A255560 -id:A255560 - 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.

A255555 Square array A(row,col) read by downwards antidiagonals: A(1,1) = 1, A(row,1) = A055938(row-1), and for col > 1, A(row,col) = A005187(1+A(row,col-1)).

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 15, 8, 10, 6, 31, 16, 19, 11, 9, 63, 32, 38, 22, 18, 12, 127, 64, 74, 42, 35, 23, 13, 255, 128, 146, 82, 70, 46, 25, 14, 511, 256, 290, 162, 138, 89, 49, 26, 17, 1023, 512, 578, 322, 274, 176, 97, 50, 34, 20, 2047, 1024, 1154, 642, 546, 350, 193, 98, 67, 39, 21, 4095, 2048, 2306, 1282, 1090, 695, 385, 194, 134, 78, 41, 24

Offset: 1

Antti Karttunen, Apr 13 2015

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Provided that I understand Kimberling's terminology correctly, this array is the dispersion of sequence b(n) = A005187(n+1), for n>=1: A005187[2..] = [3, 4, 7, 8, 10, 11, ...]. The left column is the complement of that sequence, which is {1} followed by A055938. - Antti Karttunen, Apr 17 2015

			The top left corner of the array:
   1,  3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095
   2,  4,  8,  16,  32,  64,  128,  256,  512, 1024,  2048,  4096
   5, 10, 19,  38,  74, 146,  290,  578, 1154, 2306,  4610,  9218
   6, 11, 22,  42,  82, 162,  322,  642, 1282, 2562,  5122, 10242
   9, 18, 35,  70, 138, 274,  546, 1090, 2178, 4354,  8706, 17410
  12, 23, 46,  89, 176, 350,  695, 1387, 2770, 5535, 11067, 22128
  13, 25, 49,  97, 193, 385,  769, 1537, 3073, 6145, 12289, 24577
  14, 26, 50,  98, 194, 386,  770, 1538, 3074, 6146, 12290, 24578
  17, 34, 67, 134, 266, 530, 1058, 2114, 4226, 8450, 16898, 33794
  20, 39, 78, 153, 304, 606, 1207, 2411, 4818, 9631, 19259, 38512
  ...

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 permutation: A255556.
Transpose: A255557.
Row 1: A000225.
Cf. A005187, A055938.
Cf. A255559 (column index), A255560 (row index).
Cf. also A254105, A256995 (variants), A233275-A233278.

(define (A255555 n) (A255555bi (A002260 n) (A004736 n)))
(define (A255555bi row col) (if (= 1 col) (if (= 1 row) 1 (A055938 (- row 1))) (A005187 (+ 1 (A255555bi row (- col 1))))))

A(1,1) = 1, A(row,1) = A055938(row-1), and for col > 1, A(row,col) = A005187(1+A(row,col-1)).

A255557 Square array A(row,col): A(1,1) = 1, A(1,col) = A055938(col-1), and for row > 1: A(row,col) = A005187(1+A(row-1,col)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 8, 15, 9, 11, 19, 16, 31, 12, 18, 22, 38, 32, 63, 13, 23, 35, 42, 74, 64, 127, 14, 25, 46, 70, 82, 146, 128, 255, 17, 26, 49, 89, 138, 162, 290, 256, 511, 20, 34, 50, 97, 176, 274, 322, 578, 512, 1023, 21, 39, 67, 98, 193, 350, 546, 642, 1154, 1024, 2047, 24, 41, 78, 134, 194, 385, 695, 1090, 1282, 2306, 2048, 4095

Offset: 1

Antti Karttunen, Apr 13 2015

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

This is transpose of array A255555, see comments and links given there.

			The top left corner of the array:
     1,    2,    5,    6,    9,   12,   13,   14,   17,   20,    21,    24
     3,    4,   10,   11,   18,   23,   25,   26,   34,   39,    41,    47
     7,    8,   19,   22,   35,   46,   49,   50,   67,   78,    81,    94
    15,   16,   38,   42,   70,   89,   97,   98,  134,  153,   161,   184
    31,   32,   74,   82,  138,  176,  193,  194,  266,  304,   321,   365
    63,   64,  146,  162,  274,  350,  385,  386,  530,  606,   641,   726
   127,  128,  290,  322,  546,  695,  769,  770, 1058, 1207,  1281,  1447
   255,  256,  578,  642, 1090, 1387, 1537, 1538, 2114, 2411,  2561,  2891
   511,  512, 1154, 1282, 2178, 2770, 3073, 3074, 4226, 4818,  5121,  5778
  1023, 1024, 2306, 2562, 4354, 5535, 6145, 6146, 8450, 9631, 10241, 11551
  ...

