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A355809 a(n) is the number at the apex of a triangle whose base contains the distinct powers of 2 summing to n (in ascending order), and each number in a higher row is the sum of the two numbers directly below it; a(0) = 0.

%I A355809 #9 Jul 19 2022 10:43:15
%S A355809 0,1,2,3,4,5,6,9,8,9,10,13,12,17,18,27,16,17,18,21,20,25,26,35,24,33,
%T A355809 34,47,36,53,54,81,32,33,34,37,36,41,42,51,40,49,50,63,52,69,70,97,48,
%U A355809 65,66,87,68,93,94,129,72,105,106,153,108,161,162,243,64,65
%N A355809 a(n) is the number at the apex of a triangle whose base contains the distinct powers of 2 summing to n (in ascending order), and each number in a higher row is the sum of the two numbers directly below it; a(0) = 0.
%H A355809 Rémy Sigrist, <a href="/A355809/b355809.txt">Table of n, a(n) for n = 0..8192</a>
%H A355809 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A355809 a(n) >= n with equality iff n = 0 or n belongs to A048645.
%F A355809 a(2*n) = 2*a(n).
%e A355809 For n = 27:
%e A355809 - we have the following triangle:
%e A355809           47
%e A355809         13  34
%e A355809        3  10  24
%e A355809      1   2   8  16
%e A355809 - so a(27) = 47.
%o A355809 (PARI) a(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=b[k]=2^valuation(n, 2)); while (#b>1, b=vector(#b-1, k, b[k+1]+b[k])); if (#b, b[1], 0) }
%Y A355809 See A355807 for similar sequences.
%Y A355809 Cf. A048645, A348296.
%K A355809 nonn,base
%O A355809 0,3
%A A355809 _Rémy Sigrist_, Jul 18 2022