A255559 -id:A255559 - cp's OEIS frontend

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)).

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)).

A256989 One-based column index of n in array A256995.

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 4, 2, 1, 2, 5, 1, 1, 1, 3, 2, 1, 3, 6, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 1, 4, 7, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 1, 3, 2, 1, 5, 8, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 1, 1, 2, 6, 1, 3, 2, 1, 1, 2, 3, 1, 3, 2, 1, 1, 1, 3, 5, 1, 2, 2, 1, 1, 2, 2, 1, 2, 4

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Also one-based row 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, A256995, A256997.

Cf. A256990 (corresponding row index), A255559.

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

A256993 a(1) = 0; for n > 1, a(n) = 1 + a(A256992(n)).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 3, 4, 4, 5, 3, 4, 5, 4, 5, 4, 5, 6, 5, 5, 4, 5, 6, 6, 5, 4, 5, 6, 5, 6, 5, 6, 6, 7, 5, 6, 6, 6, 7, 5, 6, 6, 6, 5, 7, 7, 6, 6, 5, 7, 7, 6, 7, 6, 6, 7, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 7, 6, 7, 8, 7, 7, 6, 7, 7, 8, 6, 7, 7, 7, 8, 6, 7, 6, 7, 8, 8, 7, 7, 6, 8, 7, 7, 8, 6, 8, 7, 7, 8, 7, 6, 8, 7, 8, 8, 7, 7, 8, 8, 6, 7, 7, 7, 8, 7, 8, 8, 7, 6, 7, 8, 7, 8, 7, 8, 7

Offset: 1

Antti Karttunen, Apr 15 2015

nonn

Number of iterations of A256992 needed to reach one when starting from n.

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

Cf. A070939, A071542, A254110, A255559, A256991, A256992, A257264, A257265, A279341, A279343, A279345, A279346.

a(1) = 0; for n > 1, a(n) = 1 + a(A256992(n)).
Other observations. For all n >= 1 it holds that:
a(n) >= A254110(n).
a(n) >= A256989(n).
a(n) >= A255559(n)-1.
Also it seems that a(n) - A070939(n) = -1, 0 or +1 for all n >= 1. [Compare A256991 and A256992 to see the connection.]
It is also very likely that a(n) <= A071542(n) for all n.
From Antti Karttunen, Dec 10 2016: (Start)
For all n >= 2, a(n) = A070939(A279341(n)) = A070939(A279343(n)).
For all n >= 2, a(n) = A279345(n) + A279346(n) - 1.
(End)

A255560 One-based row index of n in array A255555.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

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

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

Cf. A005187, A055938, A213714, A234017, A255555, A255557.
Cf. also A255559 (corresponding column index).
Cf. A254112, A256990.

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+A234017(n), otherwise a(n) = a(A213714(n)-1).

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

A256478 a(0) = 0; and for n >= 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 3, 3, 2, 2, 3, 1, 2, 3, 4, 4, 3, 3, 3, 2, 2, 4, 2, 3, 3, 4, 1, 2, 3, 4, 5, 5, 4, 4, 4, 3, 3, 4, 3, 3, 3, 5, 2, 2, 4, 3, 4, 2, 4, 5, 3, 3, 2, 3, 4, 4, 5, 1, 2, 3, 4, 5, 6, 6, 5, 5, 5, 4, 4, 5, 4, 4, 4, 5, 3, 3, 4, 4, 4, 3, 4, 6, 3, 3, 3, 3, 5, 5, 4, 2, 2, 4, 3, 5, 3, 4, 5, 6, 2, 4, 4, 4, 5, 3, 4, 3, 3, 2, 5, 5, 3, 6, 2, 4, 4, 3, 4, 5, 5, 6, 1, 2, 3, 4, 5, 6, 7, 7

Offset: 0

Antti Karttunen, Apr 15 2015

nonn

a(n) tells how many nonzero terms of A005187 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n. This count includes both n (in case it is a term of A005187) and 1 (but not 0). See also comments in A256479 and A256991.

The 1's (seem to) occur at positions given by A000325.

Antti Karttunen, Table of n, a(n) for n = 0..16384

One more than A257248.

Cf. A000120, A000325, A005187, A070939, A079559, A080791, A213714, A234017, A233275, A233276, A233277, A255559, A256479, A256991.

a(0) = 0; and for n >= 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).
a(n) = A000120(A233277(n)). [Binary weight of A233277(n).]
Other identities and observations. For all n >= 1:
a(n) = 1 + A257248(n) = 1 + A080791(A233275(n)).
a(n) = A070939(n) - A256479(n).
a(n) >= A255559(n).

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).

A257248 a(1) = 0; and for n > 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 19 2015

nonn

a(n) tells how many nonzero terms of A005187 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n and before 1 is reached. This count includes both n (in case it is a term of A005187) but excludes the 1 and 0 at the root. See also comments in A257249, A256478 and A256991.

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

One less than A256478.

Cf. A000120, A000325, A005187, A070939, A079559, A080791, A213714, A234017, A233275, A233276, A233277, A255559, A257249, A256991.

a(1) = 0; and for n > 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).
a(n) = A080791(A233275(n)). [Number of nonleading zeros in the binary representation of A233275(n).]
Other identities. For all n >= 1:
a(n) = A256478(n)-1 = A000120(A233277(n))-1.
a(n) = A070939(n) - A257249(n).

cp's OEIS Frontend

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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A256989 One-based column index of n in array A256995.

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A256993 a(1) = 0; for n > 1, a(n) = 1 + a(A256992(n)).

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A255560 One-based row index of n in array A255555.

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A256478 a(0) = 0; and for n >= 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

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A255556 Inverse permutation to A255555.

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A255558 Inverse permutation to A255557.

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A257248 a(1) = 0; and for n > 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

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