A015460 -id:A015460 - cp's OEIS frontend

A015459 q-Fibonacci numbers for q=2, scaling a(n-2).

0, 1, 1, 3, 7, 31, 143, 1135, 10287, 155567, 2789039, 82439343, 2938415279, 171774189743, 12207523172527, 1419381685547183, 201427441344229551, 46711726513354322095, 13247460522448782176431, 6135846878080826487812271

Offset: 0

Olivier Gérard

From Gary W. Adamson, Apr 17 2009: (Start)
Preface the series with another "1": (1, 1, 1, 3, 7, ..., a(n)).
Then a(n+2) = (1, 1, 1, 3, 7, ..., a(n)) dot (1, 2, 4, 8, ...).
Example: (143) = (1, 1, 1, 3, 7) dot (1, 2, 4, 8, 16) = (1 + 2 + 4 + 24 + 112).
Analogous procedures apply to other q-Fibonacci sequences for q(n). (End)
The difference equation y(n, x, s) = x*y(n-1, x, s) + q^(n-2)*s*y(n-2, x, s) yields a type of two variable q-Fibonacci polynomials in the form F(n, x, s, q) = Sum_{j=0..floor((n-1)/2)} q-binomial(n-j-1,j, q)*q^(j^2)*x^(n-2*j)*s^j. When x=s=1 these polynomials reduce to q-Fibonacci numbers. This family of q-Fibonacci numbers is different from that of the q-Fibonacci numbers defined in A015473. - G. C. Greubel, Dec 17 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..100.
L. Carlitz, Fibonacci notes 3: q-Fibonacci Numbers, Fibonacci Quarterly 12 (1974), pp. 317-322.
Eric Weisstein's World of Mathematics, q-Analog.

q-Fibonacci numbers: A000045 (q=1), this sequence (q=2), A015460 (q=3), A015461 (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).

Differs from A015473.

q:=2;; a:=[0,1];; for n in [3..30] do a[n]:=a[n-1]+q^(n-3)*a[n-2]; od; a; # G. C. Greubel, Dec 16 2019

[0] cat [n le 2 select 1 else Self(n-1) + Self(n-2)*(2^(n-2)): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

q:=2; 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

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*2^(n-2)}, a, {n, 20}] (* Vincenzo Librandi, Nov 08 2012 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 2], {n, 0, 20}] (* G. C. Greubel, Dec 16 2019 *)

q=2; 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

def a():
    a, b, p = 0, 1, 1
    while True:
        yield a
        p, a, b = p + p, b, b + p * a
A015463 = a()
print([next(A015463) for  in range(20)]) # _Peter Luschny, Dec 05 2017

from ore_algebra import *
R. = QQ['x']
A. = OreAlgebra(R, 'Qx', q=2)
print((Qx^2 - Qx - x).to_list([0,1], 10))  # Ralf Stephan, Apr 24 2014

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,2) for n in (0..20)] # G. C. Greubel, Dec 16 2019

a(n) = a(n-1) + 2^(n-2)*a(n-2).
Associated constant: C_2 = lim_{n->oo} a(2*n)*a(2*n-2)/a(2*n-1)^2 = 1.225306147422043724739386133... (C_1=1). - Benoit Cloitre, Aug 30 2003 [Formula corrected by Vaclav Kotesovec, Dec 05 2017]
a(n)*a(n+3) - a(n)*a(n+2) - 2*a(n+1)*a(n+2) + 2*a(n+1)^2 = 0. - Emanuele Munarini, Dec 05 2017
From Vaclav Kotesovec, Dec 05 2017: (Start)
a(n) ~ c * 2^(n*(n-2)/4), where
c = 2.815179026313038425026160599838001991828247939843695... if n is even and
c = 3.024413799639405763259604599843170276573526808693115... if n is odd. (End)

A015474 q-Fibonacci numbers for q=3, scale a(n-1).

Original entry on oeis.org

0, 1, 3, 28, 759, 61507, 14946960, 10896395347, 23830431570849, 156351472432735636, 3077466055723967094237, 181721293280796005380336249, 32191381943890636020834392595840, 17107820211824904790829046440906141689

Offset: 0

Olivier Gérard

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

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), this sequence (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

Differs from A015460.

q:=3;; a:=[0,1];; for n in [3..20] do a[n]:=q^(n-2)*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 17 2019

q:=3; I:=[0,1]; [n le 2 select I[n] else q^(n-2)*Self(n-1) + Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 17 2019

q:=3; seq(add((product((1-q^(2*(n-j-1-k)))/(1-q^(2*k+2)), k=0..j-1))* q^binomial(n-2*j,2), j = 0..floor((n-1)/2)), n = 0..20); # G. C. Greubel, Dec 17 2019

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*3^(n-1) + a[n-2]},  a, {n, 20}] (* Vincenzo Librandi, Nov 09 2012 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q^2]*q^Binomial[n-2*j,2], {j, 0, Floor[(n-1)/2]}]; Table[F[n, 3], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)

q=3; m=20; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=q^(n-2)*v[n-1]+v[n-2]); v \\ G. C. Greubel, Dec 17 2019

def F(n,q): return sum( q_binomial(n-j-1, j, q^2)*q^binomial(n-2*j,2) for j in (0..floor((n-1)/2)))
[F(n,3) for n in (0..20)] # G. C. Greubel, Dec 17 2019

a(n) = 3^(n-1)*a(n-1) + a(n-2).

A015463 q-Fibonacci numbers for q=6, scaling a(n-2).

