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 1-9 of 9 results.

A107479 a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 12, 20, 31, 50, 79, 126, 200, 318, 506, 804, 1279, 2033, 3233, 5140, 8173, 12995, 20662, 32853, 52236, 83056, 132059, 209975, 333861, 530841, 844040, 1342028, 2133832, 3392804, 5394577, 8577406, 13638122, 21684687, 34478769
Offset: 0

Views

Author

Roger L. Bagula, May 27 2005

Keywords

Links

Crossrefs

Cf. A013984.
Cf. A107480, A109538, A109543, A109544, A114749, A125950, A130844, A143335, A147851.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1+x+x^2)*(x^2-x+1)/(1-x^2-x^3-x^4-x^5-x^6-x^7) )); // G. C. Greubel, Nov 03 2018
  • Mathematica
    LinearRecurrence[{0,1,1,1,1,1,1}, {0,1,1,2,3,5,8}, 40] (* Harvey P. Dale, Sep 26 2012 *)
CoefficientList[Series[x (1 + x) (1 + x + x^2) (x^2 - x + 1)/(1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 16 2014 *)
  • PARI
    concat(0, Vec(x*(1+x)*(1+x+x^2)*(x^2-x+1)/(1-x^2-x^3-x^4-x^5-x^6-x^7) + O(x^60))) \\ Michel Marcus, Oct 16 2014

Formula

Lim_{n->infinity} a(n)/a(n-1) = 1.5900053739...
G.f.: x*(1 + x)*(1 - x + x^2)*(1 + x + x^2)/(1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).

Extensions

Definition replaced by recurrence - The Associate Editors of the OEIS, Oct 02 2009
Spelling and formatting corrected, index link added - Charles R Greathouse IV, Jan 26 2011

A107480 a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-7).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 14, 25, 42, 71, 121, 207, 353, 601, 1025, 1748, 2980, 5080, 8661, 14767, 25176, 42922, 73178, 124762, 212707, 362644, 618273, 1054096, 1797131, 3063933, 5223708, 8905915, 15183719, 25886764, 44134416, 75244889, 128285220, 218713827
Offset: 0

Views

Author

Roger L. Bagula, May 27 2005

Keywords

Comments

Lim_{n->infinity} a(n)/a(n-1) = 1.70490277..., the real root of x^5 = x^4 + x^3 + 1.

Links

Crossrefs

Cf. A013984.
Cf. A107479, A109538, A109543, A109544, A114749, A125950, A130844, A143335, A147851.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1 +x^2-x^5)/((1+x^2)*(1-x-x^2-x^5)))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    LinearRecurrence[{1,0,1,1,1,0,1}, {0,1,1,2,3,5,8}, 50] (* Harvey P. Dale, May 21 2012 *)
  • PARI
    concat([0], Vec(x*(1 + x^2 - x^5) / ((1 + x^2)*(1 - x - x^2 - x^5)) + O(x^40))) \\ Colin Barker, Dec 17 2017

Formula

G.f.: x*(1 + x^2 - x^5) / ((1 + x^2)*(1 - x - x^2 - x^5)). - Colin Barker, Dec 17 2017

Extensions

Entry rewritten by Charles R Greathouse IV, Jan 26 2011

A109543 a(n) = a(n-1) + a(n-3) + a(n-5), with a(1..5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 7, 11, 17, 27, 43, 67, 105, 165, 259, 407, 639, 1003, 1575, 2473, 3883, 6097, 9573, 15031, 23601, 37057, 58185, 91359, 143447, 225233, 353649, 555281, 871873, 1368969, 2149483, 3375005, 5299255, 8320611, 13064585, 20513323, 32208939
Offset: 0

Views

Author

Roger L. Bagula, Jun 20 2005

Keywords

Links

Crossrefs

Cf. A107479, A107480, A109538, A109544, A114749, A125950, A130844, A143335, A147851.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x^3-x^4)/(1-x-x^3-x^5))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    LinearRecurrence[{1, 0, 1, 0, 1}, {1, 1, 1, 1, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2012 *)
  • PARI
    Vec((1 - x^3 - x^4) / (1 - x - x^3 - x^5) + O(x^50)) \\ Colin Barker, Dec 17 2017
  • PARI
    my(p=Mod('x,'x^5-'x^4-'x^2-1)); a(n) = vecsum(Vec(lift(p^n))); \\ Kevin Ryde, Jan 15 2021

Formula

G.f.: (1 - x^3 - x^4) / (1 - x - x^3 - x^5). - Colin Barker, Dec 17 2017

A147851 Expansion of 1/(1 - x^3 - x^4 - x^5 + x^8)^2.

