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

0,5

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.

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.

Example

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}].

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 *)

Crossrefs

Cf. A016116, A052284, A276420.

Sequence in context: A323184 A102614 A307149 * A330547 A338357 A338358

Adjacent sequences: A276418 A276419 A276420 * A276422 A276423 A276424

Keyword

nonn

Author

Emeric Deutsch, Sep 03 2016

Status

approved