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 A015465 q-Fibonacci numbers for q=8. 12

%I

%S 0,1,1,9,73,4681,303689,153690697,79763939913,322392516534857,

%T 1338539241447957065,43272129632752387301961,

%U 1437288838737538572434088521,371706200490726725394268777423433,98770108622737228265012391281001570889

%N q-Fibonacci numbers for q=8.

%H Vincenzo Librandi, <a href="/A015465/b015465.txt">Table of n, a(n) for n = 0..60</a>

%F a(n) = a(n-1) + 8^(n-2) * a(n-2).

%p q:=8; 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]*8^(n-2)}, a, {n, 20}] (* _Vincenzo Librandi_, Nov 09 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, 8], {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)*(8^(n-2)): n in [1..15]]; // _Vincenzo Librandi_, Nov 09 2012

%o (PARI) q=8; 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,8) for n in (0..20)] # _G. C. Greubel_, Dec 16 2019

%o (GAP) q:=8;; 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), A015461 (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), this sequence (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).

%K nonn,easy

%O 0,4