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.

User: Nicolas Bègue

Nicolas Bègue's wiki page.

Nicolas Bègue has authored 7 sequences.

A277252 a(n) = a(n-2) + a(n-3) + a(n-4) with a(0) = 0, a(1) = a(2) = 1, a(3) = 0.

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 3, 4, 7, 9, 14, 20, 30, 43, 64, 93, 137, 200, 294, 430, 631, 924, 1355, 1985, 2910, 4264, 6250, 9159, 13424, 19673, 28833, 42256, 61930, 90762, 133019, 194948, 285711, 418729, 613678, 899388, 1318118, 1931795, 2831184, 4149301, 6081097, 8912280, 13061582, 19142678, 28054959
Offset: 0

Views

Author

Nicolas Bègue, Oct 07 2016

Keywords

Comments

Limit a(n+1)/a(n) = 1.465571... (as in Narayana's cows sequence A000930).

Links

Crossrefs

Summed to A277253 equals A000930.

Programs

  • Maple
    a:= n-> (<<0|1|0|0>, <0|0|1|0>,
          <0|0|0|1>, <1|1|1|0>>^n. <<0,1,1,0>>)[1,1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 07 2016
  • Mathematica
    CoefficientList[Series[x*(1 + x - x^2)/(1 - x^2 - x^3 - x^4), {x, 0, 50}], x]
RecurrenceTable[{a[n] == a[n - 2] + a[n - 3] + a[n - 4], a[1] == 0,  a[2] == a[3] == 1, a[4] == 0}, a, {n, 52}]
LinearRecurrence[{0, 1, 1,1}, {0,1, 1, 0},52]
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(1+x-x^2)/(1-x^2-x^3-x^4))) \\ G. C. Greubel, May 02 2017

Formula

a(n) = a(n - 2) + a(n - 3) + a(n - 4).
G.f.: x*(1 + x - x^2)/(1 - x^2 - x^3 - x^4).
a(n) + A277253(n) = A000930(n).

Extensions

Corrected G.f. - G. C. Greubel, May 02 2017

A277253 a(n) = a(n-2) + a(n-3) + a(n-4) for n>3, a(0)=1, a(1)=a(2)=0, a(3)=2.

Original entry on oeis.org

1, 0, 0, 2, 1, 2, 3, 5, 6, 10, 14, 21, 30, 45, 65, 96, 140, 206, 301, 442, 647, 949, 1390, 2038, 2986, 4377, 6414, 9401, 13777, 20192, 29592, 43370, 63561, 93154, 136523, 200085, 293238, 429762, 629846, 923085, 1352846, 1982693, 2905777, 4258624, 6241316, 9147094, 13405717, 19647034, 28794127, 42199845
Offset: 0

Views

Author

Nicolas Bègue, Oct 07 2016

Keywords

Comments

a(n+1)/a(n) = 1.465571... like Narayana's cows sequence A000930.

Links

Crossrefs

Cf. A000930.

Programs

  • Magma
    I:=[1,0,0,2]; [n le 4 select I[n] else Self(n-2)+Self(n-3)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Nov 07 2016
  • Maple
    a:= n-> (<<0|1|0|0>, <0|0|1|0>,
          <0|0|0|1>, <1|1|1|0>>^n. <<1,0,0,2>>)[1,1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 07 2016
  • Mathematica
    RecurrenceTable[{a[n] == a[n - 2] + a[n - 3] + a[n - 4], a[1] ==1,  a[2] == a[3] == 0, a[4] == 2}, a, {n, 50}]
LinearRecurrence[{0, 1, 1, 1}, {1, 0, 0, 2}, 52]
CoefficientList[Series[(-1 + x^2 - x^3)/(-1 + x^2 + x^3 + x^4), {x, 0, 52}], x]
nxt[{a_,b_,c_,d_}]:={b,c,d,a+b+c}; NestList[nxt,{1,0,0,2},50][[;;,1]] (* Harvey P. Dale, Jun 10 2023 *)

Formula

G.f.: (1 - x^2 + x^3)/((1 + x)*(1 - x - x^3)).

