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A035505 Active part of Kimberling's expulsion array as a triangular array.

%I A035505 #29 Jul 02 2025 16:01:56
%S A035505 4,2,6,2,7,4,8,7,9,2,10,6,6,2,11,9,12,7,13,8,13,12,8,9,14,11,15,2,16,
%T A035505 6,2,11,16,14,6,9,17,8,18,12,19,13,18,17,12,9,19,6,13,14,20,16,21,11,
%U A035505 22,2,16,14,21,13,11,6,22,19,2,9,23,12,24,17,25,18,23,2,12,19,24,22,17,6
%N A035505 Active part of Kimberling's expulsion array as a triangular array.
%C A035505 Active or shuffle part of Kimberling's expulsion array (A035486) is given by the elements K(i,j), where j < 2*i-3. [_Enrique Pérez Herrero_, Apr 14 2010]
%D A035505 R. K. Guy, Unsolved Problems Number Theory, Sect. E35.
%H A035505 Enrique Pérez Herrero, <a href="/A035505/b035505.txt">Table of n, a(n) for n = 1..10000</a>
%H A035505 Clark Kimberling, <a href="https://cms.math.ca/crux/backfile/Crux_v17n02_Feb.pdf">Problem 1615</a>, Crux Mathematicorum, Vol. 17 (2) 44 1991; <a href="https://cms.math.ca/crux/backfile/Crux_v18n03_Mar.pdf">Solution to Problem 1615</a>, Crux Mathematicorum, Vol. 18, March 1992, p. 82-83.
%F A035505 From _Enrique Pérez Herrero_, Apr 14 2010: (Start)
%F A035505 a(n) = K(A000194(n)+2, A074294(n)), where
%F A035505   K(i,j) = i + j - 1; (j >= 2*i - 3)
%F A035505   K(i,j) = K(i-1, i-(j+2)/2) if j is even and j < 2*i - 3
%F A035505   K(i,j) = K(i-1, i+(j-1)/2); if j is odd and j < 2*i - 3.
%F A035505 (End)
%e A035505 4 2; 6 2 7 4; 8 7 9 2 10 6; ...
%t A035505 A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2];
%t A035505 A074294[n_] := n - 2*Binomial[Floor[1/2 + Sqrt[n]], 2];
%t A035505 K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
%t A035505 K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
%t A035505 K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
%t A035505 A035505[n_] := K[A000194[n] + 2, A074294[n]]
%t A035505 (* _Enrique Pérez Herrero_, Apr 14 2010 *)
%Y A035505 Cf. A006852, A007063, A038807, A035486.
%Y A035505 Cf. A175312, A074294, A000194, A006852, A007063. [_Enrique Pérez Herrero_, Apr 14 2010]
%K A035505 nonn,tabf,nice,easy
%O A035505 1,1
%A A035505 _N. J. A. Sloane_
%E A035505 More terms from _James Sellers_, Dec 23 1999