cp's OEIS Frontend

The sequence of row n satisfies a linear recurrence with constant coefficients of order A018804(n) for n>0.

Odd numbers of the form n*(n+1)/2.

For n such that n(n+1)/2 is odd see A042963 (congruent to 1 or 2 mod 4).

Even central polygonal numbers minus 1. - Omar E. Pol, Aug 17 2011

Odd generalized hexagonal numbers. - Omar E. Pol, Sep 24 2015

This is a permutation of natural numbers provided that the sequence A112046 contains only prime values [which is true] and every prime occurs infinitely many times there.

The sequence arises from a "hybrid" cellular automaton on the infinite square grid, which consist of two successive generations using toothpicks of length 2 (cf. A139250) followed by two successive generations using the rules of the D-toothpick sequence A220500.

In other words (and more precisely) we have that:

1) If n is congruent to 1 or 2 mod 4 (cf. A042963), for example: 1, 2, 5, 6, 9, 10, ..., the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.

2) If n is a positive integer of the form 4*k-1 (cf. A004767), for example: 3, 7, 11, 15, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) and eventually D-toothpicks of length sqrt(2)/2, in both cases the D-toothpicks are connected to the structure by their endpoints, in the same way as in the even-indexed stages of A220500.

3) If n is a positive multiple of 4 (cf. A008586) the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the odd-indexed stages of A220500.

a(n) is the total number of elements in the structure after n generations.

A289841 (the first differences) gives the number of elements added at n-th stage.

Note that after 19 generations the structure is a 72-gon which essentially looks like a diamond (as a square that has been rotated 45 degrees).

The surprising fact is that from n = 20 up to 27 the structure is gradually transformed into a square cross.

The diamond mentioned above can be interpreted as the center of the cross. The diamond has an area equal to 384 and it contains 222 polygonal regions (or enclosures) of 11 distinct shapes. Missing two heptagonal shapes which are in the arms of the square cross only.

In total the complex cross contains 13 distinct shapes of polygonal regions. There are ten polygonal shapes that have an infinite number of copies. On the other hand, three of these polygonal shapes have a finite number of copies because they are in the center of the cross only. For example: there are only four copies of the concave 14-gon, which is also the largest polygon in the structure.

For n => 27 the shape of the square cross remains forever because its four arms grow indefinitely.

Every arm has a minimum width equal to 8, and a maximum width equal to 12.

Every arm also has a periodic structure which can be dissected in infinitely many clusters of area equal to 64. Every cluster is a 30-gon that contains 40 polygonal regions of nine distinct shapes.

If n is a number of the form 8*k-3 (cf. A017101) and greater than 19, for example: 27, 35, 43, 51, ..., then at n-th stage a new cluster is finished in every arm of the cross.

The behavior is similar to A290220 and A294020 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. - Omar E. Pol, Oct 29 2017

The sequence arises from a "hybrid" cellular automaton, which consist essentially in two successive generations using the rules of the D-toothpick sequence A194270 followed by one generation using toothpicks of length 2.

On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.

For the next stages we have the following rules:

1) At stage 1 we place two D-toothpicks connected by their endpoints on the same diagonal.

2) If n is a number of the form 3*k + 2 (cf. A016789), for example: 2, 5, 8, 11, 14, ..., the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the even-indexed stages of A194270.

3) If n is a positive multiple of 3 (cf. A008585) the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.

4) If n is a number of the form 3*k + 1 (cf. A016777) and > 1, for example: 4, 7, 10, 13, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) connected to the structure by their endpoints, in the same way as in the odd-indexed stages of A194270.

a(n) is the total number of elements in the structure after n generations.

A290221 (the first differences) gives the number of elements added at n-th stage.

The surprising fact is that from n = 7 up to 9 the structure is gradually transformed into a square cross.

For n => 9 the shape of the square cross remains forever because its four arms grow indefinitely in the directions North, East, West and South.

Every arm has a width equal to 4.

Every arm also has a periodic structure which can be dissected in infinitely many clusters.

In total, the narrow cross contains five distinct shapes of polygonal regions. There are three polygonal shapes that have an infinite number of copies. On the other hand, two polygonal shapes have a finite number of copies because they are in the center of the cross only. they are the heptagon and the hexagon of area 5.

The structure looks like a square cross but it's simpler than the structure of the complex cross described in A289840.

The behavior is similar to A289840 and A294020 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. - Omar E. Pol, Oct 29 2017

Up to n=10^8, a(15) is the only zero term and a(1)=a(9) are the only terms for which a(n)=1. Can it be proved that any number can only appear a finite number of times in this sequence? [M. F. Hasler, Oct 20 2008]

Even terms occur at A014601, odd terms at A042963; A010873(a(n))=A021913(n+1). - Reinhard Zumkeller, Jun 05 2012

If squares occur, they must be at indexes != 2 or 5 (mod 8). - Roderick MacPhee, Jul 17 2017

For n = 0..6 the sequence is similar to some toothpick sequences.

The surprising fact is that for n >= 7 the sequence has periodic tail. More precisely, it has period 3: repeat [8, 16, 12]. This tail is in accordance with the expansion of the four arms of the cross.

This is essentially the first differences of A290221. The behavior is similar to A289841 and A294021 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence. - Omar E. Pol, Oct 29 2017

Quasipolynomial of order 2. - Charles R Greathouse IV, Jan 19 2012

The binomial transform is 0, 1, 5, 20,... which is A084850 with offset 1. - R. J. Mathar, Nov 26 2014

First differences in A010696.

The original name was: "The changepoint a(n) is the first predecessor from n in a binary tree with: a(n) mod 2 <> n mod 2."

In a binary tree (node(row,col)=2^(row-1)+(col-1))

_________________2__________________________________3________________

____8_______9_______10_______11_______12_______13_______14_______15__

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

any node has 2 successors and one predecessor. a(n) is the first predecessor from n (going back, step by step) with another last digit (in binary sight) as n.

The subsequences from a(2^k) to a(2^(k+1) - 1) are permutations from the natural numbers from 0 to 2^k-1.