A276658 Tribonacci-like sequence a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3, with a(0) = 1, a(1) = 2, a(2) = 0.

Original entry on oeis.org

1, 2, 0, 3, 5, 8, 16, 29, 53, 98, 180, 331, 609, 1120, 2060, 3789, 6969, 12818, 23576, 43363, 79757, 146696, 269816, 496269, 912781, 1678866, 3087916, 5679563, 10446345, 19213824, 35339732, 64999901, 119553457, 219893090
Offset: 0

Views

Author

Nicolas Bègue, Sep 11 2016

Keywords

Links

Crossrefs

Cf. A000073, A275778.

Programs

  • Mathematica
    LinearRecurrence[{1, 1, 1}, {1, 2, 0}, 35] (* or *)
RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] + a[n - 3], a[1] == 1, a[2] == 2, a[3] == 0}, a, {n, 35}] (* or *)
CoefficientList[Series[(-1 - x + 3 x^2)/(-1 + x + x^2 + x^3), {x, 0, 40}], x]
  • PARI
    a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;2;0])[1,1] \\ Charles R Greathouse IV, Sep 13 2016

Formula

G.f.: (3 x^2-x-1)/(x^3+x^2+x-1).
a(n) = A275778(n) - A000073(n).

A275778 Tribonacci-like sequence a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3, with a(0) = 1, a(1) = 2, a(2) = 1.

Original entry on oeis.org

1, 2, 1, 4, 7, 12, 23, 42, 77, 142, 261, 480, 883, 1624, 2987, 5494, 10105, 18586, 34185, 62876, 115647, 212708, 391231, 719586, 1323525, 2434342, 4477453, 8235320, 15147115, 27859888, 51242323, 94249326, 173351537, 318843186, 586444049
Offset: 0

Views

Author

Nicolas Bègue, Sep 10 2016

Keywords

Links

Crossrefs

Cf. A000073, A276658.

Programs

  • Mathematica
    CoefficientList[Series[(-1 - x + 2 x^2)/(-1 + x + x^2 + x^3), {x, 0, 35}], x]
RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] + a[n - 3], a[1] == 1, a[2] == 2, a[3] == 1}, a, {n, 35}]
LinearRecurrence[{1, 1, 1}, {1, 2, 1}, 35]
  • PARI
    a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;2;1])[1,1] \\ Charles R Greathouse IV, Sep 10 2016

Formula

G.f.: (2 x^2-x-1)/(x^3+x^2+x-1).
a(n) = A276658(n) + A000073(n).

A276477 a(n) = a(n-2) + a(n-3) for n >= 3, with a(0) = a(1) = 2, a(2) = 1.

Original entry on oeis.org

2, 2, 1, 4, 3, 5, 7, 8, 12, 15, 20, 27, 35, 47, 62, 82, 109, 144, 191, 253, 335, 444, 588, 779, 1032, 1367, 1811, 2399, 3178, 4210, 5577, 7388, 9787, 12965, 17175, 22752, 30140, 39927, 52892, 70067, 92819, 122959, 162886, 215778, 285845, 378664, 501623, 664509
Offset: 0

Views

Author

Nicolas Bègue, Sep 04 2016

Keywords

Comments

Padovan-like sequence linked to Perrin sequence.

Links

Crossrefs

Cf. A001608, A084338.