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 3, 6, 7, 10, 15, 18, 27, 38, 50, 66, 92, 126, 165, 224, 300, 400, 536, 714, 948, 1258, 1676, 2218, 2932, 3882, 5128, 6768, 8924, 11760, 15479, 20366, 26780, 35174, 46182, 60602, 79473, 104158, 136445, 178654, 233797, 305834, 399881
Offset: 0

Views

Author

Roger L. Bagula, Nov 15 2008

Keywords

Links

Crossrefs

Cf. A107479, A107480, A109538, A109543, A109544, A114749, A125950, A130844, A143335.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x^3-x^4-x^5+x^8)^2)); // G. C. Greubel, Nov 03 2018
  • Mathematica
    CoefficientList[Series[1/(1 - x^3 - x^4 - x^5 + x^8)^2, {x, 0, 50}], x]
  • PARI
    x='x+O('x^50); Vec(1/(1-x^3-x^4-x^5+x^8)^2) \\ G. C. Greubel, Nov 03 2018

Formula

a(n) = 2*a(n-3) + 2*a(n-4) + 2*a(n-5) - a(n-6) - 2*a(n-7) - 5*a(n-8) - 2*a(n-9) - a(n-10) + 2*a(n-11) + 2*a(n-12) + 2*a(n-13) - a(n-16). - Franck Maminirina Ramaharo, Nov 02 2018

Extensions

Name clarified by Franck Maminirina Ramaharo, Nov 02 2018

A109538 a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 6, 11, 16, 26, 41, 66, 106, 166, 266, 421, 671, 1066, 1696, 2696, 4286, 6816, 10836, 17231, 27396, 43561, 69261, 110126, 175101, 278411, 442676, 703856, 1119136, 1779431, 2829306, 4498611, 7152816, 11373016, 18083156, 28752316
Offset: 0

Views

Author

Roger L. Bagula, Jun 20 2005

Keywords

Links

Crossrefs

Cf. A107479, A107480, A109543, A109544, A114749, A125950, A130844, A143335, A147851.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x-x^3-2*x^4-3*x^5-4*x^6)/(1-x^2-x^3-x^4-x^5-x^6-x^7))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    LinearRecurrence[{0,1,1,1,1,1,1},{1,1,1,1,1,1,1},50] (* Harvey P. Dale, Dec 29 2012 *)
  • PARI
    Vec((1 + x - x^3 - 2*x^4 - 3*x^5 - 4*x^6) / (1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7) + O(x^50)) \\ Colin Barker, Dec 17 2017

Formula

G.f.: (1 + x - x^3 - 2*x^4 - 3*x^5 - 4*x^6) / (1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7). - Colin Barker, Dec 17 2017

A114749 a(n) = a(n-1) + 4*a(n-2) + 6*a(n-3) + 4*a(n-4) + a(n-5).

Original entry on oeis.org

0, 1, 1, 2, 3, 21, 50, 161, 501, 1532, 4723, 14551, 44800, 137971, 424901, 1308512, 4029693, 12409831, 38217250, 117693681, 362448951, 1116196192, 3437432913, 10585903361, 32600301650, 100395746291, 309178300901, 952144142322, 2932218933633, 9030048595141
Offset: 0

Views

Author

Roger L. Bagula, Feb 18 2006

Keywords

Links

Crossrefs

Cf. A107479, A107480, A109538, A109543, A109544, A125950, A130844, A143335, A147851.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(9*x^3+3*x^2-1)/((x^2+x+1)*(x^3+3*x^2+2*x-1)))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    LinearRecurrence[{1,4,6,4,1},{0,1,1,2,3},30] (* Harvey P. Dale, Oct 13 2011 *)
  • PARI
    my(x='x+O('x^50)); concat([0], Vec(x*(9*x^3+3*x^2-1)/((x^2+x+1)*(x^3+ 3*x^2+2*x-1)))) \\ G. C. Greubel, Nov 03 2018

Formula

G.f.: x*(9*x^3 + 3*x^2 - 1)/((x^2 + x + 1)*(x^3 + 3*x^2 + 2*x - 1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

A109544 Expansion of (1 + x - x^3 - 2*x^4)/(1 - x^2 - x^3 - x^4 - x^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 4, 7, 10, 16, 25, 37, 58, 88, 136, 208, 319, 490, 751, 1153, 1768, 2713, 4162, 6385, 9796, 15028, 23056, 35371, 54265, 83251, 127720, 195943, 300607, 461179, 707521, 1085449, 1665250, 2554756, 3919399, 6012976, 9224854, 14152381
Offset: 0

