A206918 Sum of binary palindromes p < 2^n.

0, 1, 4, 16, 40, 136, 328, 1096, 2632, 8776, 21064, 70216, 168520, 561736, 1348168, 4493896, 10785352, 35951176, 86282824, 287609416, 690262600, 2300875336, 5522100808, 18407002696, 44176806472, 147256021576, 353414451784, 1178048172616, 2827315614280

Offset: 0

Author

Hieronymus Fischer, Feb 18 2012

Keywords

• nonn
• base
• easy

Comments

Partial sums of A206917.

Partial sums of A052955(n) terms of A006995; for example: A052955(4)=7, the sum of the first 7 terms of A006995 is 0+1+3+5+7+15+17=40 which equals a(4).

Examples

			a(0) = 0, since p=0 is the only binary palindrome p<2^0;
a(3) = 16, since p=0, 1, 3, 5, 7 are the only binary palindromes < 2^3 and 0+1+3+5+7=16.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1, 8, -8).

Crossrefs

Cf. A006995, A206820.

See A016116 for the number of binary palindromes between 2^(n-1) and 2^n.

See A052995 for the number of binary palindromes < 2^n.

See A206917 for the sum of binary palindromes between 2^(n-1) and 2^n.

Formula

a(n) = sum(k=0..n, A206917(k)).

a(n) = sum(k=1..A052955(n), A006995(k)).

a(n) = sum(k=1..(1/2)*(5-(-1)^n)*2^floor(n/2)-1, A006995(k)).

a(n) = (8/7)*((3/4)*((4-(-1)^n)/(3+(-1)^n))*2^(3*floor(n/2))-1).

G.f.: x*(1+3*x+4*x^2)/((x-1)*(8*x^2-1)). - Alois P. Heinz, Feb 28 2012