A233275 -id:A233275 - 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)).

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).

A256479 a(1) = 0, and for n > 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 15 2015

nonn

a(n) tells how many terms of A055938 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n. This count includes also n in case it itself is a term of A055938. See also comments in A256478 and A256991.

Antti Karttunen, Table of n, a(n) for n = 1..16384

One less than A257249.
Cf. A000120, A055938, A070939, A079559, A080791, A213714, A234017, A233275, A233276, A233277, A256478, A256991.
Cf. also A000225 (gives the positions zeros).

a(1) = 0, and for n > 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).
a(n) = A080791(A233277(n)). [Number of nonleading zeros in the binary representation of A233277(n).]
Other identities. For all n >= 1:
a(n) = A257249(n) - 1 = A000120(A233275(n)) - 1.
a(n) = A070939(n) - A256478(n).
a(A000225(n)) = 0.

A276441 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = 1 + 2a(n), a(A088359(n)) = 2a(n), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 8, 13, 10, 9, 11, 31, 30, 28, 24, 16, 29, 26, 20, 25, 18, 17, 27, 22, 21, 19, 23, 63, 62, 60, 56, 48, 32, 61, 58, 52, 40, 57, 50, 36, 49, 34, 33, 59, 54, 44, 53, 42, 41, 51, 38, 37, 35, 55, 46, 45, 43, 39, 47, 127, 126, 124, 120, 112, 96, 64, 125, 122, 116, 104, 80, 121, 114, 100, 72, 113, 98

Offset: 1

Antti Karttunen, Sep 03 2016

Antti Karttunen, Table of n, a(n) for n = 1..8191
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for sequences related to binary expansion of n
Index entries for Hofstadter-type sequences
Index entries for sequences that are permutations of the natural numbers

Inverse: A276442.
Cf. A000079, A000225, A004001, A080677, A087686, A088359, A093879.
Related or similar permutations: A006068, A054429, A233275, A233277, A267111, A276343, A276345, A276443.
Cf. also arrays A265901, A265903.

(definec (A276441 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (+ 1 (* 2 (A276441 (+ -1 (A080677 n)))))) (else (* 2 (A276441 (+ -1 (A004001 n)))))))

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 1 + 2*a(A080677(n)-1), otherwise [when n is in A088359], a(n) = 2*a(A004001(n)-1).
As a composition of other permutations:
a(n) = A054429(A267111(n)).
a(n) = A233277(A276343(n)).
a(n) = A233275(A276345(n)).
a(n) = A006068(A276443(n)).
Other identities. For all n >= 1:
a(A000079(n-1)) = A000225(n).

A234016 Partial sums of the characteristic function of A055938.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

Also: a(0) = a(1) = 0, and thereafter, a(n) = the largest k such that A055938(k) <= n.

Conjecture: partial sums of A308187 (i.e, A308187 is the characteristic function of A055938). - Sean A. Irvine, Jul 16 2022

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A046699, A055938, A079559, A234017, A233275, A308187.

from sympy import factorial
def a046699(n):
    if n<3: return 1
    s=1
    while factorial(2*s)%(2**(n - 1)): s+=1
    return s
def a(n): return n - (a046699(n + 2) - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

If n < 2, a(n)=0, otherwise a(n) = a(n-1) + (1-A079559(n)).

a(n) = n - (A046699(n+2)-1) [With A046699's October 2012 starting offset].

A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 15, 13, 14, 12, 11, 10, 9, 8, 31, 28, 29, 30, 23, 25, 27, 26, 22, 24, 21, 20, 19, 18, 17, 16, 63, 59, 60, 61, 50, 52, 62, 53, 55, 56, 58, 41, 44, 49, 57, 51, 46, 54, 48, 40, 43, 47, 45, 39, 42, 38, 37, 36, 35, 34, 33, 32, 127, 122, 123, 124, 108, 110, 125, 111, 113, 114, 126, 89, 92, 117, 115, 118, 94, 119, 121

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276343.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276346, A276442.

(definec (A276344 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A087686 (+ 1 (A276344 (- (A213714 n) 1))))) (else (A088359 (A276344 (A234017 n))))))

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A087686(1+a(A213714(n)-1)), otherwise a(n) = A088359(a(A234017(n))).
As a composition of other permutations:
a(n) = A267112(A233275(n)).
a(n) = A276442(A233277(n)).

A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 12, 9, 13, 8, 15, 14, 11, 19, 24, 22, 18, 27, 21, 23, 17, 29, 28, 25, 16, 31, 30, 26, 20, 36, 45, 43, 40, 54, 51, 35, 49, 42, 39, 41, 58, 48, 53, 34, 52, 38, 50, 44, 61, 60, 33, 59, 56, 55, 46, 32, 63, 62, 57, 47, 37, 69, 83, 81, 78, 102, 99, 74, 97, 93, 91, 68, 116, 112, 80, 88, 77, 109, 73, 75, 96, 90

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276345.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276344, A276442.

(definec (A276346 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A088359 (A276346 (- (A213714 n) 1)))) (else (A087686 (+ 1 (A276346 (A234017 n)))))))

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A088359(a(A213714(n)-1)), otherwise a(n) = A087686(1+a(A234017(n))).
As a composition of other permutations:
a(n) = A276442(A233275(n)).
a(n) = A267112(A233277(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).

A257249 a(0) = 1, and for n >= 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 19 2015

nonn

Because A233275(n) = A003188(n) for n = 1 .. 9, a(n) = A005811(n) for n = 1 .. 9.

Antti Karttunen, Table of n, a(n) for n = 0..8191

One more than A256479.

Cf. A000120, A003188, A005811, A055938, A070939, A079559, A080791, A213714, A234017, A233275, A233277, A233278, A257248, A256991.

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

A286581 Restricted growth sequence transform of A286580.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 2, 3, 6, 4, 5, 7, 6, 5, 2, 3, 6, 6, 4, 8, 9, 5, 7, 10, 6, 5, 11, 9, 6, 5, 2, 3, 6, 6, 6, 8, 12, 4, 8, 9, 9, 5, 13, 13, 10, 7, 10, 14, 6, 5, 12, 12, 11, 9, 10, 6, 5, 15, 14, 9, 6, 5, 2, 3, 6, 6, 6, 8, 12, 6, 8, 12, 12, 4, 13, 16, 9, 8, 9, 16, 9, 5, 13, 17, 13, 13, 10, 10, 7, 18, 19, 12, 14, 10, 14, 20, 6, 5, 18, 12, 12, 12, 10, 11, 9

Offset: 0

Antti Karttunen, Jun 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A233275, A286580, A286537, A286539, A286617, A286619, A286622.

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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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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A256479 a(1) = 0, and for n > 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

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A276441 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = 1 + 2*a(n), a(A088359(n)) = 2*a(n), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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A234016 Partial sums of the characteristic function of A055938.

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A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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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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A257249 a(0) = 1, and for n >= 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

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A276441 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = 1 + 2a(n), a(A088359(n)) = 2a(n), where A088359 & A087686 = numbers that occur only once & more than once in A004001.