cp's OEIS Frontend

Also number of distinct binary linear codes of length n and any dimension.

Equivalently, number of subgroups of the Abelian group (C_2)^n.

Let V_n be an n-dimensional vector space over a field with 2 elements. Let P(V_n) be the collection of all subspaces of V_n. Then a(n-1) is the number of times any given nonzero vector of V_n appears in P(V_n). - Geoffrey Critzer, Jun 05 2017

With V_n and P(V_n) as above, a(n) is also the cardinality of P(V_n). - Vaia Patta, Jun 25 2019

The spiral is also called the Ulam spiral, cf. A174344, A274923 (x and y coordinates). - M. F. Hasler, Oct 20 2019

The n-th positive integer occupies the point whose x- and y-coordinates are represented in the sequence by a(2n-1) and a(2n), respectively. - Robert G. Wilson v, Dec 03 2017

From Robert G. Wilson v, Dec 05 2017: (Start)

The cover of the March 1964 issue of Scientific American (see link) depicts the Ulam Spiral with a heavy black line separating the numbers from their non-sequential neighbors. The pairs of coordinates for the points on this line, assuming it starts at the origin, form this sequence, negated.

The first number which has an abscissa value of k beginning at 0: 1, 2, 10, 26, 50, 82, 122, 170, 226, 290, 362, 442, 530, 626, 730, 842, 962, ...; g.f.: -(x^3 +7x^2 -x +1)/(x-1)^3;

The first number which has an abscissa value of -k beginning at 0: 1, 5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, 677, 785, 901, ...; g.f.: -(5x^2 +2x +1)/(x-1)^3;

The first number which has an ordinate value of k beginning at 0: 1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, ...; g.f.: -(7x^2+1)/(x-1)^3;

The first number which has an ordinate value of -k beginning at 0: 1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, ...; g.f.: -(3x^2+4x+1)/(x-1)^3;

The union of the four sequences above is A033638.

(End)

Sequences A174344, A268038 and A274923 start with the integer 0 at the origin (0,0). One might then prefer offset 0 as to have (a(2n), a(2n+1)) as coordinates of the integer n. - M. F. Hasler, Oct 20 2019

This sequence can be read as an infinite table with 2 columns, where row n gives the x- and y-coordinate of the n-th point on the spiral. If the point at the origin has number 0, then the points with coordinates (n,n), (-n,n), (n,-n) and (n,-n) have numbers given by A002939(n) = 2n(2n-1): (0, 2, 12, 30, ...), A016742(n) = 4n^2: (0, 4, 16, 36, ...), A002943(n) = 2n(2n+1): (0, 6, 20, 42, ...) and A033996(n) = 4n(n+1): (0, 8, 24, 48, ...), respectively. - M. F. Hasler, Nov 02 2019

a(0)=2, a(1)=4; for n > 2, a(n) = a(n-1) + 1 if n is already in the sequence, a(n) = a(n-1) + 2 otherwise. [corrected by Jon E. Schoenfield, Jun 24 2018]

For n > 0, a(n) = ceiling((1 + sqrt(n))^2). Empirical observation. - Ronald S. Tiberio, Jun 24 2018

For n > 0, a(n) is the minimal number of thumbtacks needed to secure n square sheets of paper on a bulletin board - one thumbtack in each corner of each sheet - slight overlap allowed so that one thumbtack can secure up to four sheets. - Ronald S. Tiberio, Jun 24 2018

From Ya-Ping Lu, Mar 17 2022: (Start)

Integers m such that connecting the external lattice points of a square spiral with m lattice points forms a polyomino (m is indicated by * in the figure below).

.

37--36*-35*-34*-33*-32*-31

| |

38* 17--16*-15*-14*-13 30*

| | | |

39* 18* 5---4*--3 12* 29*

| | | | | |

40* 19* 6* 1---2* 11* 28*

| | | | |

41* 20* 7---8*--9*-10 27*

| | |

42* 21--22*-23*-24*-25*-26

|

43--44*-45*-46*-47*-48*-49* (End)

Fractal - deleting the first occurrence of each integer leaves the original sequence. Also, deleting all the 1's leaves the original sequence plus 1. New values occur at square indices. 1's occur at indices m^2+1 and m^2+m+1. Ordinal transform of A122196.

