cp's OEIS Frontend

Number of elements in the set {k: 1 <= 2k <= n}.

Dimension of the space of weight 2n+4 cusp forms for Gamma_0(2).

Dimension of the space of weight 1 modular forms for Gamma_1(n+1).

Number of ways 2^n is expressible as r^2 - s^2 with s > 0. Proof: (r+s) and (r-s) both should be powers of 2, even and distinct hence a(2k) = a(2k-1) = (k-1) etc. - Amarnath Murthy, Sep 20 2002

Lengths of sides of Ulam square spiral; i.e., lengths of runs of equal terms in A063826. - Donald S. McDonald, Jan 09 2003

Number of partitions of n into two parts. A008619 gives partitions of n into at most two parts, so A008619(n) = a(n) + 1 for all n >= 0. Partial sums are A002620 (Quarter-squares). - Rick L. Shepherd, Feb 27 2004

a(n+1) is the number of 1's in the binary expansion of the Jacobsthal number A001045(n). - Paul Barry, Jan 13 2005

Number of partitions of n+1 into two distinct (nonzero) parts. Example: a(8) = 4 because we have [8,1],[7,2],[6,3] and [5,4]. - Emeric Deutsch, Apr 14 2006

Complement of A000035, since A000035(n)+2*a(n) = n. Also equal to the partial sums of A000035. - Hieronymus Fischer, Jun 01 2007

Number of binary bracelets of n beads, two of them 0. For n >= 2, a(n-2) is the number of binary bracelets of n beads, two of them 0, with 00 prohibited. - Washington Bomfim, Aug 27 2008

Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i]:=1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n+1) = (-1)^n det(A). - Milan Janjic, Jan 24 2010

From Clark Kimberling, Mar 10 2011: (Start)

Let RT abbreviate rank transform (A187224). Then

RT(this sequence) = A187484;

RT(this sequence without 1st term) = A026371;

RT(this sequence without 1st 2 terms) = A026367;

RT(this sequence without 1st 3 terms) = A026363. (End)

The diameter (longest path) of the n-cycle. - Cade Herron, Apr 14 2011

For n >= 3, a(n-1) is the number of two-color bracelets of n beads, three of them are black, having a diameter of symmetry. - Vladimir Shevelev, May 03 2011

Pelesko (2004) refers erroneously to this sequence instead of A008619. - M. F. Hasler, Jul 19 2012

Number of degree 2 irreducible characters of the dihedral group of order 2(n+1). - Eric M. Schmidt, Feb 12 2013

For n >= 3 the sequence a(n-1) is the number of non-congruent regions with infinite area in the exterior of a regular n-gon with all diagonals drawn. See A217748. - Martin Renner, Mar 23 2013

a(n) is the number of partitions of 2n into exactly 2 even parts. a(n+1) is the number of partitions of 2n into exactly 2 odd parts. This just rephrases the comment of E. Deutsch above. - Wesley Ivan Hurt, Jun 08 2013

Number of the distinct rectangles and square in a regular n-gon is a(n/2) for even n and n >= 4. For odd n, such number is zero, see illustration in link. - Kival Ngaokrajang, Jun 25 2013

x-coordinate from the image of the point (0,-1) after n reflections across the lines y = n and y = x respectively (alternating so that one reflection is applied on each step): (0,-1) -> (0,1) -> (1,0) -> (1,2) -> (2,1) -> (2,3) -> ... . - Wesley Ivan Hurt, Jul 12 2013

a(n) is the number of partitions of 2n into exactly two distinct odd parts. a(n-1) is the number of partitions of 2n into exactly two distinct even parts, n > 0. - Wesley Ivan Hurt, Jul 21 2013

a(n) is the number of permutations of length n avoiding 213, 231 and 312, or avoiding 213, 312 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014

Also a(n) is the number of different patterns of 2-color, 2-partition of n. - Ctibor O. Zizka, Nov 19 2014

Minimum in- and out-degree for a directed K_n (see link). - Jon Perry, Nov 22 2014

a(n) is also the independence number of the triangular graph T(n). - Luis Manuel Rivera Martínez, Mar 12 2015

For n >= 3, a(n+4) is the least positive integer m such that every m-element subset of {1,2,...,n} contains distinct i, j, k with i + j = k (equivalently, with i - j = k). - Rick L. Shepherd, Jan 24 2016

More generally, the ordinary generating function for the integers repeated k times is x^k/((1 - x)(1 - x^k)). - Ilya Gutkovskiy, Mar 21 2016

a(n) is the number of numbers of the form F(i)*F(j) between F(n+3) and F(n+4), where 2 < i < j and F = A000045 (Fibonacci numbers). - Clark Kimberling, May 02 2016

The arithmetic function v_2(n,2) as defined in A289187. - Robert Price, Aug 22 2017

a(n) is also the total domination number of the (n-3)-gear graph. - Eric W. Weisstein, Apr 07 2018

Consider the numbers 1, 2, ..., n; a(n) is the largest integer t such that these numbers can be arranged in a row so that all consecutive terms differ by at least t. Example: a(6) = a(7) = 3, because of respectively (4, 1, 5, 2, 6, 3) and (1, 5, 2, 6, 3, 7, 4) (see link BMO - Problem 2). - Bernard Schott, Mar 07 2020

a(n-1) is also the number of integer-sided triangles whose sides a < b < c are in arithmetic progression with a middle side b = n (see A307136). Example, for b = 4, there exists a(3) = 1 such triangle corresponding to Pythagorean triple (3, 4, 5). For the triples, miscellaneous properties and references, see A336750. - Bernard Schott, Oct 15 2020

