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A290259 Triangle read by rows: row n (>=1) contains in increasing order the integers for which the binary representation has length n, the first run of 1's has odd length, and all the other runs of 1's have even length.

%I A290259 #42 Aug 26 2024 09:53:50
%S A290259 1,2,4,7,8,11,14,16,19,22,28,31,32,35,38,44,47,56,59,62,64,67,70,76,
%T A290259 79,88,91,94,112,115,118,124,127,128,131,134,140,143,152,155,158,176,
%U A290259 179,182,188,191,224,227,230,236,239,248,251,254,256,259,262,268,271,280,283,286,304,307,310,316,319,352,355,358,364,367,376,379,382,448,451,454,460,463,472,475,478,496,499,502,508,511
%N A290259 Triangle read by rows: row n (>=1) contains in increasing order the integers for which the binary representation has length n, the first run of 1's has odd length, and all the other runs of 1's have even length.
%C A290259 The viabin numbers of integer partitions having only odd parts. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [7,5]. The southeast border of its Ferrers board yields 11111011 (length is 8), leading to the viabin number 251 (a term in row 8).
%C A290259 Number of entries in row n is the Fibonacci number F(n) = A000045(n).
%C A290259 T(n,k) = A290258(n+1,k) - 2^n.
%C A290259 Last entry in row n = A140253(n).
%F A290259 The entries in row n (n>=3) are (i) 2x, where x is in row n-1, and (ii) 4y + 3, where y is in row n-2. The Maple program is based on this.
%e A290259 115 is in the sequence; indeed, its binary representation, namely 1110011, has first run of 1's of odd length and the other runs of 1's have even length.
%e A290259 Triangle begins:
%e A290259    1;
%e A290259    2;
%e A290259    4,  7;
%e A290259    8, 11, 14;
%e A290259   16, 19, 22, 28, 31;
%e A290259   32, 35, 38, 44, 47, 56, 59, 62;
%e A290259   ...
%p A290259 A[1] := {1}; A[2] := {2}; for n from 3 to 10 do A[n] := `union`(map(proc (x) 2*x end proc, A[n-1]), map(proc (x) 4*x+3 end proc, A[n-2])) end do; # yields sequence in triangular form
%t A290259 nmax = 10; A[1] = {1}; A[2] = {2}; For[n = 3, n <= nmax, n++, A[n] = Union[2 A[n-1], 4 A[n-2] + 3]]; Table[A[n], {n, 1, nmax}] // Flatten (* _Jean-François Alcover_, Aug 26 2024, after Maple program *)
%Y A290259 Cf. A000045, A140253, A290258.
%K A290259 nonn,base,tabf
%O A290259 1,2
%A A290259 _Emeric Deutsch_, Sep 12 2017