A015473 -id:A015473 - 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).

A015475 q-Fibonacci numbers for q=4, scaling a(n-1).

Original entry on oeis.org

0, 1, 4, 65, 4164, 1066049, 1091638340, 4471351706689, 73258627454030916, 4801077413298721817665, 1258573637505038759624004676, 1319710110525284599824799048959041, 5535265395417901871821058989004725507140

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..50

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), this sequence (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 A015461.

q:=4;; 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:=4; 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:=4; 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]*4^(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, 4], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)

q=4; 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,4) for n in (0..20)] # G. C. Greubel, Dec 17 2019

a(n) = 4^(n-1)*a(n-1) + a(n-2).

A015482 q-Fibonacci numbers for q=10, scaling a(n-1).

Original entry on oeis.org

0, 1, 10, 1001, 1001010, 10010101001, 1001010101101010, 1001010101111020101001, 10010101011111202020111101010, 1001010101111121203021211212020101001, 1001010101111121213031312223131303021111101010

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40

q-Fibonacci numbers: A280222 (q=-3), A280221 (q=-2), A280261 (q=-1), A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), this sequence (q=10), A015484 (q=11), A015485 (q=12).

Differs from A015468.

q:=10;; 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 19 2019

q:=10; 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 19 2019

q:=10; 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 19 2019

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*10^(n-1)+ a[n-2]}, a, {n, 40}] (* Vincenzo Librandi, Nov 10 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, 10], {n, 0, 20}] (* G. C. Greubel, Dec 19 2019 *)

q=10; 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 19 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,10) for n in (0..20)] # G. C. Greubel, Dec 19 2019

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

A015476 q-Fibonacci numbers for q=5, scaling a(n-1).

Original entry on oeis.org

0, 1, 5, 126, 15755, 9847001, 30771893880, 480810851722001, 37563347821553222005, 14673182743275038197425126, 28658560045496622327167502440755, 279868750444317625596488416061195472001, 13665466330288975220888581437110387323801268880

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..50

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), this sequence (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 A015462.

q:=5;; 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:=5; 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:=5; 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]*5^(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, 5], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)

q=5; 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,5) for n in (0..20)] # G. C. Greubel, Dec 17 2019

a(n) = 5^(n-1)*a(n-1) + a(n-2).

A015477 q-Fibonacci numbers for q=6, scaling a(n-1).

Original entry on oeis.org

0, 1, 6, 217, 46878, 60754105, 472423967358, 22041412681808953, 6170184900967295034366, 10363541282645125629123492409, 104440618529953822157016270251244030, 6315124821581059445960128077000914860421689

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..50

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), this sequence (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

Differs from A015463.

q:=6;; 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:=6; 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:=6; 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]*6^(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, 6], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)

q=6; 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,6) for n in (0..20)] # G. C. Greubel, Dec 17 2019

a(n) = 6^(n-1)*a(n-1) + a(n-2).

A015479 q-Fibonacci numbers for q=7, scaling a(n-1).

Original entry on oeis.org

0, 1, 7, 344, 117999, 283315943, 4761691172000, 560208204977943943, 461355545756912579822049, 2659622911535555605275705841192, 107325377740302038777488717075646201593, 30316762801210878398501692486189317906592712849

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), this sequence (q=7), A015480 (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

Differs from A015464.

q:=7;; 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 18 2019

q:=7; 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 18 2019

q:=7; 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 18 2019

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*7^(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, 7], {n, 0, 20}] (* G. C. Greubel, Dec 18 2019 *)

q=7; 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 18 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,7) for n in (0..20)] # G. C. Greubel, Dec 18 2019

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

A015480 q-Fibonacci numbers for q=8, scaling a(n-1).

Original entry on oeis.org

0, 1, 8, 513, 262664, 1075872257, 35254182380040, 9241672386909078017, 19381191729586400963887624, 325162439984693881306137776652801, 43642563925681986905603214423711358943752

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), this sequence (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

Differs from A015465.

q:=8;; 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 18 2019

q:=8; 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 18 2019

q:=8; 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 18 2019

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*8^(n-1)+a[n-2]}, a, {n, 20}] (* Vincenzo Librandi, Nov 10 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, 8], {n, 0, 20}] (* G. C. Greubel, Dec 18 2019 *)

q=8; 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 18 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,8) for n in (0..20)] # G. C. Greubel, Dec 18 2019

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

A015481 q-Fibonacci numbers for q=9, scaling a(n-1).

Original entry on oeis.org

0, 1, 9, 730, 532179, 3491627149, 206177092053480, 109570959981485091829, 524074504891889945272313781, 22559688995294431207802541840253930, 8740085742244887761578226267084082717085551

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), this sequence (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

Differs from A015467.

q:=9;; 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 18 2019

q:=9; 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 18 2019

q:=9; 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 18 2019

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==9^(n-1) a[n-1]+a[n-2]},a[n],{n,10}] (* Harvey P. Dale, Aug 24 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, 9], {n, 0, 20}] (* G. C. Greubel, Dec 18 2019 *)

q=9; 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 18 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,9) for n in (0..20)] # G. C. Greubel, Dec 18 2019

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

A015484 q-Fibonacci numbers for q=11, scaling a(n-1).

Original entry on oeis.org

0, 1, 11, 1332, 1772903, 25957074155, 4180412751509808, 7405856194503424044443, 144319186063701664852323850561, 30936099231445891001437365359291226684, 72945703751334713422596099393765798208419237205

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40

q-Fibonacci numbers: A280222 (q=-3), A280221 (q=-2), A280261 (q=-1), A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), A015482 (q=10), this sequence (q=11), A015485 (q=12).

Differs from A015469.

q:=11;; 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 19 2019

q:=11; 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 19 2019

q:=11; 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 19 2019

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*11^(n-1)+ a[n-2]}, a, {n, 20}] (* Vincenzo Librandi, Nov 10 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, 11], {n, 0, 20}] (* G. C. Greubel, Dec 19 2019 *)

q=11; 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 19 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,11) for n in (0..20)] # G. C. Greubel, Dec 19 2019

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

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