Except for its initial 1, A122197 is the natural fractal sequence of A002620; that is, A122197(n+1) is the number of the row of A194061 that contains n. See A194029 for definition of natural fractal sequence. - Clark Kimberling, Aug 12 2011

From Johannes W. Meijer, Sep 09 2013: (Start)

Triangle read by rows formed from antidiagonals of triangle A002260.

The row sums equal A008805(n-1) and the antidiagonal sums equal A211534(n+5). (End)

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.

In the example for n=6 detailed below, the orders of [6, k]_q are 1, 6, 9, 10, 9, 6, 1 for k = 0..6,

the maximum order 10 defining the row length.

Note that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.

A083480 is a variation illustrating a relationship with numeric partitions, A000041.

The rows are formed by the nonzero entries of the columns of A049597.

The coefficient of q^j in the Gaussian polynomial [n, m]q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n, k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer, May 14 2017

If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - Michael Somos, Jun 25 2017

First differences of A076644. Fractal - deleting the first occurrence of each integer leaves the original sequence. Also, original sequence plus 1. 1's occur at square indices. New values occur at indices m^2+1 and m^2+m+1.

Ordinal transform of A122197.

Row sums give A002620. - Gary W. Adamson, Nov 29 2008

From Gary W. Adamson, Dec 05 2009: (Start)

A122196 considered as an infinite lower triangular matrix * [1,2,3,...] =

A006918 starting (1, 2, 5, 8, 14, 20, 30, 40, ...).

Let A122196 = an infinite lower triangular matrix M; then lim_{n->infinity} M^n = A171238, a left-shifted vector considered as a matrix. (End)

A122196 is the fractal sequence associated with the dispersion A082156; that is, A122196(n) is the number of the row of A082156 that contains n. - Clark Kimberling, Aug 12 2011

From Johannes W. Meijer, Sep 09 2013: (Start)

The alternating row sums lead to A004524(n+2).

The antidiagonal sums equal A001840(n). (End)

The complement of a partition p[1] >= p[2] >=...>= p[k] is p[1]-p[2], p[1]-p[3], ..., p[1]-p[k]. Its Ferrers board emerges naturally from the Ferrers board of the given partition. The weight of a partition of n is n.

Sum of entries in row n is A000041(n) (the partition numbers).

Apparently, number of entries in row n is A033638(n-1) = 1 + floor((n-1)^2/4).

T(n,0) = A000005(n) = number of divisors of n.

T(n,1) = A070824(n+1).

Sum(k*T(n,k),k>0) = A188814(n).

Number of possible values is 1,2,3,5,7,10,13,17,21,... = A033638. Maximum distance divided by 2 is the same minus one, i.e., 0,1,2,4,6,9,12,16,20,... = A002620.

T. Kyle Petersen and Bridget Eileen Tenner proved that T(n,k) is also the number of permutations of n for which the sum of descent differences equals k. - Susanne Wienand, Sep 11 2014

A more detailed definition from Antti Karttunen, Dec 09 2015: (Start)

After a(1) = 1, for the next term always choose the smallest number k >= 1 such that neither k and a(n-1) nor k and a(A265400(n)) [in case A265400(n) > 0] are equal, and neither of these pairs occur anywhere adjacent to each other (horizontally or vertically) in so far constructed spiral. Here A265400(n) gives the index of the nearest horizontally or vertically adjacent inner neighbor of the n-th term in spiral, or 0 if n is one of the corner cases A033638.

The condition "... do not occur anywhere adjacent to each other (horizontally or vertically) in so far constructed spiral" can be more formally stated as: there is no such 1 < j < n, for which either the unordered pair {a(j),a(j-1)} or [in case A265400(j) > 0] also the unordered pair {a(j),a(A265400(j))} would be equal to either of the unordered pair {k,a(n-1)} or the unordered pair {k,a(A265400(n))} [in case A265400(n) > 0], where k is the term chosen for a(n). (See also my reference Scheme-implementation.)

(End)

This is the square spiral analogy of Pascal's triangle thought of as a table read by antidiagonals.