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A343757 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.

%I A343757 #36 May 25 2022 08:42:18
%S A343757 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 A343757 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 A343757 4,5,16,17,20,21,0,6,16,22,0,7,16,23,0,24
%N A343757 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 A343757 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 A343757 The n-th row has 2^A069010(n) terms.
%C A343757 This sequence has similarities with A295989.
%H A343757 Rémy Sigrist, <a href="/A343757/b343757.txt">Table of n, a(n) for n = 0..8120</a>
%H A343757 Rémy Sigrist, <a href="/A343757/a343757.png">Scatterplot of (n, T(n, k)) for n < 2^10</a>
%H A343757 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A343757 T(n, 0) = 0.
%F A343757 T(n, 1) = A342410(n) for any n > 0.
%F A343757 T(n, 2^A069010(n)-1) = n.
%e A343757 Table begins:
%e A343757      0:    [0]
%e A343757      1:    [0, 1]
%e A343757      2:    [0, 2]
%e A343757      3:    [0, 3]
%e A343757      4:    [0, 4]
%e A343757      5:    [0, 1, 4, 5]
%e A343757      6:    [0, 6]
%e A343757      7:    [0, 7]
%e A343757      8:    [0, 8]
%e A343757      9:    [0, 1, 8, 9]
%e A343757     10:    [0, 2, 8, 10]
%e A343757     11:    [0, 3, 8, 11]
%e A343757     12:    [0, 12]
%e A343757     13:    [0, 1, 12, 13]
%e A343757     14:    [0, 14]
%e A343757     15:    [0, 15]
%e A343757 Table begins in binary:
%e A343757        0:   [0]
%e A343757        1:   [0, 1]
%e A343757       10:   [0, 10]
%e A343757       11:   [0, 11]
%e A343757      100:   [0, 100]
%e A343757      101:   [0, 1, 100, 101]
%e A343757      110:   [0, 110]
%e A343757      111:   [0, 111]
%e A343757     1000:   [0, 1000]
%e A343757     1001:   [0, 1, 1000, 1001]
%e A343757     1010:   [0, 10, 1000, 1010]
%e A343757     1011:   [0, 11, 1000, 1011]
%e A343757     1100:   [0, 1100]
%e A343757     1101:   [0, 1, 1100, 1101]
%e A343757     1110:   [0, 1110]
%e A343757     1111:   [0, 1111]
%o A343757 (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 A343757 Cf. A069010, A295989, A343835.
%K A343757 nonn,base,tabf
%O A343757 0,5
%A A343757 _Rémy Sigrist_, May 01 2021