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A015465
q-Fibonacci numbers for q=8.
13
0, 1, 1, 9, 73, 4681, 303689, 153690697, 79763939913, 322392516534857, 1338539241447957065, 43272129632752387301961, 1437288838737538572434088521, 371706200490726725394268777423433, 98770108622737228265012391281001570889
OFFSET
0,4
LINKS
FORMULA
a(n) = a(n-1) + 8^(n-2) * a(n-2).
MAPLE
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
MATHEMATICA
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 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 8], {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)*(8^(n-2)): n in [1..15]]; // Vincenzo Librandi, Nov 09 2012
(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
(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, 8) for n in (0..20)] # G. C. Greubel, Dec 16 2019
(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
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), this sequence (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).
Sequence in context: A197676 A367977 A197534 * A144782 A218872 A075232
KEYWORD
nonn,easy
STATUS
approved