cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A000217 Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also referred to as T(n) or C(n+1, 2) or binomial(n+1, 2) (preferred).
Also generalized hexagonal numbers: n*(2*n-1), n=0, +-1, +-2, +-3, ... Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. In this case k = 6. - Omar E. Pol, Sep 13 2011 and Aug 04 2012
Number of edges in complete graph of order n+1, K_{n+1}.
Number of legal ways to insert a pair of parentheses in a string of n letters. E.g., there are 6 ways for three letters: (a)bc, (ab)c, (abc), a(b)c, a(bc), ab(c). Proof: there are C(n+2,2) ways to choose where the parentheses might go, but n + 1 of them are illegal because the parentheses are adjacent. Cf. A002415.
For n >= 1, a(n) is also the genus of a nonsingular curve of degree n+2, such as the Fermat curve x^(n+2) + y^(n+2) = 1. - Ahmed Fares (ahmedfares(AT)my_deja.com), Feb 21 2001
From Harnack's theorem (1876), the number of branches of a nonsingular curve of order n is bounded by a(n-1)+1, and the bound can be achieved. See also A152947. - Benoit Cloitre, Aug 29 2002. Corrected by Robert McLachlan, Aug 19 2024
Number of tiles in the set of double-n dominoes. - Scott A. Brown, Sep 24 2002
Number of ways a chain of n non-identical links can be broken up. This is based on a similar problem in the field of proteomics: the number of ways a peptide of n amino acid residues can be broken up in a mass spectrometer. In general, each amino acid has a different mass, so AB and BC would have different masses. - James A. Raymond, Apr 08 2003
Triangular numbers - odd numbers = shifted triangular numbers; 1, 3, 6, 10, 15, 21, ... - 1, 3, 5, 7, 9, 11, ... = 0, 0, 1, 3, 6, 10, ... - Xavier Acloque, Oct 31 2003 [Corrected by Derek Orr, May 05 2015]
Centered polygonal numbers are the result of [number of sides * A000217 + 1]. E.g., centered pentagonal numbers (1,6,16,31,...) = 5 * (0,1,3,6,...) + 1. Centered heptagonal numbers (1,8,22,43,...) = 7 * (0,1,3,6,...) + 1. - Xavier Acloque, Oct 31 2003
Maximum number of lines formed by the intersection of n+1 planes. - Ron R. King, Mar 29 2004
Number of permutations of [n] which avoid the pattern 132 and have exactly 1 descent. - Mike Zabrocki, Aug 26 2004
Number of ternary words of length n-1 with subwords (0,1), (0,2) and (1,2) not allowed. - Olivier Gérard, Aug 28 2012
Number of ways two different numbers can be selected from the set {0,1,2,...,n} without repetition, or, number of ways two different numbers can be selected from the set {1,2,...,n} with repetition.
Conjecturally, 1, 6, 120 are the only numbers that are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005
Binomial transform is {0, 1, 5, 18, 56, 160, 432, ...}, A001793 with one leading zero. - Philippe Deléham, Aug 02 2005
Each pair of neighboring terms adds to a perfect square. - Zak Seidov, Mar 21 2006
Number of transpositions in the symmetric group of n+1 letters, i.e., the number of permutations that leave all but two elements fixed. - Geoffrey Critzer, Jun 23 2006
With rho(n):=exp(i*2*Pi/n) (an n-th root of 1) one has, for n >= 1, rho(n)^a(n) = (-1)^(n+1). Just use the triviality a(2*k+1) == 0 (mod (2*k+1)) and a(2*k) == k (mod (2*k)).
