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 triples are displayed in increasing order of perimeter (equivalently in increasing order of middle side) and if perimeters coincide then by increasing order of the smallest side; also, each triple (a, b, c) is in increasing order.

When b is prime, all the corresponding triples in A336750 are primitive triples.

The only right integer triangle in the data corresponds to the triple (3, 4, 5).

The number of primitive such triangles whose middle side = b is equal to A023022(b) for b >= 3.

For all the triples (primitive or not), miscellaneous properties and references, see A336750.

This sequence a(n) = f(D(n)) := ceiling(sqrt(4*D(n))), with D(n) > 0, not a square, given in A000037, is important i) for finding out whether an indefinite binary quadratic form with discriminant 4*D(n) is reduced and also ii) for finding the principal reduced form for discriminant 4*D(n). See the W. Lang link under A225953 for the definition of reduced in eq. (1), and the principal reduced form [1, b(n), - (D(n) - (b(n)/2)^2] with eq. b(n) given in eq. (5) (there the discriminant D = 4*D(n)).

Even a(n) appear (a(n) - 2)/2 times, odd a(n) appear (a(n) - 1)/2 times. See the second formula below.

Middle side of integer-sided triangles whose sides a < b < c are in arithmetic progression. For the corresponding triples and miscellaneous properties and references, see A336750. - Bernard Schott, Oct 07 2020

The triples of sides (a,b,c) with a < b < c are in increasing order of perimeter = 3*b, and if perimeter coincide, then by increasing order of the smallest side. This sequence lists the a's.

Equivalently: smallest side of integer-sided triangles such that b = (a+c)/2 with a < c.

a >= 2 and each side a appears a-1 times but not consecutively.

For each a = 3*k, k>=1, there exists exactly one right triangle (3*k, 4*k, 5*k) whose sides a < b < c are in arithmetic progression.

This sequence is not increasing a(6) = 5 for triangle with perimeter = 18 and a(7) = 4 for triangle with perimeter = 21. The smallest side is not an increasing function of the perimeter of these triangles.

For the corresponding triples and miscellaneous properties and references, see A336750.

Equivalently: perimeters of integer-sided triangles such that b = (a+c)/2 with a < c.

As perimeter = 3 * middle side, these perimeters p are all multiple of 3, and each term p appears floor((p-3)/6) = A004526((p-3)/3) consecutively.

For each perimeter = 12*k with k>0, there exists one right integer triangle whose triple is (3k, 4k, 5k).

For the corresponding primitive triples, miscellaneous properties and references, see A336750.

From Bernard Schott, Oct 10 2020: (Start)

Equivalently: number of integer-sided triangles whose sides a < b < c are in arithmetic progression with perimeter n.

Equivalently: number of integer-sided triangles such that b = (a+c)/2 with a < c and perimeter n.

All the perimeters are multiple of 3 because each perimeter = 3 * middle side b.

For each perimeter n = 12*k with k>0, there exists one and only one such right integer triangle whose triple is (3k, 4k, 5k).

For the corresponding primitive triples and miscellaneous properties and references, see A336750. (End)

The triples of sides (a,b,c) with a < b < c are in increasing order of perimeter = 3*b, and if perimeters coincide, then by increasing order of the smallest side. This sequence lists the c's.

Equivalently: largest side of integer-sided triangles such that b = (a+c)/2 with a < c.

c >= 4 and each largest side c appears floor((c-1)/3) = A002264(c-1) times but not consecutively.

For each c = 5*k, k>=1, there exists exactly one right triangle (3*k, 4*k, 5*k) whose sides a < b < c are in arithmetic progression.

For the corresponding primitive triples and miscellaneous properties and references, see A336750.

Equivalently: perimeters of primitive integer-sided triangles such that b = (a+c)/2 with a < c.

As perimeter = 3 * middle side, these perimeters p are all multiples of 3 and each term p appears consecutively A023022(p/3) = phi(p/3)/2 times for p >= 9.

Remark, when the middle side is prime, then the number of primitive triangles with a perimeter p = 3*b equals phi(p/3)/2 = (b-1)/2 = (p-3)/6 and in this case, all the triangles are primitive (see A336754).

For the corresponding primitive triples, miscellaneous properties, and references, see A336750.

Equivalently: number of primitive integer-sided triangles such that b = (a+c)/2 with a < c and perimeter = n.

As the perimeter of these triangles = 3*b where b is the middle side, a(n) >= 1 iff n = 3*b, with b >= 3.

When b is prime, all the triangles of perimeter n = 3*b are primitive, hence in this case: a(n) = A024164(n).

For the corresponding triples (primitive or not), miscellaneous properties and references, see A336750.

The only triangles with integer sides that have an angle equal to a whole number of degrees are triangles which have an angle of 60° (A335893), or an angle of 90° (A263728) or an angle of 120° as here (see Keith Selkirk link, p. 251).

The triples are displayed in nondecreasing order of largest side c, and if largest sides coincide then by increasing order of the smallest side a, hence, each triple (a, b, c) is in increasing order.

The corresponding metric relation between sides is c^2 = a^2 + a*b + b^2.

The triples (a, b, c) can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u:

-> a = u^2 - v^2

-> b = 2*u*v + v^2

-> c = u^2 + u*v + v^2.

Note that side c cannot be even when the triple is primitive as here.

The (3, 5, 7) triangle is the only primitive triangle with a 120-degree angle and with its integer sides in arithmetic progression (A336750). This smallest triple is obtained for u = 2 and v = 1.

The Fermat point of these triangles is vertex C, then distance FA+FB+FC = CA+CB = b+a is an integer.

If (a,b,c) is a primitive 120-triple, then both (a,a+b,c) and (a+b,b,c) are 60-triples in A335893, see Emrys Read link, lemma 2 p. 302.