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A015462
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q-Fibonacci numbers for q=5.
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13
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0, 1, 1, 6, 31, 781, 20156, 2460781, 317398281, 192565913906, 124176269429531, 376229476867085781, 1213035110624630757656, 18371792960261297531148281, 296169521847801754865890523281, 22426801247965814514582357345601406
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) = a(n-1) + 5^(n-2)*a(n-2).
Associated constant: C_5 = lim_{n->oo} a(n)*a(n-2)/a(n-1)^2 = 1.064478080430862119874641125... . - Benoit Cloitre, Aug 30 2003
a(n)*a(n+3) - a(n)*a(n+2) - 5*a(n+1)*a(n+2) + 5*a(n+1)^2 = 0. - Emanuele Munarini, Dec 05 2017
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MAPLE
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q:=5; 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
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MATHEMATICA
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RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*5^(n-2)}, a, {n, 20}] (* Vincenzo Librandi, Nov 09 2012 *)
nxt[{n_, a_, b_}]:={n+1, b, b+a*5^(n-1)}; NestList[nxt, {1, 0, 1}, 20][[All, 2]] (* Harvey P. Dale, Aug 19 2019 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
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PROG
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(Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(5^(n-2)): n in [1..20]]; // Vincenzo Librandi, Nov 09 2012
(PARI) q=5; 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)))
(GAP) q:=5;; 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
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CROSSREFS
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q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4), this sequence (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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