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
KEYWORD
nonn
AUTHOR
Enrique Pérez Herrero, Mar 28 2010
STATUS
approved