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 1-6 of 6 results.

A113440 First row of A113439.

Original entry on oeis.org

1, 2, 8, 34, 146, 627, 2689, 11521, 49337, 211233, 904306, 3871305, 16572812, 70947073, 303719624, 1300203634, 5566087073, 23828058969, 102006385362, 436682772844, 1869410868456, 8002827727921, 34259590954322
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Links

Crossrefs

Cf. A113439.

Programs

  • Mathematica
    CoefficientList[Series[-(1 - 7*x + 18*x^2 - 20*x^3 + 8*x^4)/(-1 + 9*x - 28*x^2 + 38*x^3 - 20*x^4 + x^5), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
LinearRecurrence[{9,-28,38,-20,1},{1,2,8,34,146},30] (* Harvey P. Dale, Jul 20 2024 *)
  • PARI
    x='x+O('x^50); Vec(-(1-7*x+18*x^2-20*x^3+8*x^4)/(-1+9*x-28*x^2+38*x^3-20*x^4+x^5)) \\ G. C. Greubel, Mar 11 2017

Formula

a(n) = A113439(4*n).
a(n) = 9*a(n-1) - 28*a(n-2) + 38*a(n-3) - 20*a(n-4) + a(n-5).
G.f.: -(1-7*x+18*x^2-20*x^3+8*x^4)/(-1+9*x-28*x^2+38*x^3-20*x^4+x^5).

A113441 Second row of A113439.

Original entry on oeis.org

1, 3, 12, 50, 212, 905, 3872, 16576, 70968, 303832, 1300737, 5568473, 23838453, 102051167, 436874885, 1870233780, 8006350999, 34274673894, 146727674181, 628131735844, 2688991567300, 11511399994065, 49279563214531
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Links

Crossrefs

Cf. A113439.

Programs

  • Mathematica
    LinearRecurrence[{9,-28,38,-20,1},{1,3,12,50,212},30] (* Harvey P. Dale, Apr 06 2013 *)
CoefficientList[Series[-(1 - 6*x + 13*x^2 - 12*x^3 + 4*x^4)/(-1 + 9*x - 28*x^2 + 38*x^3 - 20*x^4 + x^5), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
  • PARI
    x='x+O('x^50); Vec(-(1-6*x+13*x^2-12*x^3+4*x^4)/(-1+9*x-28*x^2+38*x^3-20*x^4+x^5)) \\ G. C. Greubel, Mar 11 2017

Formula

a(n) = A113439(4*n+1).
a(n) = 9*a(n-1) - 28*a(n-2) + 38*a(n-3) - 20*a(n-4) + a(n-5).
G.f.: -(1-6*x+13*x^2-12*x^3+4*x^4)/(-1+9*x-28*x^2+38*x^3-20*x^4+x^5).

A113442 Third row of A113439.

Original entry on oeis.org

1, 4, 17, 72, 306, 1305, 5577, 23858, 102108, 437080, 1871065, 8009866, 34289712, 146792138, 628407994, 2690174883, 11516466985, 49301256865, 211055514017, 903515089697, 3867890010850, 16558188448465, 70884540117396
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Links

Crossrefs

Cf. A113439.

Programs

  • Mathematica
    CoefficientList[Series[-(1 - 5*x + 9*x^2 - 7*x^3 + 2*x^4)/(-1 + 9*x - 28*x^2 + 38*x^3 - 20*x^4 + x^5), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
  • PARI
    x='x+O('x^50); Vec(-(1-5*x+9*x^2-7*x^3+2*x^4)/(-1+9*x-28*x^2+38*x^3-20*x^4+x^5)) \\ G. C. Greubel, Mar 11 2017

Formula

a(n) = A113439(4*n+2).
a(n) = 9*a(n-1) - 28*a(n-2) + 38*a(n-3) - 20*a(n-4) + a(n-5).
G.f.: -(1-5*x+9*x^2-7*x^3+2*x^4)/(-1+9*x-28*x^2+38*x^3-20*x^4+x^5).

A113443 Fourth row of A113439.

