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A015459 q-Fibonacci numbers for q=2, scaling a(n-2).

%I A015459 #69 Feb 16 2025 08:32:33
%S A015459 0,1,1,3,7,31,143,1135,10287,155567,2789039,82439343,2938415279,
%T A015459 171774189743,12207523172527,1419381685547183,201427441344229551,
%U A015459 46711726513354322095,13247460522448782176431,6135846878080826487812271
%N A015459 q-Fibonacci numbers for q=2, scaling a(n-2).
%C A015459 From _Gary W. Adamson_, Apr 17 2009: (Start)
%C A015459 Preface the series with another "1": (1, 1, 1, 3, 7, ..., a(n)).
%C A015459 Then a(n+2) = (1, 1, 1, 3, 7, ..., a(n)) dot (1, 2, 4, 8, ...).
%C A015459 Example: (143) = (1, 1, 1, 3, 7) dot (1, 2, 4, 8, 16) = (1 + 2 + 4 + 24 + 112).
%C A015459 Analogous procedures apply to other q-Fibonacci sequences for q(n). (End)
%C A015459 The difference equation y(n, x, s) = x*y(n-1, x, s) + q^(n-2)*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)*q^(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 A015473. - _G. C. Greubel_, Dec 17 2019
%H A015459 Vincenzo Librandi, <a href="/A015459/b015459.txt">Table of n, a(n) for n = 0..100</a>.
%H A015459 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Issues/12-4.pdf">Fibonacci notes 3: q-Fibonacci Numbers</a>, Fibonacci Quarterly 12 (1974), pp. 317-322.
%H A015459 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Analog.html">q-Analog</a>.
%F A015459 a(n) = a(n-1) + 2^(n-2)*a(n-2).
%F A015459 Associated constant: C_2 = lim_{n->oo} a(2*n)*a(2*n-2)/a(2*n-1)^2 = 1.225306147422043724739386133... (C_1=1). - _Benoit Cloitre_, Aug 30 2003 [Formula corrected by _Vaclav Kotesovec_, Dec 05 2017]
%F A015459 a(n)*a(n+3) - a(n)*a(n+2) - 2*a(n+1)*a(n+2) + 2*a(n+1)^2 = 0. - _Emanuele Munarini_, Dec 05 2017
%F A015459 From _Vaclav Kotesovec_, Dec 05 2017: (Start)
%F A015459 a(n) ~ c * 2^(n*(n-2)/4), where
%F A015459 c = 2.815179026313038425026160599838001991828247939843695... if n is even and
%F A015459 c = 3.024413799639405763259604599843170276573526808693115... if n is odd. (End)
%p A015459 q:=2; seq(add((product((1-q^(n-j-1-k))/(1-q^(k+1)), k=0..j-1))*q^(j^2), j = 0..floor((n-1)/2)), n = 0..20); # _G. C. Greubel_, Dec 16 2019
%t A015459 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*2^(n-2)}, a, {n, 20}] (* _Vincenzo Librandi_, Nov 08 2012 *)
%t A015459 F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
%t A015459 Table[F[n, 2], {n, 0, 20}] (* _G. C. Greubel_, Dec 16 2019 *)
%o A015459 (Magma) [0] cat [n le 2 select 1 else Self(n-1) + Self(n-2)*(2^(n-2)): n in [1..20]]; // _Vincenzo Librandi_, Nov 08 2012
%o A015459 (Sage)
%o A015459 from ore_algebra import *
%o A015459 R.<x> = QQ['x']
%o A015459 A.<Qx> = OreAlgebra(R, 'Qx', q=2)
%o A015459 print((Qx^2 - Qx - x).to_list([0,1], 10))  # _Ralf Stephan_, Apr 24 2014
%o A015459 (Sage)
%o A015459 def F(n,q): return sum( q_binomial(n-j-1, j, q)*q^(j^2) for j in (0..floor((n-1)/2)))
%o A015459 [F(n,2) for n in (0..20)] # _G. C. Greubel_, Dec 16 2019
%o A015459 (Python)
%o A015459 def a():
%o A015459     a, b, p = 0, 1, 1
%o A015459     while True:
%o A015459         yield a
%o A015459         p, a, b = p + p, b, b + p * a
%o A015459 A015463 = a()
%o A015459 print([next(A015463) for _ in range(20)]) # _Peter Luschny_, Dec 05 2017
%o A015459 (GAP) q:=2;; a:=[0,1];; for n in [3..30] do a[n]:=a[n-1]+q^(n-3)*a[n-2]; od; a; # _G. C. Greubel_, Dec 16 2019
%o A015459 (PARI) q=2; m=20; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=v[n-1]+q^(n-3)*v[n-2]); v \\ _G. C. Greubel_, Dec 16 2019
%Y A015459 q-Fibonacci numbers: A000045 (q=1), this sequence (q=2), A015460 (q=3), A015461 (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).
%Y A015459 Differs from A015473.
%K A015459 nonn,easy
%O A015459 0,4
%A A015459 _Olivier Gérard_