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.

A369840 Number of compositions of 5*n into parts 2 and 5.

Original entry on oeis.org

1, 1, 2, 7, 23, 68, 194, 555, 1601, 4633, 13404, 38752, 112004, 323728, 935737, 2704817, 7818464, 22599701, 65325542, 188826693, 545813094, 1577700612, 4560424135, 13182138184, 38103641048, 110140512968, 318366757185, 920255312908, 2660044812499, 7688994894381
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A369803, A369842, A369843, A369844.
Cf. A099131, A369836, A369845.
Cf. A001687.

Programs

  • Mathematica
    LinearRecurrence[{5, -9, 10, -5, 1}, {1, 1, 2, 7, 23}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n+3*k, n-2*k));

Formula

a(n) = A001687(5*n+1).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,n-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x^2).
a(n) = A369843(n) - A369843(n-1). - R. J. Mathar, Feb 14 2024

A369803 Expansion of 1/(1 - x^2/(1-x)^5).

Original entry on oeis.org

1, 0, 1, 5, 16, 45, 126, 361, 1046, 3032, 8771, 25348, 73252, 211724, 612009, 1769080, 5113647, 14781237, 42725841, 123501151, 356986401, 1031887518, 2982723523, 8621714049, 24921502864, 72036871920, 208226244217, 601888555723, 1739789499591, 5028950081882
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 5*n-2 into parts 2 and 5.

Links

Crossrefs

Cf. A079675, A368475, A369804.
Cf. A369840, A369842, A369843, A369844.
Cf. A001687.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^5))
  • PARI
    a(n) = sum(k=0, n\2, binomial(n-1+3*k, n-2*k));

Formula

a(n) = A001687(5*n-1) for n > 0.
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+3*k,n-2*k).
a(n) = A369840(n)-A369840(n-1). - R. J. Mathar, Feb 14 2024

A369843 Number of compositions of 5*n-3 into parts 2 and 5.

Original entry on oeis.org

1, 2, 4, 11, 34, 102, 296, 851, 2452, 7085, 20489, 59241, 171245, 494973, 1430710, 4135527, 11953991, 34553692, 99879234, 288705927, 834519021, 2412219633, 6972643768, 20154781952, 58258423000, 168398935968, 486765693153, 1407021006061, 4067065818560
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A369803, A369840, A369842, A369844.
Cf. A001687.

Programs

  • Mathematica
    LinearRecurrence[{5, -9, 10, -5, 1}, {1, 2, 4, 11, 34}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n+3*k, n-1-2*k));

Formula

a(n) = A001687(5*n-2).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,n-1-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x^2).

A369844 Number of compositions of 5*n-4 into parts 2 and 5.

Original entry on oeis.org

0, 1, 4, 11, 29, 81, 235, 685, 1986, 5739, 16577, 47904, 138472, 400285, 1157071, 3344567, 9667590, 27944604, 80775310, 233485250, 674901117, 1950836005, 5638990526, 16299788815, 47115369056, 136189372297, 393662311506, 1137900943868, 3289160582291, 9507486039274
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A369803, A369840, A369842, A369843.
Cf. A001687.

Programs

  • Mathematica
    LinearRecurrence[{5, -9, 10, -5, 1}, {0, 1, 4, 11, 29}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n+1+3*k, n-2-2*k));

Formula

a(n) = A001687(5*n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+1+3*k,n-2-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^2*(1-x)/((1-x)^5 - x^2).
Showing 1-4 of 4 results.

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