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%I #38 Jan 28 2023 12:17:17
%S 1,2,2,3,3,4,4,4,2,6,5,5,5,2,8,6,6,6,7,7,6,7,7,7,4,9,2,13,8,8,8,8,2,
%T 11,12,2,9,9,9,9,10,9,8,11,18,10,10,10,10,6,12,9,16,17,16,11,11,11,11,
%U 11,7,14,14,12,14,23,12,12,12,12,12,13,11,6,9,21,2,13,13,13,13,13,13,8,15
%N Kimberling's expulsion array read by antidiagonals.
%C To get the next row, start with the first element to the right of the diagonal term, then take the first to the left of the diagonal, then the second to the right, then the second to the left, the third to the right, etc.
%C It is conjectured since 1992 that the main diagonal elements (A007063) are a permutation of the positive integers.
%D R. K. Guy, Unsolved Problems Number Theory, Sect E35.
%H Enrique Pérez Herrero, <a href="/A035486/b035486.txt">Table of n, a(n) for n = 1..10000</a>
%H D. Gale, <a href="http://dx.doi.org/10.1007/978-1-4612-2192-0">Tracking the Automatic Ant: And Other Mathematical Explorations</a>, ch. 5, p. 27. Springer, 1998.
%H Enrique Pérez Herrero, <a href="/wiki/User:Enrique_Pérez_Herrero/Kimberling">Formulas and programs for Kimberling's expulsion array</a>
%H Clark Kimberling, <a href="https://cms.math.ca/crux/backfile/Crux_v18n03_Mar.pdf">Problem 1615</a>, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, p. 82-83.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KimberlingSequence.html">Kimberling Sequence</a>
%e The array starts (with elements of A007063 in brackets):
%e [1] 2 3 4 5 6 7 8 9 10 11 12 ...
%e 2 [3] 4 5 6 7 8 9 10 11 12 13 ...
%e 4 2 [5] 6 7 8 9 10 11 12 13 14 ...
%e 6 2 7 [4] 8 9 10 11 12 13 14 15 ...
%e 8 7 9 2 [10] 6 11 12 13 14 15 16 ...
%e 6 2 11 9 12 [7] 13 8 14 15 16 17 ...
%e 13 12 8 9 14 11 [15] 2 16 6 17 18 ...
%e 2 occurs as diagonal element in row 25, 27 in row 7598, and 19 in row 49595 (cf. A006852).
%t K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
%t K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
%t K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
%t K[i_] := K[i] = K[i, i]; SetAttributes[K, Listable];
%t T[n_] := n*(n + 1)/2;
%t S[n_] := Floor[1/2 (1 + Sqrt[1 + 8 (n - 1)])];
%t AJ[n_] := 1 + T[S[n]] - n;
%t AI[n_] := 1 + S[n] - AJ[n];
%t A035486[n_] := K[AI[n], AJ[n]];
%t (* _Enrique Pérez Herrero_, Mar 30 2010 *)
%o (Python)
%o def A035486(n,k):
%o if k >= 2*n-3: return n+k-1
%o q,r = divmod(k+1,2)
%o return A035486(n-1,n-1+(1-2*r)*q) # _Pontus von Brömssen_, Jan 28 2023
%Y Cf. A006852 (positions), A007063 (main diagonal), A035505 (active part), A038807.
%Y Cf. A175312 (maximum value on lower shuffle part).
%K nonn,tabl,nice,look,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Dec 23 1999
%E Edited by _Georg Fischer_, Jul 03 2020
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