Inverse permutation: A255558.
Transpose: A255555.
Column 1: A000225.
Cf. A005187, A055938.
Cf. A255559 (row index), A255560 (column index).
Cf. also A254107, A256997 (variants).

(define (A255557 n) (A255555bi (A004736 n) (A002260 n)))
(define (A255555bi row col) (if (= 1 col) (if (= 1 row) 1 (A055938 (- row 1))) (A005187 (+ 1 (A255555bi row (- col 1))))))

A(row,col): A(1,1) = 1, and for the rest of topmost row: A(1,col) = A055938(col-1), and for any row > 1: A(row,col) = A005187(1+A(row-1,col)).

A256990 One-based row index of n in array A256995.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 1, 2, 4, 3, 1, 5, 6, 7, 2, 4, 8, 3, 1, 9, 10, 5, 6, 11, 7, 2, 12, 13, 14, 15, 4, 8, 16, 3, 1, 17, 18, 9, 10, 19, 5, 6, 20, 21, 22, 11, 7, 23, 2, 12, 24, 25, 13, 14, 26, 15, 4, 27, 28, 29, 30, 31, 8, 16, 32, 3, 1, 33, 34, 17, 18, 35, 9, 10, 36, 37, 38, 19, 5, 39, 6, 20, 40, 41, 21, 22, 42, 11, 7

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Also one-based column index for array A256997.

a(1) = 0 by convention, as 1 is outside of the actual arrays A256995 & A256997.

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

Cf. A005187, A055938, A213714, A234017, A256995, A256997.

Cf. A256989 (corresponding column index), A255560.

a(1) = 0; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = A234017(n), otherwise a(n) = a(A213714(n)).

In other words, a(1) = 0, and for n > 1, if n = A055938(k) for some k, then a(n) = k, otherwise it must be that n = A005187(h) for some h, in which case a(n) = a(h).

A255556 Inverse permutation to A255555.

Original entry on oeis.org

1, 3, 2, 5, 6, 10, 4, 8, 15, 9, 14, 21, 28, 36, 7, 12, 45, 20, 13, 55, 66, 19, 27, 78, 35, 44, 91, 105, 120, 136, 11, 17, 153, 54, 26, 171, 190, 18, 65, 210, 77, 25, 231, 253, 276, 34, 90, 300, 43, 53, 325, 351, 104, 119, 378, 135, 152, 406, 435, 465, 496, 528, 16, 23, 561, 170, 64, 595, 630, 33, 189, 666, 209, 24

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A255555.

Cf. A255558, A255559, A255560.

(define (A255556 n) (let ((col (A255559 n)) (row (A255560 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A255559(n), and r = A255560(n).

A255558 Inverse permutation to A255557.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 9, 11, 8, 12, 16, 22, 29, 10, 14, 37, 17, 13, 46, 56, 18, 23, 67, 30, 38, 79, 92, 106, 121, 15, 20, 137, 47, 24, 154, 172, 19, 57, 191, 68, 25, 211, 232, 254, 31, 80, 277, 39, 48, 301, 326, 93, 107, 352, 122, 138, 379, 407, 436, 466, 497, 21, 27, 529, 155, 58, 562, 596, 32, 173, 631, 192, 26

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A255557.

Cf. A255556, A255559, A255560.

(define (A255558 n) (let ((row (A255559 n)) (col (A255560 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where r = A255559(n), and c = A255560(n).

cp's OEIS Frontend

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

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A255555 Square array A(row,col) read by downwards antidiagonals: A(1,1) = 1, A(row,1) = A055938(row-1), and for col > 1, A(row,col) = A005187(1+A(row,col-1)).

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A255557 Square array A(row,col): A(1,1) = 1, A(1,col) = A055938(col-1), and for row > 1: A(row,col) = A005187(1+A(row-1,col)).

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A256990 One-based row index of n in array A256995.

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A255556 Inverse permutation to A255555.

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A255558 Inverse permutation to A255557.

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