 10 2009
%F a(n) = (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2. - _Gary Detlefs_ Dec 20 2010
%t CoefficientList[ Series[(1+x-2x^2)/(1-2x-2x^2+x^3), {x, 0, 27}], x] (* Or *)
%t 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_ *)
%o (PARI) vector(30, n, n--; (fibonacci(n-2)^2 + fibonacci(n+2)^2 + fibonacci(2*n))/2) \\ _G. C. Greubel_, Jan 07 2019
%o (Magma) [(Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2: n in [0..30]]; // _G. C. Greubel_, Jan 07 2019
%o (SageMath) [(fibonacci(n-2)^2 + fibonacci(n+2)^2 + fibonacci(2*n))/2 for n in (0..30)] # _G. C. Greubel_, Jan 07 2019
%o (GAP) List([0..30], n -> (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2); # _G. C. Greubel_, Jan 07 2019
%Y Cf. A001906, A054486.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Jun 04 2007
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