cp's OEIS Frontend

A276421 Number of palindromic compositions of n into nonprime numbers.

%I A276421 #23 Jan 05 2025 19:51:41
%S A276421 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,
%T A276421 117,319,180,486,273,744,420,1134,639,1735,977,2648,1491,4048,2281,
%U A276421 6180,3480,9444,5321,14421,8122,22035,12412,33655,18957,51417,28966
%N A276421 Number of palindromic compositions of n into nonprime numbers.
%H A276421 Alois P. Heinz, <a href="/A276421/b276421.txt">Table of n, a(n) for n = 0..10000</a>
%H A276421 V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
%F A276421 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.
%e A276421 a(6) = 3 because we have [6], [1,4,1], and [1,1,1,1,1,1].
%e A276421 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}].
%p A276421 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);
%p A276421 # second Maple program:
%p A276421 a:= proc(n) option remember; `if`(isprime(n), 0, 1)+
%p A276421       add(`if`(isprime(j), 0, a(n-2*j)), j=1..n/2)
%p A276421     end:
%p A276421 seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 03 2016
%t A276421 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_ *)
%Y A276421 Cf. A016116, A052284, A276420.
%K A276421 nonn
%O A276421 0,5
%A A276421 _Emeric Deutsch_, Sep 03 2016