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A007063
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Main diagonal of Kimberling's expulsion array (A035486).
(Formerly M2387)
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16
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1, 3, 5, 4, 10, 7, 15, 8, 20, 9, 18, 24, 31, 14, 28, 22, 42, 35, 33, 46, 53, 6, 36, 23, 2, 55, 62, 59, 76, 65, 54, 11, 34, 48, 70, 79, 99, 95, 44, 97, 58, 84, 25, 13, 122, 83, 26, 115, 82, 91, 52, 138, 67, 90, 71, 119, 64, 37, 81, 39, 169, 88, 108, 141, 38, 16, 146, 41, 21
(list;
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refs;
listen;
history;
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OFFSET
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1,2
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COMMENTS
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R for "right side of expelled (number)",
L for "left side",
I for "inner", i.e., next to expelled, and
O for "outer", i.e., farthest from expelled. For example, the array A035486 (and diagonal A007063) are coded as RILI. For the eight codes see Example and Mathematica. It is conjectured that six of the eight diagonal sequences are permutations of the positive integers. (End)
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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 in Number Theory, 3rd ed., Springer, 2004; Section E35, p. 359.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(theta(k)) = 3*theta(k)-(k+1), where theta(k) = Sum_{i=0..k-1} 2^floor(i/3). - Enrique Pérez Herrero, Feb 23 2010
14 sets of values which predictably appear within the sequence have been found, 1 by Richard Guy (1992) and 13 by Connor Brown (2023). Below, k is any positive integer unless otherwise specified.
a(3*2^k-3) = 9*2^k - 3*k - 10. (Guy, 1992)
a(5*2^k-3) = 15*2^k - 3*k - 12.
a(4*2^k-3) = 12*2^k - 3*k - 11.
a((20/3)*2^k-(4/3)) = 20*2^k - 3*k - 13 for odd k.
a((16/3)*2^k-(4/3)) = 16*2^k - 3*k - 12 for even k.
a((40/7)*2^k-(3/7)) = (120/7)*2^k - 3*k - (93/7) for k==1 (mod 3).
a((16/5)*2^k-(2/5)) = (48/5)*2^k - 3*k - (46/5) for k==1 (mod 4).
a((12/5)*2^k-(2/5)) = (36/5)*2^k - 3*k - (41/5) for k==0 (mod 4).
a((48/13)*2^k+(8/13)) = (144/13)*2^k - 3*k - (145/13) for k==1 (mod 12).
a((64/13)*2^k+(8/13)) = (192/13)*2^k - 3*k - (158/13) for k==3 (mod 12).
a((80/13)*2^k+(8/13)) = (240/13)*2^k - 3*k - (171/13) for k==8 (mod 12).
a((64/9)*2^k+(7/9)) = (64/3)*2^k - 3*k - (35/3) for k==1 (mod 6).
a((80/9)*2^k+(7/9)) = (80/3)*2^k - 3*k - (38/3) for k==4 (mod 6).
a((64/15)*2^k+(7/15)) = (64/5)*2^k - 3*k - (63/5) for k>4, k==1 (mod 4).
(End)
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EXAMPLE
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The eight diagonals described in Comments:
A007053 = RILI = (1, 3, 5, 4, 10, 7, 15, 8, 20, 9, 18, 24, 31, 14, ... )
A282348 = ROLO = (1, 3, 5, 2, 8, 9, 4, 10, 7, 20, 12, 24, 14, 23, ... )
A356376 = LORO = (1, 3, 5, 6, 4, 11, 12, 9, 13, 15, 23, 7, 27, 16, ... )
A356026 = LIRI = (1, 3, 5, 7, 4, 12, 10, 17, 6, 22, 15, 19, 24, 33, ... )
A356777 = ROLI = (1, 3, 5, 4, 8, 6, 10, 15, 2, 9, 13, 26, 11, 12, ... )
A356778 = RILO = (1, 3, 5, 6, 2, 10, 9, 15, 8, 20, 19, 7, 21, 31, ... )
A356779 = LORI = (1, 3, 5, 7, 4, 12, 11, 17, 10, 22, 21, 9, 23, 33, ... )
A356780 = LIRO = (1, 3, 5, 6, 4, 11, 13, 2, 7, 14, 24, 9, 10, 31, ... )
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MATHEMATICA
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K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - j/2 - 1] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
(* Next program generates the 8 arrays with highlighted diagonal sequences. *)
len = 1000;
roli = Join[{{1}},
NestList[
Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1],
Range[(Length[#] - 1)/2, 1, -1]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
rili = Join[{{1}},
NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],
Range[(Length[#] - 1)/2, 1, -1]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
rolo = Join[{{1}},
NestList[Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1],
Range[1, (Length[#] - 1)/2]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
rilo = Join[{{1}},
NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],
Range[1, (Length[#] - 1)/2, 1]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
lori = Join[{{1}},
NestList[
Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
Range[(Length[#] + 3)/2, Length[#]]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
liri = Join[{{1}},
NestList[Join[#[[Riffle[Range[(Length[#] - 1)/2, 1, -1],
Range[(Length[#] + 3)/2, Length[#]]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
loro = Join[{{1}},
NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
Range[Length[#], (Length[#] + 3)/2, -1]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
liro = Join[{{1}},
NestList[
Join[#[[Riffle[Range[(Length[#] - 1)/2, 1, -1],
Range[Length[#], (Length[#] + 3)/2, -1]]]],
Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
(Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 200] &[
ToExpression[#]]} &, {"rolo", "rilo", "roli", "rili", "loro",
"liro", "lori", "liri"}]) // ColumnForm
rows = 10; Map[{#,
Grid[Map[Map[StringPadLeft[ToString[#], 2] &, #] &,
Take[ToExpression[#], rows]],
Frame -> {None, None, Map[{#, #} -> True &, Range[rows]]},
FrameStyle -> Directive[Red]]} &, {"rolo", "rilo", "roli", "rili",
"loro", "liro", "lori", "liri"}]
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PROG
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(PARI) K(i, j) = { my(i1, j1); i1=i; j1=j;
while(j1<(2*i1-3), if(j1%2, j1=i1+((j1-1)/2), j1=i1-((j1+2)/2)); i1--; );
return(i1+j1-1); }
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CROSSREFS
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Cf. A175312, A006852, A035486, A038807, A282348, A356376, A356026, A356377, A356378, A356379, A356380.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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