A291941 Number of Carlitz compositions of n that either have length 1, or have length greater than or equal to 2 and are palindromic if we exclude the first part.

1, 1, 3, 3, 5, 7, 9, 13, 19, 21, 31, 45, 53, 73, 101, 129, 171, 233, 295, 407, 533, 701, 921, 1251, 1605, 2175, 2837, 3797, 4945, 6681, 8637, 11679, 15165, 20403, 26525, 35777, 46381, 62589, 81253, 109503, 142187, 191755, 248775, 335579, 435561, 587233, 762305

Offset: 1

Author

Petros Hadjicostas, Sep 06 2017

Keywords

• nonn

Comments

Carlitz compositions are compositions where adjacent parts are distinct. They are enumerated in sequence A003242.

In Hadjicostas and Zhang (2017), compositions that either have length 1, or have length greater than or equal to 2 and are palindromic, if we exclude the first part, are called type II palindromic compositions, while the usual palindromic compositions are called type I palindromic compositions. (Type I palindromic compositions that are Carlitz are enumerated in sequence A239327.)

Since in a Carlitz composition adjacent parts are distinct, type II palindromic compositions of length > 1 that are Carlitz must have an even number of parts.

Examples

			For n=6, the a(6)=7 compositions that are type II palindromic and Carlitz are 6, 1+5, 5+1, 2+4, 4+2, 1+2+1+2, and 2+1+2+1. For n=7, the a(7)=9 compositions of this kind are 7, 1+6, 6+1, 2+5, 5+2, 3+4, 4+3, 3+1+2+1, and 2+1+3+1. (For example, the composition 1+6 becomes palindromic, i.e. 6, if we remove the first part. Similarly, the composition 2+1+3+1 becomes palindromic, i.e., 1+3+1, if we remove the first part. A composition of length one, such as 7, is considered palindromic of both types, I and II.)
		

References

• S. Heubach and T. Mansour, "Compositions of n with parts in a set," Congr. Numer. 168 (2004), 127-143.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• P. Hadjicostas, Cyclic, dihedral and symmetric Carlitz compositions of a positive integer, Journal of Integer Sequences, 20 (2017), Article 17.8.5.
• P. Hadjicostas and L. Zhang, Sommerville's symmetrical cyclic compositions of a positive integer with parts avoiding multiples of an integer, Fibonacci Quarterly 55 (2017), 54-73.

Crossrefs

Cf. A003242, A239327.

Programs

Maple

b:= proc(n, i) option remember; `if`(n<>i, 1, 0)+add(
     `if`(i=j, 0, b(n-2*j, `if`(j>n-2*j, 0, j))), j=1..(n-1)/2)
    end:
a:= n-> 1+add(b(n-j, j), j=1..n-1):
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 08 2017

Mathematica

b[n_, i_] := b[n, i] = If[n != i, 1, 0] + Sum[If[i == j, 0, b[n - 2*j, If[j > n - 2*j, 0, j]]], {j, 1, (n - 1)/2}];
a[n_] :=  1 + Sum[b[n - j, j], {j, 1, n - 1}];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jun 06 2018, after Alois P. Heinz *)

PARI

a(n) = { my(A=sum(j=1, n, x^(2*j)/(1+x^(2*j)) + O(x*x^n)), B=sum(j=1, n, x^j/(1+x^(2*j)) + O(x*x^n))); polcoeff(x/(1-x) + B^2/(1-A)-A, n) } \\ Andrew Howroyd, Oct 12 2017

Formula

G.f.: x/(1-x) + B(x)^2/(1-A(x))-A(x), where A(x) = Sum_{n>=1} x^(2*n)/(1+x^(2*n)) and B(x) = Sum_{n>=1} x^n/(1+x^(2*n)).

Extensions

a(26)-a(47) from Alois P. Heinz, Sep 07 2017