A015464 q-Fibonacci numbers for q=7.

0, 1, 1, 8, 57, 2801, 139658, 47216065, 16477840107, 38900937658402, 95030370064332109, 1569888180568718888123, 26845297334664927227358264, 3104208728255475471662060331653, 371576574614065326331102018605110717

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..70

Formula

a(n) = a(n-1) + 7^(n-2) * a(n-2).

Maple

q:=7; 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

Join[{0}, RecurrenceTable[{a[1]==1, a[2]==1, a[n]==a[n-1]+7^(n-2)a[n-2]}, a[n], {n, 20}]] (* Harvey P. Dale, May 14 2011 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*7^(n-2)},  a, {n, 25}] (* 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, 7], {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)*(7^(n-2)): n in [1..15]]; // Vincenzo Librandi, Nov 09 2012
(PARI) q=7; 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, 7) for n in (0..20)] # G. C. Greubel, Dec 16 2019
(GAP) q:=7;; 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), this sequence (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).

Sequence in context: A208596 A324205 A002402 * A144781 A026948 A111585

Adjacent sequences: A015461 A015462 A015463 * A015465 A015466 A015467

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved