A191455 -id:A191455 - cp's OEIS frontend

A191426 Dispersion of (3+[n*r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

1, 4, 2, 9, 6, 3, 17, 12, 7, 5, 30, 22, 14, 11, 8, 51, 38, 25, 20, 15, 10, 85, 64, 43, 35, 27, 19, 13, 140, 106, 72, 59, 46, 33, 24, 16, 229, 174, 119, 98, 77, 56, 41, 28, 18, 373, 284, 195, 161, 127, 93, 69, 48, 32, 21, 606, 462, 318, 263, 208, 153, 114, 80, 54, 36, 23, 983, 750, 517, 428, 339, 250, 187, 132, 90, 61, 40, 26

Offset: 1

Clark Kimberling, Jun 02 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022342 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...4...9...17..30
  2...6...12..22..38
  3...7...14..25..43
  5...11..20..35..59
  8...15..27..46..77

Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 2.

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)
c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + 3]
mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191426 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]  (* A191426 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191450 Dispersion of (3*n-1), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 14, 23, 11, 6, 41, 68, 32, 17, 7, 122, 203, 95, 50, 20, 9, 365, 608, 284, 149, 59, 26, 10, 1094, 1823, 851, 446, 176, 77, 29, 12, 3281, 5468, 2552, 1337, 527, 230, 86, 35, 13, 9842, 16403, 7655, 4010, 1580, 689, 257, 104, 38, 15, 29525

Offset: 1

Clark Kimberling, Jun 05 2011

Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A016789(n-1), t(n) = A032766(n) [from term A032766(1) onward] and u(n) = A253887(n). [Author's original comment edited by Antti Karttunen, Jan 24 2015]

For other examples of such sequences, please see the Crossrefs section.

			The northwest corner of the square array:
  1,  2,  5,  14,  41,  122,  365,  1094,  3281,   9842,  29525,   88574, ...
  3,  8, 23,  68, 203,  608, 1823,  5468, 16403,  49208, 147623,  442868, ...
  4, 11, 32,  95, 284,  851, 2552,  7655, 22964,  68891, 206672,  620015, ...
  6, 17, 50, 149, 446, 1337, 4010, 12029, 36086, 108257, 324770,  974309, ...
  7, 20, 59, 176, 527, 1580, 4739, 14216, 42647, 127940, 383819, 1151456, ...
  9, 26, 77, 230, 689, 2066, 6197, 18590, 55769, 167306, 501917, 1505750, ...
  etc.
The leftmost column is A032766, and each successive column to the right of it is obtained by multiplying the left neighbor on that row by three and subtracting one, thus the second column is (3*1)-1, (3*3)-1, (3*4)-1, (3*6)-1, (3*7)-1, (3*9)-1, ... = 2, 8, 11, 17, 20, 26, ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254047.
Transpose: A254051.
Column 1: A032766.
Cf. A007051, A057198, A199109, A199113 (rows 1-4).
Cf. A253887 (row index of n in this array) & A254046 (column index, see also A253786).
Cf. A000244, A016789, A038722, A048673, A254053, A254103, A254104.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191426-A191455.

A191450 := proc(r, c)
    option remember;
    if c = 1 then
        A032766(r) ;
    else
        A016789(procname(r, c-1)-1) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n-1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191450 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191450 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

a(n,k)=3^(n-1)*(k*3\2*2-1)\2+1 \\ =3^(n-1)*(k*3\2-1/2)+1/2, but 30% faster. - M. F. Hasler, Jan 20 2015

(define (A191450 n) (A191450bi (A002260 n) (A004736 n)))
(define (A191450bi row col) (if (= 1 col) (A032766 row) (A016789 (- (A191450bi row (- col 1)) 1))))
(define (A191450bi row col) (/ (+ 3 (* (A000244 col) (- (* 2 (A032766 row)) 1))) 6)) ;; Another implementation based on L. Edson Jeffery's direct formula.
;; Antti Karttunen, Jan 21 2015

Conjecture: A(n,k) = (3 + (2*A032766(n) - 1)*A000244(k))/6. - L. Edson Jeffery, with slight changes by Antti Karttunen, Jan 21 2015

a(n) = A254051(A038722(n)). [When both this and transposed array A254051 are interpreted as one-dimensional sequences.] - Antti Karttunen, Jan 22 2015

