A035505 -id:A035505 - cp's OEIS frontend

A035486 Kimberling's expulsion array read by antidiagonals.

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, 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, 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

Offset: 1

N. J. A. Sloane

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.

It is conjectured since 1992 that the main diagonal elements (A007063) are a permutation of the positive integers.

			The array starts (with elements of A007063 in brackets):
  [1]  2   3   4   5   6   7   8   9  10  11  12 ...
   2  [3]  4   5   6   7   8   9  10  11  12  13 ...
   4   2  [5]  6   7   8   9  10  11  12  13  14 ...
   6   2   7  [4]  8   9  10  11  12  13  14  15 ...
   8   7   9   2 [10]  6  11  12  13  14  15  16 ...
   6   2  11   9  12  [7] 13   8  14  15  16  17 ...
  13  12   8   9  14  11 [15]  2  16   6  17  18 ...
2 occurs as diagonal element in row 25, 27 in row 7598, and 19 in row 49595 (cf. A006852).

R. K. Guy, Unsolved Problems Number Theory, Sect E35.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998.
Enrique Pérez Herrero, Formulas and programs for Kimberling's expulsion array
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, p. 82-83.
Eric Weisstein's World of Mathematics, Kimberling Sequence

Cf. A006852 (positions), A007063 (main diagonal), A035505 (active part), A038807.

Cf. A175312 (maximum value on lower shuffle part).

K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
K[i_] := K[i] = K[i, i]; SetAttributes[K, Listable];
T[n_] := n*(n + 1)/2;
S[n_] := Floor[1/2 (1 + Sqrt[1 + 8 (n - 1)])];
AJ[n_] := 1 + T[S[n]] - n;
AI[n_] := 1 + S[n] - AJ[n];
A035486[n_] := K[AI[n], AJ[n]];
(* Enrique Pérez Herrero, Mar 30 2010 *)

def A035486(n,k):
    if k >= 2*n-3: return n+k-1
    q,r = divmod(k+1,2)
    return A035486(n-1,n-1+(1-2*r)*q) # Pontus von Brömssen, Jan 28 2023

More terms from James Sellers, Dec 23 1999

Edited by Georg Fischer, Jul 03 2020

A175312 Maximum value on Lower Shuffle Part of Kimberling's Expulsion Array (A035486).

Original entry on oeis.org

1, 3, 5, 7, 10, 12, 15, 17, 20, 22, 25, 28, 31, 33, 36, 39, 42, 44, 47, 50, 53, 55, 58, 61, 64, 67, 70, 73, 76, 78, 81, 84, 87, 90, 93, 96, 99, 101, 104, 107, 110, 113, 116, 119, 122, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 171

Offset: 1

Enrique Pérez Herrero, Mar 28 2010

nonn

a(n) is the maximum value on or below diagonal of Kimberling's Expulsion Array; this part could be called the Lower Shuffle.

D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998
R. K. Guy, Unsolved Problems Number Theory, Sect E35.

Enrique Pérez Herrero, Table of n, a(n) for n=1..20000
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991; Solution to Problem 1615, Crux Mathematicorum, Vol. 18, March 1992, p. 82-83.
Eric Weisstein's World of Mathematics, Kimberling Sequence.

Cf. A006852, A007063, A035486, A035505.

(* By direct computation *)
K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
K[i_] := K[i] = K[i, i]; SetAttributes[K, Listable];
A175312[n_] := Max[Table[K[n, i], {i, 1, n}]]  (* Enrique Pérez Herrero, Mar 30 2010 *)
(* By the Formula *)
\[Lambda][n_] := Floor[Log[2, (n + 2)/3]];
A175312[n_] := 1 + 3*(n - \[Lambda][n]) - Floor[(n + 2)/(2^\[Lambda][n])] (* Enrique Pérez Herrero, Mar 30 2010 *)

lambda(n)= floor(log((n + 2)/3)/log(2));
A175312(n)= 1 + 3*(n - lambda(n)) - floor((n + 2)/(2^lambda(n))); \\ Enrique Pérez Herrero, Mar 30 2010

a(n) = 1 + 3(n-lambda(n)) - floor((n+2)/2^lambda(n)), lambda(n) = floor(log_2((n+2)/3)).
a(n) >= A007063(n); a(n) = max(K(n,1),K(n,2),...,K(n,n)), where K(i,j) is an element of Kimberling's Array given by A035486.
From Enrique Pérez Herrero, Mar 30 2010: (Start)
a(theta(k)) = A007063(theta(k)), where theta(k) = Sum_{i=0..k-1} 2^floor(i/3).
At these values the maximum in the Lower Shuffle is the diagonal expelled element. (End)

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A035486 Kimberling's expulsion array read by antidiagonals.

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