a(n) is the number of terms in the expansion of (a_1 + a_2 + a_3)^(n-1). - Sergio Falcon, Feb 12 2007
a(n+1) is the number of terms in the complete homogeneous symmetric polynomial of degree n in 2 variables. - Richard Barnes, Sep 06 2017
The number of distinct handshakes in a room with n+1 people. - Mohammad K. Azarian, Apr 12 2007 [corrected, Joerg Arndt, Jan 18 2016]
Equal to the rank (minimal cardinality of a generating set) of the semigroup PT_n\S_n, where PT_n and S_n denote the partial transformation semigroup and symmetric group on [n]. - James East, May 03 2007
a(n) gives the total number of triangles found when cevians are drawn from a single vertex on a triangle to the side opposite that vertex, where n = the number of cevians drawn+1. For instance, with 1 cevian drawn, n = 1+1 = 2 and a(n)= 2*(2+1)/2 = 3 so there is a total of 3 triangles in the figure. If 2 cevians are drawn from one point to the opposite side, then n = 1+2 = 3 and a(n) = 3*(3+1)/2 = 6 so there is a total of 6 triangles in the figure. - Noah Priluck (npriluck(AT)gmail.com), Apr 30 2007
For n >= 1, a(n) is the number of ways in which n-1 can be written as a sum of three nonnegative integers if representations differing in the order of the terms are considered to be different. In other words, for n >= 1, a(n) is the number of nonnegative integral solutions of the equation x + y + z = n-1. - Amarnath Murthy, Apr 22 2001 (edited by Robert A. Beeler)
a(n) is the number of levels with energy n + 3/2 (in units of h*f0, with Planck's constant h and the oscillator frequency f0) of the three-dimensional isotropic harmonic quantum oscillator. See the comment by A. Murthy above: n = n1 + n2 + n3 with positive integers and ordered. Proof from the o.g.f. See the A. Messiah reference. - Wolfdieter Lang, Jun 29 2007
From Hieronymus Fischer, Aug 06 2007: (Start)
Numbers m >= 0 such that round(sqrt(2m+1)) - round(sqrt(2m)) = 1.
Numbers m >= 0 such that ceiling(2*sqrt(2m+1)) - 1 = 1 + floor(2*sqrt(2m)).
Numbers m >= 0 such that fract(sqrt(2m+1)) > 1/2 and fract(sqrt(2m)) < 1/2, where fract(x) is the fractional part of x (i.e., x - floor(x), x >= 0). (End)
If Y and Z are 3-blocks of an n-set X, then, for n >= 6, a(n-1) is the number of (n-2)-subsets of X intersecting both Y and Z. - Milan Janjic, Nov 09 2007
Equals row sums of triangle A143320, n > 0. - Gary W. Adamson, Aug 07 2008
a(n) is also an even perfect number in A000396 iff n is a Mersenne prime A000668. - Omar E. Pol, Sep 05 2008. Unnecessary assumption removed and clarified by Rick L. Shepherd, Apr 14 2025
Equals row sums of triangle A152204. - Gary W. Adamson, Nov 29 2008
The number of matches played in a round robin tournament: n*(n-1)/2 gives the number of matches needed for n players. Everyone plays against everyone else exactly once. - Georg Wrede (georg(AT)iki.fi), Dec 18 2008
-a(n+1) = E(2)*binomial(n+2,2) (n >= 0) where E(n) are the Euler numbers in the enumeration A122045. Viewed this way, a(n) is the special case k=2 in the sequence of diagonals in the triangle A153641. - Peter Luschny, Jan 06 2009
Equivalent to the first differences of successive tetrahedral numbers. See A000292. - Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009
The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus a(k) = |2^(-3)(P(2,1)-(-1)^k P(2,2k+1))|. - Peter Luschny, Jul 12 2009
a(n) is the smallest number > a(n-1) such that gcd(n,a(n)) = gcd(n,a(n-1)). If n is odd this gcd is n; if n is even it is n/2. - Franklin T. Adams-Watters, Aug 06 2009
Partial sums of A001477. - Juri-Stepan Gerasimov, Jan 25 2010. [A-number corrected by Omar E. Pol, Jun 05 2012]
The numbers along the right edge of Floyd's triangle are 1, 3, 6, 10, 15, .... - Paul Muljadi, Jan 25 2010
From Charlie Marion, Dec 03 2010: (Start)
More generally, a(2k+1) == j*(2j-1) (mod 2k+2j+1) and
a(2k) == [-k + 2j*(j-1)] (mod 2k+2j).