Example corrected and description clarified by Antti Karttunen, Jan 24 2015

A191452 Dispersion of (4,8,12,16,...), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 16, 8, 3, 64, 32, 12, 5, 256, 128, 48, 20, 6, 1024, 512, 192, 80, 24, 7, 4096, 2048, 768, 320, 96, 28, 9, 16384, 8192, 3072, 1280, 384, 112, 36, 10, 65536, 32768, 12288, 5120, 1536, 448, 144, 40, 11, 262144, 131072, 49152, 20480, 6144, 1792, 576

Offset: 1

Clark Kimberling, Jun 05 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...4....16...64....256
  2...8....32...128...512
  3...12...48...192...768
  5...20...80...320...1280
  6...24...96...384...1536

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A114537, A035513, A035506, A191449.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=4n  (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191452 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191452 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(row,col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 12, 13, 9, 7, 27, 28, 20, 14, 8, 58, 59, 43, 29, 17, 10, 121, 122, 90, 60, 36, 21, 11, 248, 249, 185, 123, 75, 44, 24, 15, 503, 504, 376, 250, 154, 91, 51, 30, 16, 1014, 1015, 759, 505, 313, 186, 106, 61, 33, 18, 2037, 2038, 1526, 1016, 632, 377, 217, 124, 68, 37, 19, 4084, 4085, 3061, 2039, 1271, 760, 440, 251, 139, 76, 40, 22

Offset: 1

Antti Karttunen, Jan 26 2015

This sequence is one instance of Clark Kimberling's generic dispersion arrays. Paraphrasing his explanation in A191450, mutatis mutandis, we have the following definition:
Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A055938(n), t(n) = A005187(n) [from term A005187(1) onward] and u(n) = A254112(n).
For other examples of such sequences, see the Crossrefs section. For a general introduction, please follow the Kimberling references.
The main diagonal: 1, 6, 20, 60, 154, 377, 887, 2040, 4598, 10229, 22515, 49139, ...

			The top left corner of the array:
   1,  2,  5,  12,  27,  58,  121,  248,  503,  1014,  2037,  4084
   3,  6, 13,  28,  59, 122,  249,  504, 1015,  2038,  4085,  8180
   4,  9, 20,  43,  90, 185,  376,  759, 1526,  3061,  6132, 12275
   7, 14, 29,  60, 123, 250,  505, 1016, 2039,  4086,  8181, 16372
   8, 17, 36,  75, 154, 313,  632, 1271, 2550,  5109, 10228, 20467
  10, 21, 44,  91, 186, 377,  760, 1527, 3062,  6133, 12276, 24563
  11, 24, 51, 106, 217, 440,  887, 1782, 3573,  7156, 14323, 28658
  15, 30, 61, 124, 251, 506, 1017, 2040, 4087,  8182, 16373, 32756
  16, 33, 68, 139, 282, 569, 1144, 2295, 4598,  9205, 18420, 36851
  18, 37, 76, 155, 314, 633, 1272, 2551, 5110, 10229, 20468, 40947
etc.

Antti Karttunen, Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and Dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254106.
Transpose: A254107.
Column 1: A005187.
Cf. also A000325, A095768, A123720 (Seem to be rows 1 - 3, the last one from its second term onward.)
Columnd index of n: A254111, Row index: A254112.
Cf. A002260, A004736, A055938, A233275-A233278.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191450, A191426-A191455.

(define (A254105 n) (A254105bi (A002260 n) (A004736 n)))
(define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))

If col = 1, then A(row,col) = A005187(row), otherwise A(row,col) = A055938(A(row,col-1)).

A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 27, 18, 12, 5, 81, 54, 36, 15, 7, 243, 162, 108, 45, 21, 8, 729, 486, 324, 135, 63, 24, 10, 2187, 1458, 972, 405, 189, 72, 30, 11, 6561, 4374, 2916, 1215, 567, 216, 90, 33, 13, 19683, 13122, 8748, 3645, 1701, 648, 270, 99, 39, 14, 59049

Offset: 1

Clark Kimberling, Jun 05 2011

Transpose of A141396.
Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...3....9....27...81
  2...6....18...54...162
  4...12...36...108..324
  5...15...45...135..405
  7...21...63...189..567

Cf. A114537, A035513, A035506,
A054582: dispersion of (2,4,6,8,...).
A191450: dispersion of (2,5,8,11,...).
A191451: dispersion of (4,7,10,13,...).
A191452: dispersion of (4,8,12,16,...).

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191449 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191449 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

T(i,j)=T(i,1)*T(1,j)=floor((3i-1)/2)*3^(j-1).

