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A129905 Expansion of g.f.: (1-x)*(1+2*x)/((1+x)*(1-3*x+x^2)). 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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  ...

..................................

- Andrew Rupinski, Jan. 31, 2011 -

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) = ( (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

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

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)

(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

(Sage) [(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

Cf. A001906, A054486.

Sequence in context: A212421 A238428 A232771 * A143363 A216878 A237670

Adjacent sequences:  A129902 A129903 A129904 * A129906 A129907 A129908

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Jun 04 2007

STATUS

approved

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Last modified September 21 04:59 EDT 2019. Contains 327253 sequences. (Running on oeis4.)