Column sums of:
1 3 5 7 9 ...
1 3 5 ...
1 ...
...............
---------------
1 3 6 10 15 ...
Sum_{n>=1} 1/a(n)^2 = 4*Pi^2/3-12 = 12 less than the volume of a sphere with radius Pi^(1/3).
(End)
A004201(a(n)) = A000290(n); A004202(a(n)) = A002378(n). - Reinhard Zumkeller, Feb 12 2011
1/a(n+1), n >= 0, has e.g.f. -2*(1+x-exp(x))/x^2, and o.g.f. 2*(x+(1-x)*log(1-x))/x^2 (see the Stephen Crowley formula line). -1/(2*a(n+1)) is the z-sequence for the Sheffer triangle of the coefficients of the Bernoulli polynomials A196838/A196839. - Wolfdieter Lang, Oct 26 2011
From Charlie Marion, Feb 23 2012: (Start)
a(n) + a(A002315(k)*n + A001108(k+1)) = (A001653(k+1)*n + A001109(k+1))^2. For k=0 we obtain a(n) + a(n+1) = (n+1)^2 (identity added by N. J. A. Sloane on Feb 19 2004).
a(n) + a(A002315(k)*n - A055997(k+1)) = (A001653(k+1)*n - A001109(k))^2.
(End)
Plot the three points (0,0), (a(n), a(n+1)), (a(n+1), a(n+2)) to form a triangle. The area will be a(n+1)/2. - J. M. Bergot, May 04 2012
The sum of four consecutive triangular numbers, beginning with a(n)=n*(n+1)/2, minus 2 is 2*(n+2)^2. a(n)*a(n+2)/2 = a(a(n+1)-1). - J. M. Bergot, May 17 2012
(a(n)*a(n+3) - a(n+1)*a(n+2))*(a(n+1)*a(n+4) - a(n+2)*a(n+3))/8 = a((n^2+5*n+4)/2). - J. M. Bergot, May 18 2012
a(n)*a(n+1) + a(n+2)*a(n+3) + 3 = a(n^2 + 4*n + 6). - J. M. Bergot, May 22 2012
In general, a(n)*a(n+1) + a(n+k)*a(n+k+1) + a(k-1)*a(k) = a(n^2 + (k+2)*n + k*(k+1)). - Charlie Marion, Sep 11 2012
a(n)*a(n+3) + a(n+1)*a(n+2) = a(n^2 + 4*n + 2). - J. M. Bergot, May 22 2012
In general, a(n)*a(n+k) + a(n+1)*a(n+k-1) = a(n^2 + (k+1)*n + k-1). - Charlie Marion, Sep 11 2012
a(n)*a(n+2) + a(n+1)*a(n+3) = a(n^2 + 4*n + 3). - J. M. Bergot, May 22 2012
Three points (a(n),a(n+1)), (a(n+1),a(n)) and (a(n+2),a(n+3)) form a triangle with area 4*a(n+1). - J. M. Bergot, May 23 2012
a(n) + a(n+k) = (n+k)^2 - (k^2 + (2n-1)*k -2n)/2. For k=1 we obtain a(n) + a(n+1) = (n+1)^2 (see below). - Charlie Marion, Oct 02 2012
In n-space we can define a(n-1) nontrivial orthogonal projections. For example, in 3-space there are a(2)=3 (namely point onto line, point onto plane, line onto plane). - Douglas Latimer, Dec 17 2012
From James East, Jan 08 2013: (Start)
For n >= 1, a(n) is equal to the rank (minimal cardinality of a generating set) and idempotent rank (minimal cardinality of an idempotent generating set) of the semigroup P_n\S_n, where P_n and S_n denote the partition monoid and symmetric group on [n].
For n >= 3, a(n-1) is equal to the rank and idempotent rank of the semigroup T_n\S_n, where T_n and S_n denote the full transformation semigroup and symmetric group on [n].
(End)
For n >= 3, a(n) is equal to the rank and idempotent rank of the semigroup PT_n\S_n, where PT_n and S_n denote the partial transformation semigroup and symmetric group on [n]. - James East, Jan 15 2013
Conjecture: For n > 0, there is always a prime between A000217(n) and A000217(n+1). Sequence A065383 has the first 1000 of these primes. - Ivan N. Ianakiev, Mar 11 2013
The formula, a(n)*a(n+4k+2)/2 + a(k) = a(a(n+2k+1) - (k^2+(k+1)^2)), is a generalization of the formula a(n)*a(n+2)/2 = a(a(n+1)-1) in Bergot's comment dated May 17 2012. - Charlie Marion, Mar 28 2013
The series Sum_{k>=1} 1/a(k) = 2, given in a formula below by Jon Perry, Jul 13 2003, has partial sums 2*n/(n+1) (telescopic sum) = A022998(n)/A026741(n+1). - Wolfdieter Lang, Apr 09 2013
For odd m = 2k+1, we have the recurrence a(m*n + k) = m^2*a(n) + a(k). Corollary: If number T is in the sequence then so is 9*T+1. - Lekraj Beedassy, May 29 2013
Euler, in Section 87 of the Opera Postuma, shows that whenever T is a triangular number then 9*T + 1, 25*T + 3, 49*T + 6 and 81*T + 10 are also triangular numbers. In general, if T is a triangular number then (2*k + 1)^2*T + k*(k + 1)/2 is also a triangular number. - Peter Bala, Jan 05 2015
Using 1/b and 1/(b+2) will give a Pythagorean triangle with sides 2*b + 2, b^2 + 2*b, and b^2 + 2*b + 2. Set b=n-1 to give a triangle with sides of lengths 2*n,n^2-1, and n^2 + 1. One-fourth the perimeter = a(n) for n > 1. - J. M. Bergot, Jul 24 2013
a(n) = A028896(n)/6, where A028896(n) = s(n) - s(n-1) are the first differences of s(n) = n^3 + 3*n^2 + 2*n - 8. s(n) can be interpreted as the sum of the 12 edge lengths plus the sum of the 6 face areas plus the volume of an n X (n-1) X (n-2) rectangular prism. - J. M. Bergot, Aug 13 2013
Dimension of orthogonal group O(n+1). - Eric M. Schmidt, Sep 08 2013
Number of positive roots in the root system of type A_n (for n > 0). - Tom Edgar, Nov 05 2013
A formula for the r-th successive summation of k, for k = 1 to n, is binomial(n+r,r+1) [H. W. Gould]. - Gary Detlefs, Jan 02 2014
Also the alternating row sums of A095831. Also the alternating row sums of A055461, for n >= 1. - Omar E. Pol, Jan 26 2014
For n >= 3, a(n-2) is the number of permutations of 1,2,...,n with the distribution of up (1) - down (0) elements 0...011 (n-3 zeros), or, the same, a(n-2) is up-down coefficient {n,3} (see comment in A060351). - Vladimir Shevelev, Feb 14 2014
a(n) is the dimension of the vector space of symmetric n X n matrices. - Derek Orr, Mar 29 2014
Non-vanishing subdiagonal of A132440^2/2, aside from the initial zero. First subdiagonal of unsigned A238363. Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices of complete graphs. - Tom Copeland, Apr 05 2014
The number of Sidon subsets of {1,...,n+1} of size 2. - Carl Najafi, Apr 27 2014
Number of factors in the definition of the Vandermonde determinant V(x_1,x_2,...,x_n) = Product_{1 <= i < k <= n} x_i - x_k. - Tom Copeland, Apr 27 2014
Number of weak compositions of n into three parts. - Robert A. Beeler, May 20 2014
