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

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

1,2

Comments

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

References

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.

Links

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.

Formula

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)

Mathematica

(* 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 *)

Prog

(PARI) 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

Crossrefs

Cf. A006852, A007063, A035486, A035505.

Sequence in context: A206334 A184741 A020959 * A057640 A182136 A335408

Adjacent sequences: A175309 A175310 A175311 * A175313 A175314 A175315

Keyword

nonn

Author

Enrique Pérez Herrero, Mar 28 2010

Status

approved