|
%I #11 Sep 08 2022 08:44:40
%S 0,1,1,11,111,11111,1121111,1112221111,1122223221111,
%T 11123333333221111,112233445444433221111,11123445566666555433221111,
%U 1122345577889898877665433221111,1112345679012233433220988765433221111
%N q-Fibonacci numbers for q=10.
%H Vincenzo Librandi, <a href="/A015468/b015468.txt">Table of n, a(n) for n = 0..60</a>
%F a(n) = a(n-1) + 10^(n-2) a(n-2).
%p q:=10; 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 17 2019
%t RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*10^(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, 10], {n, 0, 20}] (* _G. C. Greubel_, Dec 17 2019 *)
%o (Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(10^(n-2)): n in [1..15]]; // _Vincenzo Librandi_, Nov 09 2012
%o (PARI) q=10; 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 17 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,10) for n in (0..20)] # _G. C. Greubel_, Dec 17 2019
%o (GAP) q:=10;; 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 17 2019
%Y q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4),
%Y A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), this sequence (q=10), A015469 (q=11), A015470 (q=12).
%K nonn,easy
%O 0,4
%A _Olivier Gérard_
|