Suppose a bag contains a(n) red marbles and a(n+1) blue marbles, where a(n), a(n+1) are consecutive triangular numbers. Then, for n > 0, the probability of choosing two marbles at random and getting two red or two blue is 1/2. In general, for k > 2, let b(0) = 0, b(1) = 1 and, for n > 1, b(n) = (k-1)*b(n-1) - b(n-2) + 1. Suppose, for n > 0, a bag contains b(n) red marbles and b(n+1) blue marbles. Then the probability of choosing two marbles at random and getting two red or two blue is (k-1)/(k+1). See also A027941, A061278, A089817, A053142, A092521. - Charlie Marion, Nov 03 2014
Let O(n) be the oblong number n(n+1) = A002378 and S(n) the square number n^2 = A000290(n). Then a(4n) = O(3n) - O(n), a(4n+1) = S(3n+1) - S(n), a(4n+2) = S(3n+2) - S(n+1) and a(4n+3) = O(3n+2) - O(n). - Charlie Marion, Feb 21 2015
Consider the partition of the natural numbers into parts from the set S=(1,2,3,...,n). The length (order) of the signature of the resulting sequence is given by the triangular numbers. E.g., for n=10, the signature length is 55. - David Neil McGrath, May 05 2015
a(n) counts the partitions of (n-1) unlabeled objects into three (3) parts (labeled a,b,c), e.g., a(5)=15 for (n-1)=4. These are (aaaa),(bbbb),(cccc),(aaab),(aaac),(aabb),(aacc),(aabc),(abbc),(abcc),(abbb),(accc),(bbcc),(bccc),(bbbc). - David Neil McGrath, May 21 2015
Conjecture: the sequence is the genus/deficiency of the sinusoidal spirals of index n which are algebraic curves. The value 0 corresponds to the case of the Bernoulli Lemniscate n=2. So the formula conjectured is (n-1)(n-2)/2. - Wolfgang Tintemann, Aug 02 2015
Conjecture: Let m be any positive integer. Then, for each n = 1,2,3,... the set {Sum_{k=s..t} 1/k^m: 1 <= s <= t <= n} has cardinality a(n) = n*(n+1)/2; in other words, all the sums Sum_{k=s..t} 1/k^m with 1 <= s <= t are pairwise distinct. (I have checked this conjecture via a computer and found no counterexample.) - Zhi-Wei Sun, Sep 09 2015
The Pisano period lengths of reading the sequence modulo m seem to be A022998(m). - R. J. Mathar, Nov 29 2015
For n >= 1, a(n) is the number of compositions of n+4 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
In this sequence only 3 is prime. - Fabian Kopp, Jan 09 2016
Suppose you are playing Bulgarian Solitaire (see A242424 and Chamberland's and Gardner's books) and, for n > 0, you are starting with a single pile of a(n) cards. Then the number of operations needed to reach the fixed state {n, n-1,...,1} is a(n-1). For example, {6}->{5,1}->{4,2}->{3,2,1}. - Charlie Marion, Jan 14 2016
Numbers k such that 8k + 1 is a square. - Juri-Stepan Gerasimov, Apr 09 2016
Every perfect cube is the difference of the squares of two consecutive triangular numbers. 1^2-0^2 = 1^3, 3^2-1^2 = 2^3, 6^2-3^2 = 3^3. - Miquel Cerda, Jun 26 2016
For n > 1, a(n) = tau_n(k*) where tau_n(k) is the number of ordered n-factorizations of k and k* is the square of a prime. For example, tau_3(4) = tau_3(9) = tau_3(25) = tau_3(49) = 6 (see A007425) since the number of divisors of 4, 9, 25, and 49's divisors is 6, and a(3) = 6. - Melvin Peralta, Aug 29 2016
In an (n+1)-dimensional hypercube, number of two-dimensional faces congruent with a vertex (see also A001788). - Stanislav Sykora, Oct 23 2016
