cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A015473 q-Fibonacci numbers for q=2, scale a(n-1).

Original entry on oeis.org

0, 1, 2, 9, 74, 1193, 38250, 2449193, 313534954, 80267397417, 41097221012458, 42083634584154409, 86187324725569242090, 353023324159566199755049, 2891967157702491033962603498, 47381990264820937260009495466281
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

a(1) = 1, a(n+1) = denominator of continued fraction [1;2,4,8,...,2^n]. - Amarnath Murthy, May 02 2001
The difference equation y(n, x, s) = q^(n-1)*x*y(n-1, x, s) + 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^2)*q^binomial(n-2*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 A015459. - G. C. Greubel, Dec 17 2019

Links

Crossrefs

Cf. A061377.
q-Fibonacci numbers: A000045 (q=1), this sequence (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), A015484 (q=11), A015485 (q=12).

Programs

  • GAP
    q:=2;; 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
  • Magma
    [0] cat [n le 2 select n else 2^(n-1)*Self(n-1) + Self(n-2): n in [1..16]]; // Vincenzo Librandi, Nov 09 2012
  • Maple
    q:=2; 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
  • Mathematica
    RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*2^(n-1)+a[n-2]},  a, {n, 30}] (* Vincenzo Librandi, Nov 09 2012 *)
Join[{0},Denominator[Table[FromContinuedFraction[2^Range[0,n]],{n,0,20}]]] (* Harvey P. Dale, Feb 09 2013 *)
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, 2], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)
  • PARI
    q=2; 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
  • Sage
    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,2) for n in (0..20)] # G. C. Greubel, Dec 17 2019

Formula

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

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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

a(n) = 5^(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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

A015485 q-Fibonacci numbers for q=12, scaling a(n-1).

Original entry on oeis.org

0, 1, 12, 1729, 2987724, 61953446593, 15416000025617100, 46031929420554204172993, 1649407256866864913519509578444, 709214929702322267749941478181800334017, 3659393259623103647557638545139154960967463412428
Offset: 0

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Author

Olivier Gérard

Keywords

Links

Crossrefs

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), A015484 (q=11), this sequence (q=12).
Differs from A015470.

Programs

  • GAP
    q:=12;; 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
  • Magma
    q:=12; 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
  • Maple
    q:=12; 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
  • Mathematica
    RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*12^(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, 12], {n, 0, 20}] (* G. C. Greubel, Dec 19 2019 *)
  • PARI
    q=12; 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
  • Sage
    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,12) for n in (0..20)] # G. C. Greubel, Dec 19 2019

Formula

a(n) = 12^(n-1)*a(n-1) + a(n-2).
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