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A191797 a(n) = binomial(F(n), 2) where F(n) = A000045(n). 6
0, 0, 0, 1, 3, 10, 28, 78, 210, 561, 1485, 3916, 10296, 27028, 70876, 185745, 486591, 1274406, 3337236, 8738290, 22879230, 59901985, 156830905, 410597496, 1074972528, 2814337800, 7368069528, 19289917153, 50501756955, 132215475106, 346144864780, 906219437046 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
a(n) = +3*a(n-1) + 1*a(n-2) - 5*a(n-3) - 1*a(n-4) + 1*a(n-5).
G.f.: x^3/(1-3*x-x^2+5*x^3+x^4-x^5) = x^3/((1+x)*(1-x-x^2)*(1-3*x+x^2)).
a(n) + a(n+1) = A056014(n+1). - R. J. Mathar, Jun 24 2011
a(n) = (2*F(n)^2 - F(n+4) + 3*F(n+1))/4, F(n) = A000045(n). - Gary Detlefs, Jan 05 2013
a(n) = Sum_{k=1..n-2} A122931(k). - J. M. Bergot, Apr 05 2013
a(n) = A000217(A000071(n)). - Peter M. Chema, Mar 26 2017
a(n) = (2^(-1-n)*(-(-1)^n*2^(1+n) + sqrt(5)*(1-sqrt(5))^n + (3-sqrt(5))^n - sqrt(5)*(1+sqrt(5))^n + (3+sqrt(5))^n)) / 5. - Colin Barker, Mar 26 2017
EXAMPLE
a(7) = binomial(13,2) = 78.
MAPLE
with(combinat): seq(binomial(fibonacci(n), 2), n = 0 .. 30);
MATHEMATICA
Table[Binomial[Fibonacci[n], 2], {n, 0, 39}] (* Alonso del Arte, Apr 04 2013 *)
PROG
(PARI) a(n) = binomial(fibonacci(n), 2); \\ Michel Marcus, Sep 07 2015
(PARI) concat(vector(3), Vec(x^3 / ((1+x)*(1-x-x^2)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 26 2017
(Python)
from sympy import binomial, fibonacci
def a(n): return binomial(fibonacci(n), 2) # Indranil Ghosh, Mar 26 2017
CROSSREFS
Cf. A000045, A094825 (binomial transform), A000071, A000217.
Sequence in context: A320244 A128135 A350551 * A355356 A027252 A104574
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 21 2011
STATUS
approved

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Last modified April 15 02:01 EDT 2024. Contains 371667 sequences. (Running on oeis4.)