Programs

  • Magma
    I:=[2,2,1]; [n le 3 select I[n] else Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Sep 10 2016
  • Mathematica
    RecurrenceTable[{a[n] == a[n - 2] + a[n - 3], a[1] == a[2] == 2, a[3] == 1}, a, {n, 42}]
CoefficientList[Series[(x^2 - 2 x - 2)/(x^3 + x^2 - 1), {x, 0, 41}], x] (* Michael De Vlieger, Sep 06 2016 *)
LinearRecurrence[{0, 1, 1}, {2, 2, 1}, 60] (* Vincenzo Librandi, Sep 10 2016 *)
  • PARI
    x='x+O('x^99); Vec((x^2-2*x-2)/(x^3+x^2-1)) \\ Altug Alkan, Sep 10 2016

Formula

a(n) = A001608(n) + A084338(n-7).
G.f.: (x^2-2*x-2)/(x^3+x^2-1).

A276276 a(n) = a(n-2)+a(n-3) with a(1)=2 a(2)=1 a(3)=0.

Original entry on oeis.org

2, 1, 0, 3, 1, 3, 4, 4, 7, 8, 11, 15, 19, 26, 34, 45, 60, 79, 105, 139, 184, 244, 323, 428, 567, 751, 995, 1318, 1746, 2313, 3064, 4059, 5377, 7123, 9436, 12500, 16559, 21936, 29059, 38495, 50995, 67554, 89490
Offset: 1

Views

Author

Nicolas Bègue, Aug 26 2016

Keywords

Comments

Sequence is obtained from modulo 3 periodic sequence of Padovan numbers 1112201210010, it satisfies the same recurrence a(n) = a(n-2) + a(n-3).

Links

Crossrefs

Cf. A000931, A060006.

Programs

  • Mathematica
    RecurrenceTable[{a[n] == a[n - 2] + a[n - 3], a[1] == 2, a[2] == 1, a[3] == 0}, a, {n, 1, 43}] (* or *)
CoefficientList[Series[(2 x^2 - x - 2)/(x^3 + x^2 - 1), {x, 0, 42}], x] (* Michael De Vlieger, Aug 28 2016 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; 1,1,0]^(n-1)*[2;1;0])[1,1] \\ Charles R Greathouse IV, Aug 28 2016

Formula

G.f.: (2x^2-x-2)/(x^3+x^2-1).
a(n) = A000931(n) - A000931(n-9), for n>2.

Extensions

More terms from Charles R Greathouse IV, Aug 28 2016

A276275 Padovan like sequence: a(n) = a(n-2) + a(n-3) for n>3, a(1)=2, a(2)=2, a(3)=0.

Original entry on oeis.org

2, 2, 0, 4, 2, 4, 6, 6, 10, 12, 16, 22, 28, 38, 50, 66, 88, 116, 154, 204, 270, 358, 474, 628, 832, 1102, 1460, 1934, 2562, 3394, 4496, 5956, 7890, 10452, 13846, 18342, 24298, 32188, 42640, 56486, 74828, 99126, 131314, 173954, 230440, 305268, 404394, 535708
Offset: 1

Views

Author

Nicolas Bègue, Aug 26 2016

Keywords

Comments

Obtained from Padovan Spiral number (A134816) modulo 3 reduction periodic sequence 1112201210010, 111 112 122 220 ... fourth initialization values 220, it satisfies the same recurrence a(n) = a(n-2) + a(n-3).

Links

Crossrefs

Cf. A134816, A084338.

Programs

  • Mathematica
    RecurrenceTable[{a[n] == a[n - 2] + a[n - 3], a[1] == 2, a[2] == 2, a[3] == 0}, a, {n, 1, 48}] (* or *) CoefficientList[Series[2 x (1 + x - x^2)/(1 - x^2 - x^3), {x, 0, 47}], x] (* Michael De Vlieger, Sep 02 2016 *)
LinearRecurrence[{0,1,1},{2,2,0},60] (* Harvey P. Dale, Jan 27 2023 *)

Formula

G.f.: 2*x*(1 + x - x^2)/(1 - x^2 - x^3).
a(n) = A134816(n) + A007307(n-3) for n>=4.
a(n) = 2*A084338(n-3) for n>=4.

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