Original entry on oeis.org

1, 5, 23, 101, 436, 1871, 8014, 34309, 146868, 628708, 2691401, 11521603, 49323052, 211148530, 903912916, 3869592721, 16565477544, 70915744968, 303585747088, 1299631069856, 5563637060865, 23817572559345, 101961496787621
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Links

Crossrefs

Cf. A113439.

Programs

  • Mathematica
    CoefficientList[Series[-(1 - 4*x + 6*x^2 - 4*x^3 + x^4)/(-1 + 9*x - 28*x^2 + 38*x^3 - 20*x^4 + x^5), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
  • PARI
    x='x+O('x^50); Vec(-(1-4*x+6*x^2-4*x^3+x^4)/(-1+9*x-28*x^2 +38*x^3 -20*x^4+x^5)) \\ G. C. Greubel, Mar 11 2017

Formula

a(n) = A113439(4*n+3).
a(n) = 9*a(n-1) - 28*a(n-2) + 38*a(n-3) - 20*a(n-4) + a(n-5).
G.f.: -(1-4*x+6*x^2-4*x^3+x^4)/(-1+9*x-28*x^2+38*x^3-20*x^4+x^5).

A113435 a(n) = a(n-1) + Sum_{k=0..n/3} a(n-3k) with a(0)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 11, 16, 26, 41, 62, 98, 154, 237, 371, 581, 901, 1406, 2197, 3418, 5329, 8317, 12956, 20196, 31501, 49096, 76532, 119338, 186029, 289997, 452141, 704861, 1098826, 1713111, 2670692, 4163483, 6490879, 10119152, 15775426
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Comments

If presented in three rows a(3n), a(3n+1) and a(3n+2) each term is the sum of the previous term in the sequence and the partial sum of its row.

Links

Crossrefs

Cf. A113439, A113444, A028495.
Partial sums of A176848.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x^3)/(1 - x - 2*x^3 + x^4), {x,0,50}], x] (* G. C. Greubel, Mar 10 2017 *)
LinearRecurrence[{1,0,2,-1},{1,1,1,2},40] (* Harvey P. Dale, Dec 17 2023 *)
  • PARI
    x='x+O(x^50); Vec((1 - x^3)/(1 - x - 2*x^3 + x^4)) \\ G. C. Greubel, Mar 10 2017

Formula

a(n) = a(n-1) + 2*a(n-3) - a(n-4) = 7*a(n-3) - 5*a(n-6) + 11*a(n-9) - a(n-12).
G.f.: (1-x^3)/(1-x-2*x^3+x^4).
G.f.: 1/(1-x) + x^3*Q(0)/(2-2*x) , where Q(k) = 1 + 1/(1 - x*(4*k+1 + 2*x^2 - x^3)/( x*(4*k+3 + 2*x^2 - x^3 ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 11 2013

A113444 a(n) = a(n-1) + Sum_{0

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 13, 18, 24, 31, 43, 60, 83, 113, 151, 206, 283, 389, 532, 721, 982, 1342, 1837, 2512, 3422, 4665, 6367, 8699, 11886, 16218, 22126, 30195, 41226, 56299, 76849, 104883, 143147, 195404, 266776, 364175, 497092, 678503, 926164
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Comments

If presented in five rows a(5n) a(5n+1).. a(5n+4) each term is the sum of the previous term in the sequence and the partial sum of its row.

Links

Crossrefs

Cf. A113435, A113439, A028495.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x^5)/(1 - x - 2*x^5 + x^6), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
  • PARI
    x='x+O('x^50); Vec((1-x^5)/(1-x-2*x^5+x^6)) \\ G. C. Greubel, Mar 11 2017

Formula

G.f.: (1-x^5)/(1-x-2*x^5+x^6).
a(n) = a(n-1) + 2*a(n-5) - a(n-6).
a(n) = 11*a(n-5) -45*a(n-10) +90*a(n-15) -90*a(n-20) +37*a(n-25)-a(n-30).
Showing 1-6 of 6 results.

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