Generalizations of the familiar formulas, a(n) + a(n+1) = (n+1)^2 (Feb 19 2004) and a(n)^2 + a(n+1)^2 = a((n+1)^2) (Nov 22 2006), follow: a(n) + a(n+2k-1) + 4a(k-1) = (n+k)^2 + 6a(k-1) and a(n)^2 + a(n+2k-1)^2 + (4a(k-1))^2 + 3a(k-1) = a((n+k)^2 + 6a(k-1)). - Charlie Marion, Nov 27 2016
a(n) is also the greatest possible number of diagonals in a polyhedron with n+4 vertices. - Vladimir Letsko, Dec 19 2016
For n > 0, 2^5 * (binomial(n+1,2))^2 represents the first integer in a sum of 2*(2*n + 1)^2 consecutive integers that equals (2*n + 1)^6. - Patrick J. McNab, Dec 25 2016
Does not satisfy Benford's law (cf. Ross, 2012). - N. J. A. Sloane, Feb 12 2017
Number of ordered triples (a,b,c) of positive integers not larger than n such that a+b+c = 2n+1. - Aviel Livay, Feb 13 2017
Number of inequivalent tetrahedral face colorings using at most n colors so that no color appears only once. - David Nacin, Feb 22 2017
Also the Wiener index of the complete graph K_{n+1}. - Eric W. Weisstein, Sep 07 2017
Number of intersections between the Bernstein polynomials of degree n. - Eric Desbiaux, Apr 01 2018
a(n) is the area of a triangle with vertices at (1,1), (n+1,n+2), and ((n+1)^2, (n+2)^2). - Art Baker, Dec 06 2018
For n > 0, a(n) is the smallest k > 0 such that n divides numerator of (1/a(1) + 1/a(2) + ... + 1/a(n-1) + 1/k). It should be noted that 1/1 + 1/3 + 1/6 + ... + 2/(n(n+1)) = 2n/(n+1). - Thomas Ordowski, Aug 04 2019
Upper bound of the number of lines in an n-homogeneous supersolvable line arrangement (see Theorem 1.1 in Dimca). - Stefano Spezia, Oct 04 2019
For n > 0, a(n+1) is the number of lattice points on a triangular grid with side length n. - Wesley Ivan Hurt, Aug 12 2020
From Michael Chu, May 04 2022: (Start)
Maximum number of distinct nonempty substrings of a string of length n.
Maximum cardinality of the sumset A+A, where A is a set of n numbers. (End)
a(n) is the number of parking functions of size n avoiding the patterns 123, 132, and 312. - Lara Pudwell, Apr 10 2023
Suppose two rows, each consisting of n evenly spaced dots, are drawn in parallel. Suppose we bijectively draw lines between the dots of the two rows. For n >= 1, a(n - 1) is the maximal possible number of intersections between the lines. Equivalently, the maximal number of inversions in a permutation of [n]. - Sela Fried, Apr 18 2023
The following equation complements the generalization in Bala's Comment (Jan 05 2015). (2k + 1)^2*a(n) + a(k) = a((2k + 1)*n + k). - Charlie Marion, Aug 28 2023
a(n) + a(n+k) + a(k-1) + (k-1)*n = (n+k)^2. For k = 1, we have a(n) + a(n+1) = (n+1)^2. - Charlie Marion, Nov 17 2023
a(n+1)/3 is the expected number of steps to escape from a linear row of n positions starting at a random location and randomly performing steps -1 or +1 with equal probability. - Hugo Pfoertner, Jul 22 2025
a(n+1) is the number of nonnegative integer solutions to p + q + r = n. By Sylvester's law of inertia, it is also the number of congruence classes of real symmetric n-by-n matrices or equivalently, the number of symmetric bilinear forms on a real n-dimensional vector space. - Paawan Jethva, Jul 24 2025

Examples

			G.f.: x + 3*x^2 + 6*x^3 + 10*x^4 + 15*x^5 + 21*x^6 + 28*x^7 + 36*x^8 + 45*x^9 + ...
When n=3, a(3) = 4*3/2 = 6.
