Revision History for A267700
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#58 by Michael De Vlieger at Mon Oct 10 09:37:46 EDT 2022
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#57 by Jon E. Schoenfield at Mon Oct 10 09:36:13 EDT 2022
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#56 by Jon E. Schoenfield at Mon Oct 10 09:36:03 EDT 2022
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| NAME
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"Tree" sequence in a 90 -degree sector of the cellular automaton of A160720.
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| COMMENTS
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Conjecture: this is also the "tree" sequence in a 120 -degree sector of the cellular automaton of A266532.
a(n) is also the total number of ON cells after n-th stage in the tree that arises from one of the four spokes in a 90 -degree sector of the cellular automaton A160720 on the square grid.
Conjecture: a(n) is also the total number of Y-toothpicks after n-th stage in the tree that arises from one of the three spokes in a 120 -degree sector of the cellular automaton of A266532 on the triangular grid.
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| FORMULA
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All of the above conjectures are true - _. - _N. J. A. Sloane_, Jan 23 2016
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approved
editing
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#55 by Bruno Berselli at Thu Sep 26 04:18:08 EDT 2019
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#54 by Michel Marcus at Thu Sep 26 04:13:54 EDT 2019
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#53 by Michel Marcus at Thu Sep 26 04:13:50 EDT 2019
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a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the of the second. For example, the a(1) = 1 through a(6) = 12 pairs are:
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approved
editing
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#52 by N. J. A. Sloane at Sat Jul 27 14:57:51 EDT 2019
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a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary digitsexpansion of the first are a subset of the positions of 1's in the reversed binary digitsexpansion of the of the second. For example, the a(1) = 1 through a(6) = 12 pairs are:
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Discussion
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| Sat Jul 27
| 14:57
| OEIS Server: https://oeis.org/edit/global/2822
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#51 by N. J. A. Sloane at Sun Mar 31 02:27:52 EDT 2019
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#50 by Gus Wiseman at Sun Mar 31 01:12:33 EDT 2019
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#49 by Gus Wiseman at Sun Mar 31 00:15:39 EDT 2019
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From Gus Wiseman, Mar 31 2019: (Start)
a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary digits of the first are a subset of the positions of 1's in the reversed binary digits of the of the second. For example, the a(1) = 1 through a(6) = 12 pairs are:
(1,1) (1,1) (1,1) (1,1) (1,1) (1,1)
(2,2) (1,3) (1,3) (1,3) (1,3)
(2,2) (2,2) (1,5) (1,5)
(2,3) (2,3) (2,2) (2,2)
(3,3) (3,3) (2,3) (2,3)
(4,4) (3,3) (2,6)
(4,4) (3,3)
(4,5) (4,4)
(5,5) (4,5)
(4,6)
(5,5)
(6,6)
(End)
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| FORMULA
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(Conjecture) a(n) = A267610(n) + n. - Gus Wiseman, Mar 31 2019
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| MATHEMATICA
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Accumulate[Table[2^DigitCount[n, 2, 1]-1, {n, 0, 30}]] (* based on conjecture confirmed by Sloane, Gus Wiseman, Mar 31 2019 *)
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| CROSSREFS
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Cf. A006218, A019565, A070939, A080572, A267610, A267700.
Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325124.
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approved
editing
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