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Irregular table read by rows; the n-th row contains the sums of distinct terms of the n-th row of table A343835, in ascending order.
2

%I #36 May 25 2022 08:42:18

%S 0,0,1,0,2,0,3,0,4,0,1,4,5,0,6,0,7,0,8,0,1,8,9,0,2,8,10,0,3,8,11,0,12,

%T 0,1,12,13,0,14,0,15,0,16,0,1,16,17,0,2,16,18,0,3,16,19,0,4,16,20,0,1,

%U 4,5,16,17,20,21,0,6,16,22,0,7,16,23,0,24

%N Irregular table read by rows; the n-th row contains the sums of distinct terms of the n-th row of table A343835, in ascending order.

%C In other words, the n-th row contains the numbers k whose runs of 1's in the binary expansion also appear in that of n.

%C The n-th row has 2^A069010(n) terms.

%C This sequence has similarities with A295989.

%H Rémy Sigrist, <a href="/A343757/b343757.txt">Table of n, a(n) for n = 0..8120</a>

%H Rémy Sigrist, <a href="/A343757/a343757.png">Scatterplot of (n, T(n, k)) for n < 2^10</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F T(n, 0) = 0.

%F T(n, 1) = A342410(n) for any n > 0.

%F T(n, 2^A069010(n)-1) = n.

%e Table begins:

%e 0: [0]

%e 1: [0, 1]

%e 2: [0, 2]

%e 3: [0, 3]

%e 4: [0, 4]

%e 5: [0, 1, 4, 5]

%e 6: [0, 6]

%e 7: [0, 7]

%e 8: [0, 8]

%e 9: [0, 1, 8, 9]

%e 10: [0, 2, 8, 10]

%e 11: [0, 3, 8, 11]

%e 12: [0, 12]

%e 13: [0, 1, 12, 13]

%e 14: [0, 14]

%e 15: [0, 15]

%e Table begins in binary:

%e 0: [0]

%e 1: [0, 1]

%e 10: [0, 10]

%e 11: [0, 11]

%e 100: [0, 100]

%e 101: [0, 1, 100, 101]

%e 110: [0, 110]

%e 111: [0, 111]

%e 1000: [0, 1000]

%e 1001: [0, 1, 1000, 1001]

%e 1010: [0, 10, 1000, 1010]

%e 1011: [0, 11, 1000, 1011]

%e 1100: [0, 1100]

%e 1101: [0, 1, 1100, 1101]

%e 1110: [0, 1110]

%e 1111: [0, 1111]

%o (PARI) row(n) = { my (rr=[]); while (n, my (z=valuation(n, 2), o=valuation(n/2^z+1, 2), r=(2^o-1)*2^z); n-=r; rr = concat(rr, r)); vector(2^#rr, k, vecsum(vecextract(rr, k-1))) }

%Y Cf. A069010, A295989, A343835.

%K nonn,base,tabf

%O 0,5

%A _Rémy Sigrist_, May 01 2021