%I #19 Jul 03 2023 20:10:13

%S 0,1,4,16,40,136,328,1096,2632,8776,21064,70216,168520,561736,1348168,

%T 4493896,10785352,35951176,86282824,287609416,690262600,2300875336,

%U 5522100808,18407002696,44176806472,147256021576,353414451784,1178048172616,2827315614280

%N Sum of binary palindromes p < 2^n.

%C Partial sums of A206917.

%C 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).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 8, -8).

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

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

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

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

%F G.f.: x*(1+3*x+4*x^2)/((x-1)*(8*x^2-1)). - _Alois P. Heinz_, Feb 28 2012

%e a(0) = 0, since p=0 is the only binary palindrome p<2^0;

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

%Y Cf. A006995, A206820.

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

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

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

%K nonn,base,easy

%O 0,3

%A _Hieronymus Fischer_, Feb 18 2012