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%I #12 Jul 19 2022 10:43:34
%S 0,1,2,1,4,3,2,1,8,7,6,5,4,1,2,1,16,15,14,13,12,9,10,9,8,1,2,3,4,3,2,
%T 1,32,31,30,29,28,25,26,25,24,17,18,13,20,19,18,17,16,1,2,11,4,5,6,3,
%U 8,7,6,3,4,1,2,1,64,63,62,61,60,57,58,57,56,49,50
%N 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 absolute difference of the two numbers directly below it; a(0) = 0.
%C This sequence has similarities with A334387.
%H Rémy Sigrist, <a href="/A355807/b355807.txt">Table of n, a(n) for n = 0..8191</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F a(n) <= n with equality iff n = 0 or n is a power of 2.
%F a(2*n) = 2*a(n).
%e For n = 27:
%e - we have the following triangle:
%e 3
%e 5 2
%e 1 6 8
%e 1 2 8 16
%e - so a(27) = 3.
%o (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, abs(b[k+1]-b[k]))); if (#b, b[1], 0) }
%Y See A355808, A355809, A355810 and A355811 for other variants.
%Y Cf. A133457, A334387, A348296.
%K nonn,base
%O 0,3
%A _Rémy Sigrist_, Jul 18 2022