A015467 q-Fibonacci numbers for q=9.

0, 1, 1, 10, 91, 7381, 604432, 436445101, 321656391613, 2087825044676482, 13848340772676227455, 808880048095782179467153, 48286987465947852695801396608, 25383561292811993463191359951919785, 13637696871632801620185917930189837576233

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Formula

a(n) = a(n-1) + 9^(n-2) * a(n-2).

Maple

q:=9; 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

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1] + a[n-2]*9^(n-2)},  a, {n, 20}] (* Vincenzo Librandi, Nov 09 2012 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 9], {n, 0, 20}] (* G. C. Greubel, Dec 16 2019 *)

Prog

(Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(9^(n-2)): n in [1..15]]; // Vincenzo Librandi, Nov 09 2012
(PARI) q=9; 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
(Sage)
def F(n, q): return sum( q_binomial(n-j-1, j, q)*q^(j^2) for j in (0..floor((n-1)/2)))
[F(n, 9) for n in (0..20)] # G. C. Greubel, Dec 16 2019
(GAP) q:=9;; 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

Crossrefs

q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), this sequence (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).

Sequence in context: A347260 A267833 A354380 * A144783 A365217 A227512

Adjacent sequences: A015464 A015465 A015466 * A015468 A015469 A015470

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved