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A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.

%I A228083 #20 Jun 19 2021 21:25:23
%S A228083 1,2,4,3,5,6,5,7,8,13,7,10,9,16,15,10,12,11,17,19,18,12,14,14,19,22,
%T A228083 20,21,14,17,17,22,25,22,24,23,17,19,19,25,28,25,26,27,30,19,22,22,28,
%U A228083 31,28,29,31,34,32,22,25,25,31,36,31,33,36,36,33,37
%N A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.
%H A228083 Antti Karttunen, <a href="/A228083/b228083.txt">The first 141 antidiagonals of the table, flattened</a>
%H A228083 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F A228083 T(r,1) are those numbers not of form n + sum of binary digits of n (binary Self numbers) = A010061(r);
%F A228083 T(r,c) = T(r,c-1) + sum of binary digits of T(r,c-1) = A092391(T(r,c-1)).
%e A228083 The top-left corner of the square array:
%e A228083    1,  2,  3,  5,  7, 10, 12, 14, ...
%e A228083    4,  5,  7, 10, 12, 14, 17, 19, ...
%e A228083    6,  8,  9, 11, 14, 17, 19, 22, ...
%e A228083   13, 16, 17, 19, 22, 25, 28, 31, ...
%e A228083   15, 19, 22, 25, 28, 31, 36, 38, ...
%e A228083   18, 20, 22, 25, 28, 31, 36, 38, ...
%e A228083   21, 24, 26, 29, 33, 35, 38, 41, ...
%e A228083   23, 27, 31, 36, 38, 41, 44, 47, ...
%e A228083   ...
%e A228083 The non-initial terms on each row are obtained by adding to the preceding term the number of 1-bits in its binary representation (A000120).
%t A228083 nmax0 = 100;
%t A228083 nmax := Length[col[1]];
%t A228083 col[1] = Table[n + DigitCount[n, 2, 1], {n, 0, nmax0}] // Complement[Range[Last[#]], #]&;
%t A228083 col[k_] := col[k] = col[k - 1] + DigitCount[col[k-1], 2, 1];
%t A228083 T[n_, k_] := col[k][[n]];
%t A228083 Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 28 2020 *)
%o A228083 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A228083 (define (A228083 n) (A228083bi (A002260 n) (A004736 n)))
%o A228083 (define (A228083bi row col) ((rowfun-for-A228083 row) col))
%o A228083 (definec (rowfun-for-A228083 n) (implement-cached-function 0 (rowfun-n k) (cond ((= 1 k) (A010061 n)) (else (A092391 (rowfun-n (- k 1)))))))
%Y A228083 First column: A010061. First row: A010062. Transpose: A228084. See A151942 for decimal analog.
%Y A228083 Cf. also A000120, A092391, A227643, A228082, A228085, A228086, A228087, A228088, A228091, A218254.
%K A228083 nonn,base,tabl
%O A228083 1,2
%A A228083 _Antti Karttunen_, Aug 09 2013