This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A206919 #32 Oct 26 2024 02:51:46
%S A206919 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,
%T A206919 78,78,105,105,105,105,136,136,169,169,169,169,169,169,169,169,169,
%U A206919 169,169,169,214,214,214,214,214,214,265,265,265,265,265,265,265,265
%N A206919 Sum of binary palindromes <= n.
%C A206919 Sum of binary palindromes A006995(k) <= n.
%C A206919 Different from A206920.
%F A206919 a(n) = Sum_{k=1..A206915(A206913(n))} A006995(k).
%F A206919 a(n) = A206920(A206915(A206913(n))).
%F A206919 Let p = A206913(n) > 3, m = floor(log_2(p)), then
%F A206919 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]
%e A206919 a(2)=1, since the only binary palindromes <= 1 are p=0 and p=1;
%e A206919 a(5)=9, since the sum of all binary palindromes <= 5 is 9 = 0 + 1 + 3 + 5.
%o A206919 (PARI) a(n) = sum(k=1, n, my(b=binary(k)); if (b==Vecrev(b), k)); \\ _Michel Marcus_, Sep 09 2018
%Y A206919 Cf. A006995, A206913, A206915.
%K A206919 nonn,base
%O A206919 0,4
%A A206919 _Hieronymus Fischer_, Feb 18 2012