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%I #14 Sep 08 2022 08:44:40
%S 0,1,10,1001,1001010,10010101001,1001010101101010,
%T 1001010101111020101001,10010101011111202020111101010,
%U 1001010101111121203021211212020101001,1001010101111121213031312223131303021111101010
%N q-Fibonacci numbers for q=10.
%H Vincenzo Librandi, <a href="/A015482/b015482.txt">Table of n, a(n) for n = 0..40</a>
%F a(n) = 10^(n-1) a(n-1) + a(n-2).
%p q:=10; 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 19 2019
%t RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*10^(n-1)+ a[n-2]}, a, {n, 40}] (* _Vincenzo Librandi_, Nov 10 2012 *)
%t 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, 10], {n, 0, 20}] (* _G. C. Greubel_, Dec 19 2019 *)
%o (PARI) q=10; 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 19 2019
%o (Magma) q:=10; I:=[0,1]; [n le 2 select I[n] else q^(n-2)*Self(n-1) + Self(n-2): n in [1..20]]; // _G. C. Greubel_, Dec 19 2019
%o (Sage)
%o 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)))
%o [F(n,10) for n in (0..20)] # _G. C. Greubel_, Dec 19 2019
%o (GAP) q:=10;; 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 19 2019
%Y q-Fibonacci numbers: A280222 (q=-3), A280221 (q=-2), A280261 (q=-1), A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), this sequence (q=10), A015484 (q=11), A015485 (q=12).
%Y Differs from A015468.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
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