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

%I A175312 #25 Jun 11 2025 02:50:07
%S A175312 1,3,5,7,10,12,15,17,20,22,25,28,31,33,36,39,42,44,47,50,53,55,58,61,
%T A175312 64,67,70,73,76,78,81,84,87,90,93,96,99,101,104,107,110,113,116,119,
%U A175312 122,124,127,130,133,136,139,142,145,148,151,154,157,160,163,166,169,171
%N A175312 Maximum value on Lower Shuffle Part of Kimberling's Expulsion Array (A035486).
%C A175312 a(n) is the maximum value on or below diagonal of Kimberling's Expulsion Array; this part could be called the Lower Shuffle.
%D A175312 D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998
%D A175312 R. K. Guy, Unsolved Problems Number Theory, Sect E35.
%H A175312 Enrique Pérez Herrero, <a href="/A175312/b175312.txt">Table of n, a(n) for n=1..20000</a>
%H A175312 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.
%H A175312 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KimberlingSequence.html">Kimberling Sequence</a>.
%F A175312 a(n) = 1 + 3(n-lambda(n)) - floor((n+2)/2^lambda(n)), lambda(n) = floor(log_2((n+2)/3)).
%F A175312 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.
%F A175312 From _Enrique Pérez Herrero_, Mar 30 2010: (Start)
%F A175312 a(theta(k)) = A007063(theta(k)), where theta(k) = Sum_{i=0..k-1} 2^floor(i/3).
%F A175312 At these values the maximum in the Lower Shuffle is the diagonal expelled element. (End)
%t A175312 (* By direct computation *)
%t A175312 K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
%t A175312 K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
%t A175312 K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
%t A175312 K[i_] := K[i] = K[i, i]; SetAttributes[K, Listable];
%t A175312 A175312[n_] := Max[Table[K[n, i], {i, 1, n}]]  (* _Enrique Pérez Herrero_, Mar 30 2010 *)
%t A175312 (* By the Formula *)
%t A175312 \[Lambda][n_] := Floor[Log[2, (n + 2)/3]];
%t A175312 A175312[n_] := 1 + 3*(n - \[Lambda][n]) - Floor[(n + 2)/(2^\[Lambda][n])] (* _Enrique Pérez Herrero_, Mar 30 2010 *)
%o A175312 (PARI) lambda(n)= floor(log((n + 2)/3)/log(2));
%o A175312 A175312(n)= 1 + 3*(n - lambda(n)) - floor((n + 2)/(2^lambda(n))); \\ _Enrique Pérez Herrero_, Mar 30 2010
%Y A175312 Cf. A006852, A007063, A035486, A035505.
%K A175312 nonn
%O A175312 1,2
%A A175312 _Enrique Pérez Herrero_, Mar 28 2010