A276420 Number of palindromic compositions of n into prime parts.

1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 5, 6, 8, 7, 12, 14, 16, 17, 26, 27, 36, 40, 55, 56, 81, 88, 118, 124, 177, 189, 257, 275, 384, 404, 564, 605, 833, 880, 1233, 1314, 1813, 1929, 2685, 2850, 3956, 4215, 5845, 6203, 8629, 9185, 12731, 13531, 18807, 19994

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

Cf. A016116, A023360.

Sequence in context: A214628 A355806 A032576 * A239494 A349388 A347663

Adjacent sequences: A276417 A276418 A276419 * A276421 A276422 A276423

Keyword

nonn

Author

Emeric Deutsch, Sep 02 2016

Status

approved