A276421 Number of palindromic compositions of n into nonprime numbers.

1, 1, 1, 1, 2, 1, 3, 1, 5, 3, 7, 4, 11, 6, 16, 9, 25, 14, 38, 21, 59, 34, 89, 50, 137, 77, 208, 117, 319, 180, 486, 273, 744, 420, 1134, 639, 1735, 977, 2648, 1491, 4048, 2281, 6180, 3480, 9444, 5321, 14421, 8122, 22035, 12412, 33655, 18957, 51417, 28966

Offset: 0

Author

Emeric Deutsch, Sep 03 2016

Keywords

• nonn

Examples

			a(6) = 3 because we have [6], [1,4,1], and [1,1,1,1,1,1].
a(10) = 7 because we have [10], [1,8,1], [1,1,6,1,1], [1,4,4,1], [4,1,1,4], [1,1,1,4,1,1,1], and [1^{10}].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Crossrefs

Cf. A016116, A052284, A276420.

Programs

Maple

F:=sum(z^j,j=1..229)-(sum(z^ithprime(k),k=1..50)): g:=(1+F)/(1-subs(z = z^2, F)): gser:=series(g,z=0,53): seq(coeff(gser,z,n),n=0..50);
# second Maple program:
a:= proc(n) option remember; `if`(isprime(n), 0, 1)+
      add(`if`(isprime(j), 0, a(n-2*j)), j=1..n/2)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 03 2016

Mathematica

a[n_] := a[n] = If[PrimeQ[n], 0, 1] + Sum[If[PrimeQ[j], 0, a[n-2j]], {j, 1, n/2}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

Formula

G.f.: g(z)=(1+F(z))/(1-F(z^2)), where F(z)=Sum_{m nonprime} z^m = z + z^4 + z^6 + z^8 + z^9 + z^10 + z^12 + ... is the g.f. of A005171.