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 1-4 of 4 results.

A192914 Constant term in the reduction by (x^2 -> x + 1) of the polynomial C(n)*x^n, where C=A000285.

Original entry on oeis.org

1, 0, 5, 9, 28, 69, 185, 480, 1261, 3297, 8636, 22605, 59185, 154944, 405653, 1062009, 2780380, 7279125, 19057001, 49891872, 130618621, 341963985, 895273340, 2343856029, 6136294753, 16065028224, 42058789925, 110111341545, 288275234716, 754714362597
Offset: 0

Views

Author

Clark Kimberling, Jul 12 2011

Keywords

Comments

See A192872.

Links

Crossrefs

Cf. A192232, A192744, A192872.

Programs

  • GAP
    F:=Fibonacci; List([0..30], n -> F(n+1)^2 +F(n)*F(n-3)); # G. C. Greubel, Jan 12 2019
  • Magma
    F:=Fibonacci; [F(n+1)^2+F(n)*F(n-3): n in [0..30]]; // Bruno Berselli, Feb 15 2017
  • Mathematica
    q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 4 x;
p[n_, x_] := p[n-1, x]*x + p[n-2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192914 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
LinearRecurrence[{2,2,-1}, {1,0,5}, 30] (* or *) With[{F:= Fibonacci}, Table[F[n+1]^2 +F[n]*F[n-3], {n, 0, 30}]] (* G. C. Greubel, Jan 12 2019 *)
  • PARI
    a(n) = round((2^(-1-n)*(3*(-1)^n*2^(2+n)+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5))))/5) \\ Colin Barker, Sep 29 2016
  • PARI
    Vec((1+3*x^2-2*x)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Sep 29 2016
  • PARI
    {f=fibonacci}; vector(30, n, n--; f(n+1)^2 +f(n)*f(n-3)) \\ G. C. Greubel, Jan 12 2019
  • Sage
    f=fibonacci; [f(n+1)^2 +f(n)*f(n-3) for n in (0..30)] # G. C. Greubel, Jan 12 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1 + 3*x^2 - 2*x)/((1 + x)*(x^2 - 3*x + 1)). - R. J. Mathar, May 08 2014
a(n) = (2^(-1-n)*(3*(-1)^n*2^(2+n) + (3 + sqrt(5))^n*(-1 + 3*sqrt(5)) - (3-sqrt(5))^n*(1 + 3*sqrt(5))))/5. - Colin Barker, Sep 29 2016
a(n) = F(n+1)^2 + F(n)*F(n-3). - Bruno Berselli, Feb 15 2017

A192879 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

Original entry on oeis.org

0, 1, 4, 10, 27, 70, 184, 481, 1260, 3298, 8635, 22606, 59184, 154945, 405652, 1062010, 2780379, 7279126, 19057000, 49891873, 130618620, 341963986, 895273339, 2343856030, 6136294752, 16065028225, 42058789924, 110111341546, 288275234715, 754714362598
Offset: 0

Views

Author

Clark Kimberling, Jul 11 2011

Keywords

Comments

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x+1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.
A192879 is also generated as the coefficient sequence of x in the reduction x^2->x+1 of the polynomial v(n,x) defined by v(0,x) = 2, v(1,x) = x+1, and v(n,x) = x*v(n-1,x) + 2*(x^2)*v(n-1,x) + 1, for n>0, v(n,x) = F(n)*x^(n-1) + L(n)*x^n, where F(n) = A000045(n) (Fibonacci numbers) and L(n) = A000032(n) (Lucas numbers).

Examples

			The first six polynomials and reductions:
  p(0,x) = 3 -> 3
  p(1,x) = x -> x
  p(2,x) = 4*x^2 -> 4+4*x
  p(3,x) = 5*x^3 -> 5+10*x
  p(4,x) = 9*x^4 -> 18+27*x
  p(5,x) = 14*x^5 -> 42+27*x
In general, p(n,x) = (A104449(n))*x^n -> A192878(n) + A192879(n)*x.

Links

Crossrefs

Cf. A192872, A192878.

Programs

  • GAP
    a:=[0,1,4];; for n in [4..40] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 07 2019
  • Magma
    I:=[0,1,4]; [n le 3 select I[n] else 2*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Jan 07 2019
  • Maple
    with(combinat); seq( fibonacci(2*n) + fibonacci(n)*fibonacci(n-1), n=0..40); # G. C. Greubel, Feb 13 2020
  • Mathematica
    (See A192878.)
LinearRecurrence[{2,2,-1}, {0,1,4}, 30] (* G. C. Greubel, Jan 07 2019 *)
a[n_] := a[n] = 2*a[n-1]+2*a[n - 2]-a[n-3]; a[0] = 0; a[1]=1; a[2]=4; Table[a[n], {n,0,40}] (* Rigoberto Florez, Feb 06 2020 *)
Table[Fibonacci[n]*Fibonacci[n-1]+Fibonacci[2n], {n,0,40}] (* Rigoberto Florez, Feb 06 2020 *)
  • PARI
    a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5))))/5) \\ Colin Barker, Sep 29 2016
  • PARI
    concat(0, Vec(x*(1+2*x)/((1+x)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Sep 29 2016
  • Sage
    (x*(1+2*x)/((1+x)*(1-3*x+x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 07 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), with a(0) = 0, a(1) = 1, a(2) = 4.
G.f.: x * (1+2*x) / ((1+x) * (1-3*x+x^2)). - Colin Barker, Jun 18 2012
a(n) = (2^(-1-n) * ((-1)^n*2^(1+n) + (3+sqrt(5))^n * (-1+3*sqrt(5)) - (3-sqrt(5))^n * (1+3*sqrt(5))))/5. - Colin Barker, Sep 29 2016
a(n) = F(n-1)*F(n) + F(2n), where F(n) is a Fibonacci number. - Rigoberto Florez, Feb 06 2020
E.g.f.: (exp(-x) + exp(3*x/2) * (3*sqrt(5)*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2)))/5. - Stefano Spezia, Feb 06 2020
a(n)*F(n) = the number of ways to tile a 3-arm starfish (with n-1 cells on each arm and one cell in the center) using squares and dominos. - Greg Dresden and Hasita Kanamarlapudi, Oct 02 2023

