A362057 Number of compositions of n that are anti-palindromic modulo 3.
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
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
- Jia Huang, Partially Palindromic Compositions, Journal of Integer Sequences, Vol. 26 (2023), Article 23.4.1.
- Index entries for linear recurrences with constant coefficients, signature (1,0,3,-1).
Programs
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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
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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).
Comments