OFFSET
0,7
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_{p prime} z^p = z^2 + z^3 + z^5 + z^7 + ... .
EXAMPLE
a(9) = 2 because we have [2,5,2] and [3,3,3].
a(12) = 5 because we have [5,2,5], [2,3,2,3,2], [3,2,2,2,3], [3,3,3,3], and [2,2,2,2,2,2].
MAPLE
F := sum(z^ithprime(j), j=1..90): F2:=sum(z^(2*ithprime(j)), j=1..90): g:= (1+F)/(1-F2): gser:=series(g, z=0, 55): seq(coeff(gser, z, n), n=0..50);
# second Maple program:
a:= proc(n) option remember; `if`(n=0 or isprime(n), 1, 0)+
add(`if`(isprime(j), a(n-2*j), 0), j=1..n/2)
end:
seq(a(n), n=0..60); # Alois P. Heinz, Sep 02 2016
MATHEMATICA
Table[Count[Flatten[Map[Permutations, Select[IntegerPartitions@ n, Times @@ Boole@ Map[PrimeQ, #] > 0 &]], 1], w_ /; Reverse@ w == w], {n, 0, 40}] (* Michael De Vlieger, Sep 02 2016 *)
a[n_] := a[n] = If[n == 0 || PrimeQ[n], 1, 0] +
Sum[If[PrimeQ[j], a[n - 2*j], 0], {j, 1, n/2}];
a /@ Range[0, 60] (* Jean-François Alcover, Jun 01 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 02 2016
STATUS
approved