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.

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A130668 Diagonal of A129819.

Original entry on oeis.org

0, 0, 1, -2, 5, -11, 23, -48, 102, -220, 476, -1024, 2184, -4624, 9744, -20480, 42976, -90048, 188352, -393216, 819328, -1704192, 3539200, -7340032, 15203840, -31456256, 65010688, -134217728, 276826112, -570429440, 1174409216
Offset: 0

Views

Author

Paul Curtz, Jun 27 2007

Keywords

Comments

This sequence is connected to A124072. To see this, change the sign of every negative term and consider the differences of every line. Hence for the second line, and following lines, the four terms form periodic sequences:
0 1 0 1 0
1 0 0 1 1
1 0 1 2 1
1 1 3 3 1
2 4 6 4 2
6 10 10 6 6
16 20 16 12 16
36 36 28 28 36
72 64 56 64 72
136 120 120 136 136
256 240 256 272 256.
The lines are connected as seen by the examples: (3rd line connected to 2nd, from right to left) 1+1=2, 1+0=1, 0+0=0, 0+1=1; (11th line connected to 10th) 136+136=272, 136+120=256, 120+120=240, 120+136=256.
The 4 columns are almost known (must the first line be suppressed?): A038503 (without the first 1), A000749 (without the first 0), A038505, A038504. Like the present sequence, every sequence of A124072 beginning with a negative number (-2, -11, ...) is a "twisted" sequence (see A129339 comments, A129961 and the present 4 columns). Periodic with period 2^n.
Inverse binomial transform of A129819. - R. J. Mathar, Feb 25 2009

Links

Programs

  • GAP
    a:=[-2,5,-11,23];; for n in [5..30] do a[n]:=-6*a[n-1]+-14*a[n-2] -16*a[n-3]-8*a[n-4]; od; Concatenation([0,0,1], a); # G. C. Greubel, Mar 24 2019
  • Magma
    I:=[-2,5,-11,23]; [0,0,1] cat [n le 4 select I[n] else -6*Self(n-1) - 14*Self(n-2)-16*Self(n-3)-8*Self(n-4): n in [1..30]]; // G. C. Greubel, Mar 24 2019
  • Mathematica
    gf = x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2); CoefficientList[Series[gf, {x, 0, 30}], x] (* Jean-François Alcover, Dec 16 2014, after R. J. Mathar *)
Join[{0, 0, 1}, LinearRecurrence[{-6,-14,-16,-8}, {-2,5,-11,23}, 30]] (* Jean-François Alcover, Feb 15 2016 *)
  • PARI
    my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1+x)*(1+3*x+4*x^2+3*x^3 )/((1+2*x +2*x^2)*(1+2*x)^2))) \\ G. C. Greubel, Mar 24 2019
  • Sage
    (x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2 )).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Mar 24 2019

Formula

From R. J. Mathar, Feb 25 2009: (Start)
G.f.: x^2*(1+x)*(1 + 3*x + 4*x^2 + 3*x^3)/((1 + 2*x + 2*x^2)*(1+2*x)^2).
a(n) = ((-1)^n*A001787(n+1) - 4*A108520(n) + 4*A122803(n))/32, n > 2. (End)
a(n) = -6*a(n-1) - 14*a(n-2) - 16*a(n-3) - 8*a(n-4) for n >= 7. - G. C. Greubel, Mar 24 2019

Extensions

Extended by R. J. Mathar, Feb 25 2009

A132353 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), starting with 1, 2, 6, 20.

Original entry on oeis.org

1, 2, 6, 20, 61, 183, 547, 1640, 4920, 14762, 44287, 132861, 398581, 1195742, 3587226, 10761680, 32285041, 96855123, 290565367, 871696100, 2615088300, 7845264902, 23535794707, 70607384121, 211822152361, 635466457082
Offset: 0

Views

Author

Paul Curtz, Nov 24 2007

Keywords

Comments

A132868(n) - a(n) = A128834(n) (discovered in 1995).

Links

Crossrefs

Cf. A129339.

Programs

  • Magma
    I:=[1,2,6,20]; [n le 4 select I[n] else 3*Self(n-1) - Self(n-3) + 3*Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018
  • Mathematica
    LinearRecurrence[{3, 0, -1, 3}, {1, 2, 6, 20}, 50] (* G. C. Greubel, Jan 15 2018 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,3d-b+3a}; NestList[nxt,{1,2,6,20},50][[;;,1]] (* Harvey P. Dale, Feb 17 2025 *)
  • PARI
    x='x+O('x^30); Vec((1-x+3*x^3)/((1-3*x)*(1+x)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018

Formula

Also a(n) - 3^(n+1) = hexaperiodic 1, -1, -3, -1, 1, 3; cf. A132951.
From R. J. Mathar, Apr 04 2008: (Start)
O.g.f.: (1-x+3*x^3)/((1-3*x)*(1+x)*(x^2-x+1)).
a(n) = -(-1)^n/12 + 3^(n+1)/4 + A057079(n+2)/3. (End)

Extensions

More terms from R. J. Mathar, Apr 04 2008

A132798 Period 6: repeat [0, 2, 1, 0, -2, -1].

Original entry on oeis.org

0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1
Offset: 0

Views

Author

Paul Curtz, Nov 21 2007

Keywords

Links

Crossrefs

Cf. A002264, A080425, A117373, A129339, A130151.

Programs

Formula

G.f.: x*(2+x)/((x+1)*(x^2-x+1)) = (1/3)*(4*x+1)/(x^2-x+1)-(1/3)/(x+1). - R. J. Mathar, Nov 28 2007
a(n) + a(n+1) = A117373(n+4). - R. J. Mathar, Jul 22 2009
a(n) = (-n mod 3) * (-1)^floor(n/3) = A080425(n) * (-1)^A002264(n) = A080425(n) * A130151(n). - Wesley Ivan Hurt, Jun 20 2014
From Wesley Ivan Hurt, Jun 21 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = sin(n*Pi/3) * (3*sqrt(3) + 2*sin(2*n*Pi/3))/3. (End)

A133453 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 2,4,13,40.

Original entry on oeis.org

2, 4, 13, 40, 122, 365, 1094, 3280, 9841, 29524, 88574, 265721, 797162, 2391484, 7174453, 21523360, 64570082, 193710245, 581130734, 1743392200, 5230176601, 15690529804, 47071589414, 141214768241, 423644304722, 1270932914164
Offset: 0

Views

Author

Paul Curtz, Nov 27 2007

Keywords

Links

Crossrefs

Cf. A129339.

Programs

  • Mathematica
    LinearRecurrence[{3,0,-1,3},{2,4,13,40},30] (* Harvey P. Dale, Sep 10 2018 *)

Formula

O.g.f: -(2 - 2*x + x^2 + 3*x^3)/((3*x-1)*(x+1)*(x^2-x+1)). - R. J. Mathar, Nov 30 2007
6*a(n) = 3^(n+2) +(-1)^n +2*A057079(n+2). - R. J. Mathar, Oct 03 2021
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