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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: Jia Huang

Jia Huang's wiki page.

Jia Huang has authored 2 sequences.

A362057 Number of compositions of n that are anti-palindromic modulo 3.

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 15, 27, 43, 81, 147, 249, 449, 809, 1409, 2507, 4485, 7903, 14015, 24963, 44187, 78329, 139203, 246801, 437601, 776881, 1378081, 2444083, 4337125, 7694487, 13648655, 24215947, 42962283, 76213761, 135212947, 239883849, 425562849, 754987929
Offset: 0

Views

Author

Jia Huang, Apr 06 2023

Keywords

Comments

A composition (c(1), c(2), ..., c(k)) is anti-palindromic modulo 3 if c(i) and c(k+1-i) are not congruent modulo 3 whenever 1 <= i <= k/2.

Examples

			There are a(5) = 7 compositions of n = 5 that are anti-palindromic modulo 3: 5, 32, 23, 311, 113, 221, 122. Note that 41 and 14 are anti-palindromic but not anti-palindromic modulo 3.

Links

Crossrefs

Cf. A000213 (anti-palindromic compositions), A362055.

Programs

  • PARI
    a(n) = {sum(i=0, n\3,  sum(d=0, (n-3*i)\3, sum(s=0, (n-3*i-3*d)\2, 2^i * binomial(i+s-1,s) * binomial(i+d-1,d) * sum(r=0, (n-3*i-3*d-2*s)\2, my(j=n-3*i-3*d-2*s-2*r); (-1)^r * binomial(i+j,j) * binomial(i,r) ))))} \\ Andrew Howroyd, Apr 10 2023
  • PARI
    Vec((1 - x^3)/(1 - x - 3*x^3 + x^4) + O(x^41)) \\ Andrew Howroyd, Apr 11 2023
  • PARI
    my(p=Mod('x, 'x^4-'x^3-3*'x+1)); a(n) = vecsum(Vec(lift(p^(n+1)))); \\ Kevin Ryde, Apr 12 2023

Formula

a(n) = Sum_{3*i + j + 2*r + 2*s + 3*d = n} (-1)^r * 2^i * binomial(i+j,j) * binomial(i,r) * binomial(i+s-1,s) * binomial(i+d-1,d).
G.f.: (1 - x^3)/(1 - x - 3*x^3 + x^4).

A362055 Number of compositions of n that are anti-palindromic modulo 2.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 11, 17, 33, 49, 89, 147, 243, 423, 691, 1185, 1985, 3329, 5649, 9443, 15971, 26855, 45179, 76209, 128097, 215921, 363433, 611827, 1030611, 1734599, 2921443, 4918593, 8281473, 13945473, 23478689, 39535299, 66566851, 112082503, 188725611
Offset: 0

Views

Author

Jia Huang, Apr 06 2023

Keywords

Comments

A composition (c(1), c(2), ..., c(k)) is anti-palindromic modulo 2 if c(i) and c(k+1-i) are not congruent modulo 2 whenever 1 <= i <= k/2.

Examples

			There are a(4) = 3 compositions of n = 4 that are anti-palindromic modulo 2: 4, 211, and 112. Although 31 and 13 are anti-palindromic, they are not anti-palindromic modulo 2.

Links

Crossrefs

Cf. A000213 (number of anti-palindromic compositions), A362057.

Programs

  • PARI
    a(n) = {sum(i=0, n\3,  sum(s=0, (n-3*i)\2, sum(d=0, (n-3*i)\2-s, 2^i * binomial(i+s-1,s) * binomial(i+d-1,d) * sum(j=0, n-3*i-2*d-2*s, my(r=n-3*i-2*d-2*s-j); (-1)^r * binomial(i+j,j) * binomial(i,r) ))))} \\ Andrew Howroyd, Apr 10 2023
  • PARI
    Vec((1 + x - x^2 - x^3)/(1 - 2*x^2 - 2*x^3 + x^4) + O(x^41)) \\ Andrew Howroyd, Apr 11 2023
  • PARI
    my(p=Mod('x, 'x^4-2*'x^2-2*'x+1)); a(n) = vecsum(Vec(lift(p^(n+1)))); \\ Kevin Ryde, Apr 12 2023

Formula

a(n) = Sum_{3*i + j + r + 2*s + 2*d = n} (-1)^r * 2^i * binomial(i+j,j) * binomial(i,r) * binomial(i+s-1,s) * binomial(i+d-1,d).
G.f.: (1+x-x^2-x^3)/(1-2*x^2-2*x^3+x^4).

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