A239632 Number of parts in all palindromic compositions of n.

0, 1, 3, 4, 10, 12, 28, 32, 72, 80, 176, 192, 416, 448, 960, 1024, 2176, 2304, 4864, 5120, 10752, 11264, 23552, 24576, 51200, 53248, 110592, 114688, 237568, 245760, 507904, 524288, 1081344, 1114112, 2293760, 2359296, 4849664, 4980736, 10223616, 10485760, 21495808, 22020096, 45088768, 46137344

Offset: 0

Author

Geoffrey Critzer, Mar 22 2014

Keywords

• nonn
• easy

Examples

			a(5)=12 because we have: 5, 1+3+1, 2+1+2, 1+1+1+1+1 with a total of 12 parts.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Crossrefs

Cf. A051159.

Programs

Mathematica

nn=30; r=Solve[p==y/(1-x) - y + 1 + y^2*x^2/(1-x^2)*p, p]; CoefficientList[Series[D[p/.r,y]/.y->1, {x,0,nn}], x]
CoefficientList[Series[(x + 3 x^2 - 2 x^4)/(1 - 2 x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)

Formula

G.f.: (x + 3*x^2 - 2*x^4)/(1 - 2*x^2)^2.

a(n) = Sum_{k=1..n} A051159(n,k)*k.

a(n) = 4*a(n-2) - 4*a(n-4) for n > 3. - Giovanni Resta, Mar 23 2014

a(2k) = (2k+1)*2^(k-1) for k>0, a(2k+1) = (2k+2)*2^(k-1) for k>=0. - Gregory L. Simay, Dec 05 2022

E.g.f.: (2*(1 + x)*cosh(sqrt(2)*x) + sqrt(2)*(1 + 2*x)*sinh(sqrt(2)*x) - 2)/4. - Stefano Spezia, Apr 25 2024