A192916 Constant term in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

Original entry on oeis.org

1, 0, 6, 11, 34, 84, 225, 584, 1534, 4011, 10506, 27500, 72001, 188496, 493494, 1291979, 3382450, 8855364, 23183649, 60695576, 158903086, 416013675, 1089137946, 2851400156, 7465062529, 19543787424, 51166299750, 133955111819, 350699035714, 918141995316
Offset: 0

Views

Author

Clark Kimberling, Jul 12 2011

Keywords

Comments

See A192872.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192872, A192917.

Programs

  • GAP
    F:=Fibonacci;; List([0..30], n-> F(2*n) +2*F(n)*F(n-1) +(-1)^n); # G. C. Greubel, Jul 28 2019
  • Magma
    F:=Fibonacci; [F(2*n) +2*F(n)*F(n-1) +(-1)^n: n in [0..30]]; // G. C. Greubel, Jul 28 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192914 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[2*n] +2*F[n]*F[n-1] +(-1)^n, {n,0,30}]] (* G. C. Greubel, Jul 28 2019 *)
  • PARI
    a(n) = round((2^(-n)*(7*(-2)^n+(3+sqrt(5))^n*(-1+2*sqrt(5))-(3-sqrt(5))^n*(1+2*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016
  • PARI
    Vec((1+4*x^2-2*x)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Oct 01 2016
  • PARI
    vector(30, n, n--; f=fibonacci; f(2*n) +2*f(n)*f(n-1) +(-1)^n) \\ G. C. Greubel, Jul 28 2019
  • Sage
    f=fibonacci; [f(2*n) +2*f(n)*f(n-1) +(-1)^n for n in (0..30)] # G. C. Greubel, Jul 28 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1 -2*x +4*x^2)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, May 08 2014
a(n) + a(n+1) = A054492(n). - R. J. Mathar, May 07 2014
a(n) = (2^(-n)*(7*(-2)^n+(3+sqrt(5))^n*(-1+2*sqrt(5))-(3-sqrt(5))^n*(1+2*sqrt(5))))/5. - Colin Barker, Oct 01 2016
a(n) = Fibonacci(2*n) + 2*Fibonacci(n)*Fibonacci(n-1) + (-1)^n. - G. C. Greubel, Jul 28 2019

A192917 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

Original entry on oeis.org

0, 5, 6, 22, 51, 140, 360, 949, 2478, 6494, 16995, 44500, 116496, 304997, 798486, 2090470, 5472915, 14328284, 37511928, 98207509, 257110590, 673124270, 1762262211, 4613662372, 12078724896, 31622512325, 82788812070, 216743923894, 567442959603, 1485584954924
Offset: 0

Views

Author

Clark Kimberling, Jul 12 2011

Keywords

Comments

See A192872.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192872, A192916.

Programs

  • GAP
    F:=Fibonacci;; List([0..30], n-> F(2*n+1) +2*F(n)^2 -(-1)^n); # G. C. Greubel, Jul 29 2019
  • Magma
    F:=Fibonacci; [F(2*n+1) +2*F(n)^2 -(-1)^n: n in [0..30]]; // G. C. Greubel, Jul 29 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192914 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[2*n+1] +2*F[n]^2 -(-1)^n, {n,0,30}]] (* G. C. Greubel, Jul 28 2019 *)
  • PARI
    a(n) = round((2^(-1-n)*(-9*(-1)^n*2^(1+n)-(3-sqrt(5))^n*(-9+sqrt(5))+(3+sqrt(5))^n*(9+sqrt(5))))/5) \\ Colin Barker, Oct 01 2016
  • PARI
    concat(0, Vec((-x*(-5+4*x))/((1+x)*(x^2-3*x+1)) + O(x^40))) \\ Colin Barker, Oct 01 2016
  • PARI
    vector(30, n, n--; f=fibonacci; f(2*n+1) +2*f(n)^2 -(-1)^n) \\ G. C. Greubel, Jul 29 2019
  • Sage
    f=fibonacci; [f(2*n+1) +2*f(n)^2 -(-1)^n for n in (0..30)] # G. C. Greubel, Jul 29 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: x*(5 -4*x)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, May 08 2014
a(n) = -3*a(n-1) +a(n-2) = 9*(-1)^(n+1). - R. J. Mathar, May 08 2014
a(n) = (2^(-1-n)*(-9*(-1)^n*2^(1+n)-(3-sqrt(5))^n*(-9+sqrt(5))+(3+sqrt(5))^n*(9+sqrt(5))))/5. - Colin Barker, Oct 01 2016
a(n) = Fibonacci(2*n+1) + 2*Fibonacci(n)^2 - (-1)^n. - G. C. Greubel, Jul 29 2019
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