A254111 -id:A254111 - cp's OEIS frontend

A255559 One-based column index of n in array A255555.

1, 1, 2, 2, 1, 1, 3, 3, 1, 2, 2, 1, 1, 1, 4, 4, 1, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 5, 5, 1, 2, 3, 1, 1, 4, 2, 1, 2, 4, 1, 1, 1, 3, 2, 1, 3, 3, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 6, 6, 1, 2, 3, 1, 1, 4, 2, 1, 2, 5, 1, 1, 1, 3, 2, 1, 3, 5, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 1, 4, 4, 1, 1, 2, 2, 1, 3, 3, 1

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Equally: One-based row index of n in array A255557.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A005187, A055938, A070939, A213714, A255555, A255557.
Cf. also A255560 (corresponding row index).
Cf. A254111, A256989, A256478, A256993.

a(1) = 1; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = 1, otherwise 1 + a(A213714(n)-1).
In other words, a(1) = 1, and for n > 1, if n = A005187(k) for some k, then a(n) = 1 + a(k-1), otherwise it must be that n is in A055938, in which case a(n) = 1.
Other identities and observations. For all n >= 1:
a(n) <= A256478(n) <= A070939(n).
a(n) <= A256993(n) + 1.

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)).

A254107 Transposed dispersion of A055938.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 26 2015

This is dispersion array A254105 transposed. Please see there for a description.

			The top left corner of the array:
     1,    3,    4,    7,    8,   10,   11,   15,   16,    18,    19,    22
     2,    6,    9,   14,   17,   21,   24,   30,   33,    37,    40,    45
     5,   13,   20,   29,   36,   44,   51,   61,   68,    76,    83,    92
    12,   28,   43,   60,   75,   91,  106,  124,  139,   155,   170,   187
    27,   59,   90,  123,  154,  186,  217,  251,  282,   314,   345,   378
    58,  122,  185,  250,  313,  377,  440,  506,  569,   633,   696,   761
   121,  249,  376,  505,  632,  760,  887, 1017, 1144,  1272,  1399,  1528
   248,  504,  759, 1016, 1271, 1527, 1782, 2040, 2295,  2551,  2806,  3063
   503, 1015, 1526, 2039, 2550, 3062, 3573, 4087, 4598,  5110,  5621,  6134
  1014, 2038, 3061, 4086, 5109, 6133, 7156, 8182, 9205, 10229, 11252, 12277
etc.

Inverse: A254108.
Transpose: A254105.
Row 1: A005187. Column 1: A000325.
Row index: A254111, Column index of n: A254112.
Cf. A002260, A004736, A005187, A055938.

(define (A254107 n) (A254105bi (A004736 n) (A002260 n)))
(define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))

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

A254112 Row index of n in A254105: If A234017(n) = 0, then a(n) = A213714(n), otherwise a(n) = a(A234017(n)).

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 4, 5, 3, 6, 7, 1, 2, 4, 8, 9, 5, 10, 11, 3, 6, 12, 13, 7, 14, 15, 1, 2, 4, 8, 16, 17, 9, 18, 19, 5, 10, 20, 21, 11, 22, 23, 3, 6, 12, 24, 25, 13, 26, 27, 7, 14, 28, 29, 15, 30, 31, 1, 2, 4, 8, 16, 32, 33, 17, 34, 35, 9, 18, 36, 37, 19, 38, 39, 5, 10, 20, 40, 41, 21, 42, 43, 11, 22, 44, 45, 23, 46, 47, 3, 6, 12, 24, 48, 49, 25, 50

Offset: 1

Antti Karttunen, Jan 27 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Row index of n in A254105. Column index for A254107.
Cf. A254111 (corresponding column index).
Cf. A213714, A234017.

If A234017(n) = 0, then a(n) = A213714(n), otherwise a(n) = a(A234017(n)).

A254110 Zero-based column index of n in A254105: if A234017(n) = 0, then a(n) = 0, otherwise a(n) = 1 + a(A234017(n)).

Original entry on oeis.org

0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 4, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 5, 4, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 4, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 6, 5, 4, 3, 2, 1, 0, 0

Offset: 1

Antti Karttunen, Jan 27 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

One less than A254111.

Cf. A234017, A254105, A254107.

If A234017(n) = 0, then a(n) = 0, otherwise a(n) = 1 + a(A234017(n)).

A254106 Inverse permutation to A254105.

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 10, 15, 9, 21, 28, 7, 8, 14, 36, 45, 20, 55, 66, 13, 27, 78, 91, 35, 105, 120, 11, 12, 19, 44, 136, 153, 54, 171, 190, 26, 65, 210, 231, 77, 253, 276, 18, 34, 90, 300, 325, 104, 351, 378, 43, 119, 406, 435, 135, 465, 496, 16, 17, 25, 53, 152, 528, 561, 170, 595, 630, 64, 189, 666, 703, 209, 741, 780, 33, 76

Offset: 1

Antti Karttunen, Jan 27 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A254105.

Cf. A254108, A254111, A254112.

(define (A254106 n) (let ((col (A254112 n)) (row (A254111 n))) (* (/ 1 2) (- (expt (+ col row) 2) col row row row -2))))

A254108 Inverse permutation to A254107.

Original entry on oeis.org

1, 3, 2, 4, 6, 5, 7, 11, 8, 16, 22, 10, 9, 12, 29, 37, 17, 46, 56, 13, 23, 67, 79, 30, 92, 106, 15, 14, 18, 38, 121, 137, 47, 154, 172, 24, 57, 191, 211, 68, 232, 254, 19, 31, 80, 277, 301, 93, 326, 352, 39, 107, 379, 407, 122, 436, 466, 21, 20, 25, 48, 138, 497, 529, 155, 562, 596, 58, 173, 631, 667, 192, 704, 742, 32, 69, 212

Offset: 1

Antti Karttunen, Jan 27 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A254107.

Cf. A254106, A254111, A254112.

(define (A254108 n) (let ((col (A254111 n)) (row (A254112 n))) (* (/ 1 2) (- (expt (+ col row) 2) col row row row -2))))

cp's OEIS Frontend

A255559 One-based column index of n in array A255555.

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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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A254107 Transposed dispersion of A055938.

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A254112 Row index of n in A254105: If A234017(n) = 0, then a(n) = A213714(n), otherwise a(n) = a(A234017(n)).

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A254110 Zero-based column index of n in A254105: if A234017(n) = 0, then a(n) = 0, otherwise a(n) = 1 + a(A234017(n)).

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A254106 Inverse permutation to A254105.

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A254108 Inverse permutation to A254107.

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