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A129905 G.f. (2*x+1)*(1-x)/((x+1)*(x^2-3*x+1)). 1
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

Table of n, a(n) for n=0..27.

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) = -1/5*(3/2+1/2*sqrt(5))^n*sqrt(5)+4/5*(3/2+1/2*sqrt(5))^n+1/5*(3/2-1/2*sqrt(5))^n*sqrt(5)+4/5*(3/2-1/2*sqrt(5))^n+2/5*(-1)^n

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

a(n)= (Fibonacci(n-2)^2+Fibonacci(n+2)^2+Fibonacci(2n))/2,   [From Gary Detlefs Dec 20 2010]

MATHEMATICA

CoefficientList[ Series[(1 + x - 2 x^2)/(1 - 2 x - 2 x^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)

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 December 4 21:09 EST 2016. Contains 278755 sequences.