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A015473
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q-Fibonacci numbers for q=2.
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18
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0, 1, 2, 9, 74, 1193, 38250, 2449193, 313534954, 80267397417, 41097221012458, 42083634584154409, 86187324725569242090, 353023324159566199755049, 2891967157702491033962603498, 47381990264820937260009495466281
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OFFSET
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0,3
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COMMENTS
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a(1) = 1, a(n+1) = denominator of continued fraction [1;2,4,8,...,2^n]. - Amarnath Murthy, May 02 2001
The difference equation y(n, x, s) = q^(n-1)*x*y(n-1, x, s) + s*y(n-2, x, s) yields a type of two variable q-Fibonacci polynomials in the form F(n, x, s, q) = Sum_{j=0..floor((n-1)/2)} q-binomial(n-j-1,j, q^2)*q^binomial(n-2*j,2)* x^(n-2*j)*s^j. When x=s=1 these polynomials reduce to q-Fibonacci numbers. This family of q-Fibonacci numbers is different from that of the q-Fibonacci numbers defined in A015459. - G. C. Greubel, Dec 17 2019
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LINKS
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FORMULA
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a(n) = 2^(n-1)*a(n-1) + a(n-2).
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MAPLE
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q:=2; seq(add((product((1-q^(2*(n-j-1-k)))/(1-q^(2*k+2)), k=0..j-1))* q^binomial(n-2*j, 2), j = 0..floor((n-1)/2)), n = 0..20); # G. C. Greubel, Dec 17 2019
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MATHEMATICA
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RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*2^(n-1)+a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Nov 09 2012 *)
Join[{0}, Denominator[Table[FromContinuedFraction[2^Range[0, n]], {n, 0, 20}]]] (* Harvey P. Dale, Feb 09 2013 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q^2]*q^Binomial[n-2*j, 2], {j, 0, Floor[(n-1)/2] }]; Table[F[n, 2], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)
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PROG
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(Magma) [0] cat [n le 2 select n else 2^(n-1)*Self(n-1) + Self(n-2): n in [1..16]]; // Vincenzo Librandi, Nov 09 2012
(PARI) q=2; m=20; v=concat([0, 1], vector(m-2)); for(n=3, m, v[n]=q^(n-2)*v[n-1]+v[n-2]); v \\ G. C. Greubel, Dec 17 2019
(Sage)
def F(n, q): return sum( q_binomial(n-j-1, j, q^2)*q^binomial(n-2*j, 2) for j in (0..floor((n-1)/2)))
(GAP) q:=2;; a:=[0, 1];; for n in [3..20] do a[n]:=q^(n-2)*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 17 2019
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CROSSREFS
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q-Fibonacci numbers: A000045 (q=1), this sequence (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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