Original entry on oeis.org

0, 1, 1, 7, 43, 1555, 57283, 12148963, 2684744611, 3403616850979, 4512743621400355, 34305128668265064739, 272902655183139496957219, 12446072589202949254455565603, 594062125322746104949654522449187, 162554939850629908283324416663519980835

Offset: 0

Olivier Gérard

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

q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4), A015462 (q=5), this sequence (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).

q:=6;; 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

[0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(6^(n-2)): n in [1..20]]; // Vincenzo Librandi, Nov 09 2012

q:=6; 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

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+6^(n-2) a[n-2]},a,{n,20}] (* Harvey P. Dale, Nov 11 2011 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 6], {n, 0, 20}] (* G. C. Greubel, Dec 16 2019 *)

q=6; 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

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,6) for n in (0..20)] # G. C. Greubel, Dec 16 2019

a(n) = a(n-1) + 6^(n-2)*a(n-2).
Associated constant: C_6 = lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1.046607628427088904183396615... . - Benoit Cloitre, Aug 30 2003
a(n)*a(n+3) - a(n)*a(n+2) - 6*a(n+1)*a(n+2) + 6*a(n+1)^2 = 0. - Emanuele Munarini, Dec 05 2017

A015468 q-Fibonacci numbers for q=10, scaling a(n-2).

Original entry on oeis.org

0, 1, 1, 11, 111, 11111, 1121111, 1112221111, 1122223221111, 11123333333221111, 112233445444433221111, 11123445566666555433221111, 1122345577889898877665433221111, 1112345679012233433220988765433221111

Offset: 0

Olivier Gérard

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

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), A015467 (q=9), this sequence (q=10), A015469 (q=11), A015470 (q=12).

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

[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

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

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 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 10], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)

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

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,10) for n in (0..20)] # G. C. Greubel, Dec 17 2019

a(n) = a(n-1) + 10^(n-2) a(n-2).

A015461 q-Fibonacci numbers for q=4, scaling a(n-2).

Original entry on oeis.org

0, 1, 1, 5, 21, 341, 5717, 354901, 23771733, 5838469717, 1563742763605, 1532083548256853, 1641235215638133333, 6427665390003549698645, 27541785384957544314239573, 431380864280640133787922528853

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..80

q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), this sequence (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).

q:=4;; 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

[0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(4^(n-2)): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

q:=4; 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

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*4^(n-2)}, a, {n, 30}] (* Vincenzo Librandi, Nov 08 2012 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 4], {n, 0, 20}] (* G. C. Greubel, Dec 16 2019 *)

q=4; 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

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,4) for n in (0..20)] # G. C. Greubel, Dec 16 2019

a(n) = a(n-1) + 4^(n-2)*a(n-2).
Associated constant: C_4 = lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1.094337777197221121533242886... . - Benoit Cloitre, Aug 30 2003
a(n)*a(n+3) - a(n)*a(n+2) - 4*a(n+1)*a(n+2) + 4*a(n+1)^2 = 0. - Emanuele Munarini, Dec 05 2017

A015462 q-Fibonacci numbers for q=5, scaling a(n-2).

Original entry on oeis.org

0, 1, 1, 6, 31, 781, 20156, 2460781, 317398281, 192565913906, 124176269429531, 376229476867085781, 1213035110624630757656, 18371792960261297531148281, 296169521847801754865890523281, 22426801247965814514582357345601406

Offset: 0

Olivier Gérard

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

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).

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

[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

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

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]}];
Table[F[n, 5], {n, 0, 20}] (* G. C. Greubel, Dec 16 2019 *)

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

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,5) for n in (0..20)] # G. C. Greubel, Dec 16 2019

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

A015464 q-Fibonacci numbers for q=7, scaling a(n-2).

Original entry on oeis.org

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

Offset: 0

Olivier Gérard

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

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).

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

[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

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

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 *)

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

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

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

A015465 q-Fibonacci numbers for q=8, scaling a(n-2).

Original entry on oeis.org

0, 1, 1, 9, 73, 4681, 303689, 153690697, 79763939913, 322392516534857, 1338539241447957065, 43272129632752387301961, 1437288838737538572434088521, 371706200490726725394268777423433, 98770108622737228265012391281001570889

Offset: 0

Olivier Gérard

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

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).

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

[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

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

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 *)

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

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

a(n) = a(n-1) + 8^(n-2) * a(n-2).

A015467 q-Fibonacci numbers for q=9, scaling a(n-2).

Original entry on oeis.org

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

Offset: 0

Olivier Gérard

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

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).

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

[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

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

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 *)

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

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

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

A015469 q-Fibonacci numbers for q=11, scaling a(n-2).

Original entry on oeis.org

0, 1, 1, 12, 133, 16105, 1963358, 2595689713, 3480804151551, 50586130104323474, 746191869036731097905, 119280194867984161366496439, 19354414621214347335584253057344, 34032051023004810891710239239325511573

Offset: 0

Olivier Gérard

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

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), A015467 (q=9), A015468 (q=10), this sequence (q=11), A015470 (q=12).

q:=11;; 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

[0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(11^(n-2)): n in [1..15]]; // Vincenzo Librandi, Nov 09 2012

q:=11; 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

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*11^(n-2)},  a, {n, 61}] (* 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, 11], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)

q=11; 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

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,11) for n in (0..20)] # G. C. Greubel, Dec 17 2019

a(n) = a(n-1) + 11^(n-2)*a(n-2).

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