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A175312
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Maximum value on Lower Shuffle Part of Kimberling's Expulsion Array (A035486).
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5
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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
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OFFSET
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1,2
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COMMENTS
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a(n) is the maximum value on or below diagonal of Kimberling's Expulsion Array; this part could be called the Lower Shuffle.
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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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MATHEMATICA
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(* 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];
(* By the Formula *)
\[Lambda][n_] := Floor[Log[2, (n + 2)/3]];
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PROG
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(PARI) lambda(n)= floor(log((n + 2)/3)/log(2));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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