cp's OEIS Frontend

The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.

This array defines a binary operation on the nonnegative integers that matches up the zeros in the binary representation of each operand (starting from the right, and including as many leading zeros as necessary) and concatenates the two (possibly null) strings of ones to the right of each matched pair of zeros. See the examples. - Peter Munn, Feb 14 2021.

As such it could be useful for implementing multiplication, say, in Turing machines, with a "tape-like" unary-binary encoding of the prime factorization of n (A156552). However, such representation is not very useful if addition or subtraction is also needed.

Let b(0) be the sequence 1,2,3,4. Proceeding by induction, let b(n) be a sequence of length 2^(n+2). Quarter b(n) into four blocks, A,B,C,D each of length 2^n, so that b(n) = ABCD. Then b(n+1) = ABCDADCB. [After Dean paper.] - Sean A. Irvine, Apr 20 2015

1 together with A004767. - Omar E. Pol, Feb 23 2014

Turn sequence of the alternate paperfolding curve. Davis and Knuth define the alternate paperfolding curve by folding a long strip of paper repeatedly in half alternately to the left side or right side, then unfolding it so each crease is 90 degrees (or other angle). a(n) is their d(n) at equation 4.2. Their equation 6.2 (varied to d(2) = -1 as described there) is equivalent to the definition here. The curve is drawn by a unit step forward, turn a(1)*90 degrees left, a unit step forward, turn a(2)*90 degrees left, and so on. - Kevin Ryde, Apr 18 2020

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

Corresponding greater members: A071699(n).

Via the partial fraction decomposition 1/((4n+3)*(4n+5)) = (1/2) *(1/(4n+3) -1/(4n+5)) we find 2*Sum_{n>=0} (-1)^n/a(n) = 2*Sum_{n>=0} (-1)^n/( (4*n+3)*(4*n+5) ) = 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ... = (1/1 + 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ..)-1 = Sum_{n>=0} (-1)^n/A016813(n) + Sum_{n>=0} (-1)^n/A004767(n) -1 = -1 + Sum_{n>=0} b(n)/n^1 where b(n) = 1, 0, 1, 0, -1, 0, -1, 0 is a sequence with period length 8, one of the Dirichlet L-series modulo 8. The alternating sum becomes -1 +L(m=8,r=4,s=1) = Pi*sqrt(2)/4-1 = A093954 - 1.

Pi = 4 - 8*Sum(1/a(n)) noted by Bronstein-Semendjajew for the variant a(n) = (4n-1)*(4n+1) starting at n=1. - Frank Ellermann, Sep 18 2011

The identity (16*n^2-1)^2 - (64*n^2-8)*(2*n)^2 = 1 can be written as a(n)^2 - A158487(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 09 2012

Sequence found by reading the line from 15, in the direction 15, 63,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Essentially the least common multiple of 4*n+1 and 4*n-1. - Colin Barker, Feb 11 2017

For the points that form the Pythagorean triple (for example see illustration n = 5, on the first quadrant at coordinate (4,3) and (3,4)), the transit of circumference occurs exactly at the corners, therefore there are no additional intersecting squares on the upper or lower rows (diagonally NE & SW directions) of such points.

If the center of the circle is instead chosen at the middle of a square grid centered at (1/2,0), the sequence will be 2*A004767(n-1).