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A290258 Triangle read by rows: row n (>=2) contains in increasing order the integers for which the binary representation has length n and all runs of 1's have even length.

%I A290258 #12 Aug 19 2024 15:31:37
%S A290258 3,6,12,15,24,27,30,48,51,54,60,63,96,99,102,108,111,120,123,126,192,
%T A290258 195,198,204,207,216,219,222,240,243,246,252,255,384,387,390,396,399,
%U A290258 408,411,414,432,435,438,444,447,480,483,486,492,495,504,507,510
%N A290258 Triangle read by rows: row n (>=2) contains in increasing order the integers for which the binary representation has length n and all runs of 1's have even length.
%C A290258 The viabin numbers of integer partitions having only even 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 [6,4,4,2]. The southeast border of its Ferrers board yields 110110011 (length is 9), leading to the viabin number 435 (a term in row 9).
%C A290258 Number of entries in row n is the Fibonacci number F(n-1) = A000045(n-1).
%C A290258 T(n,k) = A290259(n-1,k) + 2^(n-1).
%C A290258 Last entry in row n = A141023(n).
%C A290258 Basically the same as A277335.
%F A290258 The entries in row n (n>=4) 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 A290258 399 is in the sequence because all the runs of 1's of its binary representation, namely 110001111, have even lengths.
%e A290258 Triangle begins:
%e A290258   3;
%e A290258   6;
%e A290258   12,15;
%e A290258   24,27,30;
%e A290258   48,51,54,60,63;
%e A290258   96,99,102,108,111,120,123,126;
%e A290258   ...
%p A290258 A[2] := {3}; A[3] := {6}; for n from 4 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 A290258 A[2] = {3}; A[3] = {6};
%t A290258 For[n = 4, n <= 10, n++, A[n] = Union[2 A[n-1], 4 A[n-2] + 3]];
%t A290258 Table[A[n], {n, 2, 10}] // Flatten (* _Jean-François Alcover_, Aug 19 2024, after Maple program *)
%Y A290258 Cf. A000045, A141023, A277335, A290259.
%K A290258 nonn,tabf
%O A290258 2,1
%A A290258 _Emeric Deutsch_, Sep 12 2017