For n >= 1, a(n-1) is the greatest remainder on division of n by any k in 1..n. - David James Sycamore, Sep 05 2021

Number of incongruent right triangles that can be formed from the vertices of a regular n-gon is given by a(n/2) for n even. For n odd such number is zero. For a regular n-gon, the number of incongruent triangles formed from its vertices is given by A069905(n). The number of incongruent acute triangles is given by A005044(n). The number of incongruent obtuse triangles is given by A008642(n-4) for n > 3 otherwise 0, with offset 0. - Frank M Jackson, Nov 26 2022

The inverse binomial transform is 0, 0, 1, -2, 4, -8, 16, -32, ... (see A122803). - R. J. Mathar, Feb 25 2023

The first term at which a step to a non-adjacent square is required is a(9) = 61; the previous square 59 has adjacent squares 31,32,33,58,60,93,94,95 of which only 31 is prime, but 31 has already been visited at a(7).

In the first 10 million terms the longest required step is from a(8165267) = 22147771, which has coordinates (-2353,1019) relative to the starting 1-square, to a(8165268) = 8236981 with coordinates (-1435,1355), a step of length sqrt(955620), approximately 977.6 units. See A331027 for the progression of step length records. If the maximum step distance between adjacent prime terms has a finite value or is unbounded as n increases is unknown. The largest difference between adjacent prime terms is for a(8176270) = 32960287 to a(8176271) = 18983957, a difference of 13976330.

In the first 10 million terms the smallest unvisited prime is 2701871, which has coordinates (44,822) relative to the starting 1-square. The smallest unvisited term is found to slowly increase as the number of steps increases, indicating that eventually all primes will be visited, although this is unknown. It may require an extremely large number of steps before all primes below a certain value are visited due to the decreasing likelihood of the walk taking the long steps required to visit those primes near the origin which were unvisited in earlier steps.

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

The eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

This sequence is the complement to A330979; here only composite numbers can be stepped to, while in A330979 only prime numbers can be stepped to. Due to the existence of many more composite numbers than primes the walk here forms a much tigher spiral and generally stays as close as possible to the origin. However the primes occasionally block this preferred path and causes the walk to detour away from the origin, which leaves gaps in the visited squares with composite numbers. Some of these gaps are eventually visited by later steps in the walk.

The first term at which a step to a non-adjacent square is required is a(154) = 74, which steps to a(155) = 158, a distance of sqrt(8) units away. The square with number 74 is surrounded by three primes 43,73,113 and five composites 44,72,75,112,114, all of which have been previously visited.

In the first 1 million terms the longest required step is from a(149464) = 64666, which has coordinates (-127,-22) relative to the starting 1-square, to a(149465) = 67774 with coordinates (-130,-43), a step of length sqrt(450), approximately 21.2 units. See A330782 for the progression of step length records. If the maximum step distance between adjacent composite terms has a finite value or is unbounded as n increases is unknown. The largest difference between adjacent composite terms is for a(650382) = 863400 to a(650383) = 939342, a difference of 75942.

In the first 1 million terms the smallest unvisited composite is 12, which is at coordinates (2,1) relative to the starting square. This square is surrounded by four primes so the walk is never required to step to it during the initial walk steps. See the image in the links. Given the composites become more frequent relative to the primes as n increases it would require a very large detour from the spiral pattern for this square to be visited, so it is likely, although unknown, this square will never be visited. However the link image for 1 million steps shows the path can make detours toward the central square when it is trapped by surrounding paths, so the possibility remains the inner unvisited squares could eventually be visited, although the number of walk steps required before such a detour occurs could be extremely large.

The eight possible orientations of the Ulam spiral can be derived by combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

In Ulam's lattice there are 8 numbers around any number.

Or "Spironacci numbers" for short. See also Spironacci polynomials, A265408. This sequence has an interesting growth rate, see A265370 and A265404. - Antti Karttunen, Dec 13 2015

Any grid point with relative coordinates (x,y) from the central grid point, which is numbered 1, and where the greatest common divisor (gcd) of |x| and |y| equals 1 will be visible from the central point. Grid points where gcd(|x|,|y|) > 1 will have another point directly between it and the central point and will thus not be visible.

In an infinite 2D square lattice the ratio of visible grid points to all points is 6/pi^2, the same as the probability of two random numbers being relative prime.

Any even number on the square spiral has 4 diagonally adjacent squares which contain an even number and thus, unless all four such squares have been previously visited, a step to one of those adjacent squares, the one containing the smallest number, will always be possible. Any visited square containing a prime number will need to step to, and be stepped to from, a square containing a multiple of that prime number.

In the first 10 million terms the longest required step is from a(97528) = 5981, a prime number which has coordinates (39,13) relative to the starting 1-square, to a(97529) = 167468 (27*5981), with coordinates (205,-18), a step of length sqrt(28517), approximately 168.9 units. This is an extremely large step length relative to the total number of steps taken up to that point - see the attached link image. It is not surpassed by any subsequent step up to 10 million steps. If the maximum step distance between adjacent terms has a finite value or is unbounded as n increases is unknown. The largest difference between terms is for a(9404208) = 8964653 to a(9404209) = 10485343, a difference of 1520690.

In the first 10 million terms the smallest unvisited square is 37, which has coordinates (-3,3) relative to the starting 1-square. It is unknown if this square, and similar unvisited squares near the origin, is eventually visited for very large values of n or is never visited. The longest run of diagonal steps in the same direction to adjacent smaller even numbers is 52, from a(3979714) = 5051162 to a(3979766) = 4594498.