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%I M2387 #68 Feb 03 2024 10:15:09
%S 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,
%T 62,59,76,65,54,11,34,48,70,79,99,95,44,97,58,84,25,13,122,83,26,115,
%U 82,91,52,138,67,90,71,119,64,37,81,39,169,88,108,141,38,16,146,41,21
%N Main diagonal of Kimberling's expulsion array (A035486).
%C From _Clark Kimberling_ Aug 05 2022, Oct 24 2022: (Start)
%C Eight such arrays (including A035486 and A356026) have been coded by _Peter J. C. Moses_ using
%C R for "right side of expelled (number)",
%C L for "left side",
%C I for "inner", i.e., next to expelled, and
%C 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)
%D D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998.
%D R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004; Section E35, p. 359.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Enrique Pérez Herrero, <a href="/A007063/b007063.txt">Table of n, a(n) for n = 1..100000</a>
%H Enrique Pérez Herrero, <a href="http://oeis.org/wiki/User:Enrique_P%C3%A9rez_Herrero/Kimberling">Kimberling's Expulsion Array</a>
%H 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, pp. 82-83.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KimberlingSequence.html">Kimberling Sequence</a>
%F 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
%F From _Connor Brown_, May 05 2023 to Feb 01 2024: (Start)
%F 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.
%F a(3*2^k-3) = 9*2^k - 3*k - 10. (Guy, 1992)
%F a(5*2^k-3) = 15*2^k - 3*k - 12.
%F a(4*2^k-3) = 12*2^k - 3*k - 11.
%F a((20/3)*2^k-(4/3)) = 20*2^k - 3*k - 13 for odd k.
%F a((16/3)*2^k-(4/3)) = 16*2^k - 3*k - 12 for even k.
%F a((40/7)*2^k-(3/7)) = (120/7)*2^k - 3*k - (93/7) for k==1 (mod 3).
%F a((16/5)*2^k-(2/5)) = (48/5)*2^k - 3*k - (46/5) for k==1 (mod 4).
%F a((12/5)*2^k-(2/5)) = (36/5)*2^k - 3*k - (41/5) for k==0 (mod 4).
%F a((48/13)*2^k+(8/13)) = (144/13)*2^k - 3*k - (145/13) for k==1 (mod 12).
%F a((64/13)*2^k+(8/13)) = (192/13)*2^k - 3*k - (158/13) for k==3 (mod 12).
%F a((80/13)*2^k+(8/13)) = (240/13)*2^k - 3*k - (171/13) for k==8 (mod 12).
%F a((64/9)*2^k+(7/9)) = (64/3)*2^k - 3*k - (35/3) for k==1 (mod 6).
%F a((80/9)*2^k+(7/9)) = (80/3)*2^k - 3*k - (38/3) for k==4 (mod 6).
%F a((64/15)*2^k+(7/15)) = (64/5)*2^k - 3*k - (63/5) for k>4, k==1 (mod 4).
%F (End)
%e The eight diagonals described in Comments:
%e A007053 = RILI = (1, 3, 5, 4, 10, 7, 15, 8, 20, 9, 18, 24, 31, 14, ... )
%e A282348 = ROLO = (1, 3, 5, 2, 8, 9, 4, 10, 7, 20, 12, 24, 14, 23, ... )
%e A356376 = LORO = (1, 3, 5, 6, 4, 11, 12, 9, 13, 15, 23, 7, 27, 16, ... )
%e A356026 = LIRI = (1, 3, 5, 7, 4, 12, 10, 17, 6, 22, 15, 19, 24, 33, ... )
%e A356777 = ROLI = (1, 3, 5, 4, 8, 6, 10, 15, 2, 9, 13, 26, 11, 12, ... )
%e A356778 = RILO = (1, 3, 5, 6, 2, 10, 9, 15, 8, 20, 19, 7, 21, 31, ... )
%e A356779 = LORI = (1, 3, 5, 7, 4, 12, 11, 17, 10, 22, 21, 9, 23, 33, ... )
%e A356780 = LIRO = (1, 3, 5, 6, 4, 11, 13, 2, 7, 14, 24, 9, 10, 31, ... )
%t K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
%t K[i_, j_] := K[i - 1, i - j/2 - 1] /; (EvenQ[j] && (j < 2 i - 3));
%t K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
%t A007063[i_] := A007063[i] = K[i, i]; SetAttributes[A007063, Listable] (* _Enrique Pérez Herrero_, Feb 09 2010 *)
%t (* Next program generates the 8 arrays with highlighted diagonal sequences. *)
%t len = 1000;
%t roli = Join[{{1}},
%t NestList[
%t Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1],
%t Range[(Length[#] - 1)/2, 1, -1]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
%t rili = Join[{{1}},
%t NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],
%t Range[(Length[#] - 1)/2, 1, -1]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
%t len]];(*A007063*)
%t rolo = Join[{{1}},
%t NestList[Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1],
%t Range[1, (Length[#] - 1)/2]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
%t len]];(*A282348*)
%t rilo = Join[{{1}},
%t NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],
%t Range[1, (Length[#] - 1)/2, 1]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
%t lori = Join[{{1}},
%t NestList[
%t Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
%t Range[(Length[#] + 3)/2, Length[#]]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
%t liri = Join[{{1}},
%t NestList[Join[#[[Riffle[Range[(Length[#] - 1)/2, 1, -1],
%t Range[(Length[#] + 3)/2, Length[#]]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
%t len]];(*A356026*)
%t loro = Join[{{1}},
%t NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
%t Range[Length[#], (Length[#] + 3)/2, -1]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
%t liro = Join[{{1}},
%t NestList[
%t Join[#[[Riffle[Range[(Length[#] - 1)/2, 1, -1],
%t Range[Length[#], (Length[#] + 3)/2, -1]]]],
%t Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
%t (Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 200] &[
%t ToExpression[#]]} &, {"rolo", "rilo", "roli", "rili", "loro",
%t "liro", "lori", "liri"}]) // ColumnForm
%t rows = 10; Map[{#,
%t Grid[Map[Map[StringPadLeft[ToString[#], 2] &, #] &,
%t Take[ToExpression[#], rows]],
%t Frame -> {None, None, Map[{#, #} -> True &, Range[rows]]},
%t FrameStyle -> Directive[Red]]} &, {"rolo", "rilo", "roli", "rili",
%t "loro", "liro", "lori", "liri"}]
%t (* _Peter J. C. Moses_, Oct 24 2022; _Clark Kimberling_, Oct 24 2022 *)
%o (PARI) K(i,j) = { my(i1,j1);i1=i; j1=j;
%o while(j1<(2*i1-3),if(j1%2,j1=i1+((j1-1)/2),j1=i1-((j1+2)/2));i1--;);
%o return(i1+j1-1);}
%o A007063(i)=K(i,i); \\ _Enrique Pérez Herrero_, Feb 21 2010
%Y Cf. A175312, A006852, A035486, A038807, A282348, A356376, A356026, A356377, A356378, A356379, A356380.
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_, _Mira Bernstein_
%E More terms from _James A. Sellers_, Dec 23 1999
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