A191448 Dispersion of the odd integers greater than 1, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 15, 11, 9, 6, 31, 23, 19, 13, 8, 63, 47, 39, 27, 17, 10, 127, 95, 79, 55, 35, 21, 12, 255, 191, 159, 111, 71, 43, 25, 14, 511, 383, 319, 223, 143, 87, 51, 29, 16, 1023, 767, 639, 447, 287, 175, 103, 59, 33, 18, 2047, 1535, 1279, 895, 575

Offset: 1

Clark Kimberling, Jun 05 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...3...7...15..31
  2...5...11..23..47
  4...9...19..39..79
  6...13..27..55..111
  8...17..35..71..143

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=2n+1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191448 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191448 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191545 Dispersion of (floor(9*n/4)), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 9, 13, 11, 7, 20, 29, 24, 15, 8, 45, 65, 54, 33, 18, 10, 101, 146, 121, 74, 40, 22, 12, 227, 328, 272, 166, 90, 49, 27, 14, 510, 738, 612, 373, 202, 110, 60, 31, 16, 1147, 1660, 1377, 839, 454, 247, 135, 69, 36, 17, 2580, 3735, 3098, 1887

Offset: 1

Clark Kimberling, Jun 09 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1   2    4    9    20
  3   6    13   29   65
  5   11   25   54   121
  7   15   33   74   166
  8   18   40   90   202

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := Floor[9n/4]   (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191545 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191545 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191438 Dispersion of ([n*sqrt(2)+n]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 16, 12, 6, 21, 38, 28, 14, 8, 50, 91, 67, 33, 19, 10, 120, 219, 161, 79, 45, 24, 11, 289, 528, 388, 190, 108, 57, 26, 13, 697, 1274, 936, 458, 260, 137, 62, 31, 15, 1682, 3075, 2259, 1105, 627, 330, 149, 74, 36, 17, 4060, 7423, 5453

Offset: 1

Clark Kimberling, Jun 04 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1....2....4....9....21
  3....7....16...38...91
  5....12...28...67...161
  6....14...33...79...190
  8....19...45...108..260

Cf. A114537, A035513, A035506, A191438.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[2];
f[n_] := Floor[n*x+n] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191438 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191438 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191536 Dispersion of (3+floor(n*sqrt(2))), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 8, 5, 3, 14, 10, 7, 6, 22, 17, 12, 11, 9, 34, 27, 19, 18, 15, 13, 51, 41, 29, 28, 24, 21, 16, 75, 60, 44, 42, 36, 32, 25, 20, 109, 87, 65, 62, 53, 48, 38, 31, 23, 157, 126, 94, 90, 77, 70, 56, 46, 35, 26, 225, 181, 135, 130, 111, 101, 82, 68, 52, 39

Offset: 1

Clark Kimberling, Jun 06 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455 and A191536-A191545.

			Northwest corner:
  1...4....8....14...22
  2...5....10...17...27
  3...7....12...19...29
  6...11...18...28...42
  9...15...24...36...54

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=3+Floor[n*Sqrt[2]]   (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]
(* A191536 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191536 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191442 Dispersion of ([n*sqrt(3)+1/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 12, 10, 8, 9, 21, 17, 14, 11, 16, 36, 29, 24, 19, 13, 28, 62, 50, 42, 33, 23, 15, 48, 107, 87, 73, 57, 40, 26, 18, 83, 185, 151, 126, 99, 69, 45, 31, 20, 144, 320, 262, 218, 171, 120, 78, 54, 35, 22, 249, 554, 454, 378, 296, 208, 135, 94

Offset: 1

Clark Kimberling, Jun 04 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1....2....3....5....9
  4....7....12...21...36
  6....10...17...29...50
  8....14...24...42...73
  11...19...33...57...99

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[3];
f[n_] := Floor[n*x+1/2] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191442 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191442 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

cp's OEIS Frontend

A191426 Dispersion of (3+[n*r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

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A191450 Dispersion of (3*n-1), read by antidiagonals.

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A191452 Dispersion of (4,8,12,16,...), by antidiagonals.

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A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(row,col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

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A191448 Dispersion of the odd integers greater than 1, by antidiagonals.

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A191545 Dispersion of (floor(9*n/4)), by antidiagonals.

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A191438 Dispersion of ([n*sqrt(2)+n]), where [ ]=floor, by antidiagonals.

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