A015461 - OEIS

%I #28 Sep 08 2022 08:44:40

%S 0,1,1,5,21,341,5717,354901,23771733,5838469717,1563742763605,

%T 1532083548256853,1641235215638133333,6427665390003549698645,

%U 27541785384957544314239573,431380864280640133787922528853

%N q-Fibonacci numbers for q=4.

%H Vincenzo Librandi, <a href="/A015461/b015461.txt">Table of n, a(n) for n = 0..80</a>

%F a(n) = a(n-1) + 4^(n-2)*a(n-2).

%F Associated constant: C_4 = lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1.094337777197221121533242886... . - _Benoit Cloitre_, Aug 30 2003

%F a(n)*a(n+3) - a(n)*a(n+2) - 4*a(n+1)*a(n+2) + 4*a(n+1)^2 = 0. - _Emanuele Munarini_, Dec 05 2017

%p q:=4; 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 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*4^(n-2)}, a, {n, 30}] (* _Vincenzo Librandi_, Nov 08 2012 *)

%t F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];

%t Table[F[n, 4], {n, 0, 20}] (* _G. C. Greubel_, Dec 16 2019 *)

%o (Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(4^(n-2)): n in [1..20]]; // _Vincenzo Librandi_, Nov 08 2012

%o (PARI) q=4; 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

%o (Sage)

%o 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 [F(n,4) for n in (0..20)] # _G. C. Greubel_, Dec 16 2019

%o (GAP) q:=4;; a:=[0,1];; for n in [3..20] do a[n]:=a[n-1]+q^(n-3)*a[n-2]; od; a; # _G. C. Greubel_, Dec 16 2019

%Y q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), this sequence (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).

%K nonn,easy

%O 0,4

%A _Olivier Gérard_