Views

Author

Roger L. Bagula, Jun 20 2005

Keywords

Links

Crossrefs

Cf. A013982.
Cf. A107479, A107480, A109538, A109543, A114749, A125950, A130844, A143335, A147851.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x-x^3-2*x^4)/(1-x^2-x^3-x^4-x^5))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    LinearRecurrence[{0, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, 50]
CoefficientList[Series[(1+x-x^3-2x^4)/(1-x^2-x^3-x^4-x^5),{x,0,50}],x] (* Harvey P. Dale, Oct 24 2021 *)
  • Maxima
    makelist(ratcoef(taylor((1 + x - x^3 - 2*x^4)/(1 - x^2 - x^3 - x^4 - x^5), x, 0, n), x, n), n, 0, 50); /* Franck Maminirina Ramaharo, Oct 31 2018 */
  • PARI
    x='x+O('x^50); Vec((1+x-x^3-2*x^4)/(1-x^2-x^3-x^4-x^5)) \\ G. C. Greubel, Nov 03 2018

Formula

a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5).

A130844 a(n) = 2*a(n-1) + a(n-2) - a(n-3) + a(n-4), with a(1) = 0, a(2) = 3, a(3) = 5 and a(4) = 17.

Original entry on oeis.org

0, 3, 5, 17, 36, 87, 198, 464, 1075, 2503, 5815, 13522, 31431, 73072, 169868, 394899, 918025, 2134153, 4961300, 11533627, 26812426, 62331332, 144902763, 336858059, 783099975, 1820486578, 4232117835, 9838480332, 22871691896, 53170232867
Offset: 1

Views

Author

Roger L. Bagula, Jul 20 2007

Keywords

Links

Crossrefs

Cf. A107479, A107480, A109538, A109543, A109544, A114749, A125950, A143335, A147851.

Programs

  • Magma
    I:=[0,3,5,17]; [n le 4 select I[n] else 2*Self(n-1) +Self(n-2) -Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Nov 03 2018
  • Mathematica
    LinearRecurrence[{2,1,-1,1},{0,3,5,17},30] (* Harvey P. Dale, Dec 20 2014 *)
  • PARI
    m=30; v=concat([0,3,5,17], vector(m-4)); for(n=5, m, v[n] = 2*v[n-1] +v[n-2] -v[n-3] +v[n-4]); v \\ G. C. Greubel, Nov 03 2018

Formula

G.f.: x^2*(3 - x + 4*x^2)/((1 + x)*(1 - 3*x + 2*x^2 - x^3)). - Colin Barker, Nov 02 2012

Extensions

New name (after Colin Barker) by Franck Maminirina Ramaharo, Nov 02 2018
Edited by N. J. A. Sloane, Nov 03 2018

A143364 Triangle read by rows: T(n,k) is the number of {0-1-2}-trees with n edges and k protected vertices (0<=k<=n-1). A {0-1-2}-tree is an ordered tree in which the outdegree of every vertex is 0, 1, or 2. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 4, 6, 9, 1, 1, 4, 19, 12, 14, 1, 1, 8, 24, 53, 20, 20, 1, 1, 8, 62, 78, 116, 30, 27, 1, 1, 16, 80, 250, 190, 220, 42, 35, 1, 1, 16, 184, 382, 735, 390, 379, 56, 44, 1, 1, 32, 240, 1020, 1270, 1785, 714, 609, 72, 54, 1, 1, 32, 512, 1580, 3900, 3390, 3808
Offset: 1

Views

Author

Emeric Deutsch, Aug 20 2008

Keywords

Comments

Row sums are the Motzkin numbers (A001006).
T(n,0) is the sequence 1,1,2,2,4,4,8,8,16,16,... (A016116).
Sum(k*T(n,k),k=0..n-1) = A143335(n).

Examples

			Triangle starts:
1;
1,1;
2,1,1;
2,5,1,1;
4,6,9,1,1;
4,19,12,14,1,1;

Links

Crossrefs

Cf. A001006, A016116, A143335.

Programs

  • Maple
    g:=((1-t*z-2*z^2-sqrt((1-t*z)^2-4*z^2*(1-z^2+t*z^2)))*1/2)/(t*z^2): gser:= simplify(series(g,z=0,16)): for n to 12 do P[n]:=sort(coeff(gser,z,n)) end do: for n to 12 do seq(coeff(P[n],t,j),j=0..n-1) end do; # yields sequence in triangular form

Formula

G.f.: g, where g=g(t,z) satisfies tz^2*g^2-(1-tz-2z^2)g+z(1+z)=0.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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