A206919 Sum of binary palindromes <= n.

0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 25, 25, 25, 25, 40, 40, 57, 57, 57, 57, 78, 78, 78, 78, 78, 78, 105, 105, 105, 105, 136, 136, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 214, 214, 214, 214, 214, 214, 265, 265, 265, 265, 265, 265, 265, 265

Offset: 0

Author

Hieronymus Fischer, Feb 18 2012

Keywords

• nonn
• base

Comments

Sum of binary palindromes A006995(k) <= n.

Different from A206920.

Examples

			a(2)=1, since the only binary palindromes <= 1 are p=0 and p=1;
a(5)=9, since the sum of all binary palindromes <= 5 is 9 = 0 + 1 + 3 + 5.
		

Crossrefs

Cf. A006995, A206913, A206915.

Programs

PARI

a(n) = sum(k=1, n, my(b=binary(k)); if (b==Vecrev(b), k)); \\ Michel Marcus, Sep 09 2018

Formula

a(n) = Sum_{k=1..A206915(A206913(n))} A006995(k).

a(n) = A206920(A206915(A206913(n))).

Let p = A206913(n) > 3, m = floor(log_2(p)), then

a(n) = (8/7)*((3/4)*(4-(-1)^m)/(3+(-1)^m)*2^(3*floor(m/2))-1) + (floor(p/2^floor(m/2)) mod 2)*p + 2^m + 1 + Sum_{k=1..floor(m/2)-1} (floor(p/2^k) mod 2)*(2^k+2^(m-k)+2^(m-floor(m/2)+1)*(4^(floor(m/2)-k-1)-1)+(2-(-1)^m)*2^floor(m/2)+2^(floor(m/2)-k)*(p-floor((p mod (2^(m-k+1)))/2^k)*2^k)). - [Corrected; missing factor to the sum term (2-(-1)^m) pasted by the author, Sep 08 2018]