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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 difference of the two numbers directly below it; a(0) = 0.
2

%I #10 Jul 19 2022 10:43:22

%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,

%U -15,8,7,6,9,4,1,2,1,64,63,62,61,60,57,58,57,56,49

%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 difference of the two numbers directly below it; a(0) = 0.

%H Rémy Sigrist, <a href="/A355808/b355808.txt">Table of n, a(n) for n = 0..8192</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, b[k+1]-b[k])); if (#b, b[1], 0) }

%Y See A355807 for similar sequences.

%Y Cf. A348296.

%K sign,base

%O 0,3

%A _Rémy Sigrist_, Jul 18 2022