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A127804 a(2n) = 2^(2n), a(2n+1) = 2^(2n+1) + a(n).

%I A127804 #63 Jul 16 2025 14:57:22
%S A127804 1,3,4,11,16,36,64,139,256,528,1024,2084,4096,8256,16384,32907,65536,
%T A127804 131328,262144,524816,1048576,2098176,4194304,8390692,16777216,
%U A127804 33558528,67108864,134225984,268435456,536887296,1073741824,2147516555,4294967296,8590000128
%N A127804 a(2n) = 2^(2n), a(2n+1) = 2^(2n+1) + a(n).
%C A127804 From _Tilman Piesk_, Jun 30 2025: (Start)
%C A127804 The binary expansion of a(n), with bits least to most significant, is row n of A115361.
%C A127804 Number of 1's in binary expansion of a(n) is A001511(n+1).
%C A127804 Row sums of triangle A128807.
%C A127804 Row sums of triangle A127803.(End)
%H A127804 Tilman Piesk, <a href="/A127804/b127804.txt">Table of n, a(n) for n = 0..1023</a> (terms 0..61 from R. J. Mathar)
%H A127804 Tilman Piesk, Illustrated triangles with 32 rows:
%H A127804  <a href="https://commons.wikimedia.org/wiki/File:Sequence_A127804_from_necklace_triangle_(diagonals).svg">related to necklaces</a> (similar to A385665),
%H A127804  <a href="https://commons.wikimedia.org/wiki/File:Sequence_A127804_from_binary_triangle.svg">binary</a> (A115361),
%H A127804  <a href="https://commons.wikimedia.org/wiki/File:Sequence_A127804_from_signed_triangle.svg">signed</a> (A127803)
%H A127804 Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Serration_of_Boolean_functions#Serrator_permutation">Serration of Boolean functions</a> (Wikiversity)
%F A127804 Conjecture: a(n) = 1 + A187767(n+1). - _Andrew Howroyd_, Jun 02 2017
%F A127804 From _Tilman Piesk_, Jun 30 2025: (Start)
%F A127804 a(n) = Sum_{i=0..A001511(n+1)-1} 2^((n+1) / 2^i - 1)
%F A127804      = Sum_{i=0..A001511(n+1)-1} 2^floor(n / 2^i).
%F A127804 a(n) = A045654(n+1) / 2. (End)
%p A127804 A127804 := proc(n)
%p A127804     add( A127803(n,k),k=0..n) ;
%p A127804 end proc: # _R. J. Mathar_, Feb 12 2024
%p A127804 # second Maple program:
%p A127804 a:= proc(n) option remember;
%p A127804       2^n+`if`(n::odd, a((n-1)/2), 0)
%p A127804     end:
%p A127804 seq(a(n), n=0..33);  # _Alois P. Heinz_, Jul 01 2025
%t A127804 rows = 30;
%t A127804 A[n_, k_] := If[k <= n, If[n <= 2 k, 1/(2*2^n - 1), 0], 0];
%t A127804 T = Table[A[n, k], {n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
%t A127804 a[n_] := T[[n+1]] // Total;
%t A127804 Table[a[n], {n, 0, rows-1}] (* _Jean-François Alcover_, Jul 03 2018 *)
%Y A127804 Cf. A045654, A128807, A127803, A115361, A187767, A001511.
%K A127804 nonn,easy
%O A127804 0,2
%A A127804 _Paul Barry_, Jan 29 2007
%E A127804 Name changed by _Tilman Piesk_, Jun 30 2025