Example(a(4)=10): ABCD where A, B, C and D are different links in a chain or different amino acids in a peptide possible fragments: A, B, C, D, AB, ABC, ABCD, BC, BCD, CD = 10.
a(2): hollyhock leaves on the Tokugawa Mon, a(4): points in Pythagorean tetractys, a(5): object balls in eight-ball billiards. - _Bradley Klee_, Aug 24 2015
From _Gus Wiseman_, Oct 28 2020: (Start)
The a(1) = 1 through a(5) = 15 ordered triples of positive integers summing to n + 2 [Beeler, McGrath above] are the following. These compositions are ranked by A014311.
  (111)  (112)  (113)  (114)  (115)
         (121)  (122)  (123)  (124)
         (211)  (131)  (132)  (133)
                (212)  (141)  (142)
                (221)  (213)  (151)
                (311)  (222)  (214)
                       (231)  (223)
                       (312)  (232)
                       (321)  (241)
                       (411)  (313)
                              (322)
                              (331)
                              (412)
                              (421)
                              (511)
The unordered version is A001399(n-3) = A069905(n), with Heinz numbers A014612.
The strict case is A001399(n-6)*6, ranked by A337453.
The unordered strict case is A001399(n-6), with Heinz numbers A007304.
(End)

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • C. Alsina and R. B. Nelson, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010. See Chapter 1.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 109ff.
  • Marc Chamberland, Single Digits: In Praise of Small Numbers, Chapter 3, The Number Three, p. 72, Princeton University Press, 2015.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 33, 38, 40, 70.
  • J. M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 309 pp 46-196, Ellipses, Paris, 2004
  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.
  • Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.
  • James Gleick, The Information: A History, A Theory, A Flood, Pantheon, 2011. [On page 82 mentions a table of the first 19999 triangular numbers published by E. de Joncort in 1762.]
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.6 Mathematical Proof and §8.6 Figurate Numbers, pp. 158-159, 289-290.
  • Cay S. Horstmann, Scala for the Impatient. Upper Saddle River, New Jersey: Addison-Wesley (2012): 171.
  • Elemer Labos, On the number of RGB-colors we can distinguish. Partition Spectra. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. Jul 06 2005.
  • A. Messiah, Quantum Mechanics, Vol.1, North Holland, Amsterdam, 1965, p. 457.
  • J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
  • Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 52-53, 129-132, 274.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 2-6, 13.
  • T. Trotter, Some Identities for the Triangular Numbers, Journal of Recreational Mathematics, Spring 1973, 6(2).
  • D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 91-93 Penguin Books 1987.

Links

Crossrefs

The figurate numbers, with parameter k as in the second Python program: A001477 (k=0), this sequence (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6), A001106 (k=7), A001107 (k=8).
Cf. A000096, A000124, A000292, A000330, A000396, A000668, A001082, A001788, A002024, A002378, A002415, A003056 (inverse function), A004526, A006011, A007318, A008953, A008954, A010054 (characteristic function), A028347, A036666, A046092, A051942, A055998, A055999, A056000, A056115, A056119, A056121, A056126, A062717, A087475, A101859, A109613, A143320, A210569, A245031, A245300, A060544, A016754.
a(n) = A110449(n, 0).
a(n) = A110555(n+2, 2).
A diagonal of A008291.
Column 2 of A195152.
Numbers of the form n*t(n+k,h)-(n+k)*t(n,h), where t(i,h) = i*(i+2*h+1)/2 for any h (for A000217 is k=1): A005563, A067728, A140091, A140681, A212331.
Boustrophedon transforms: A000718, A000746.
Iterations: A007501 (start=2), A013589 (start=4), A050542 (start=5), A050548 (start=7), A050536 (start=8), A050909 (start=9).
Cf. A002817 (doubly triangular numbers), A075528 (solutions of a(n)=a(m)/2).
Cf. A104712 (first column, starting with a(1)).
Some generalized k-gonal numbers are A001318 (k=5), this sequence (k=6), A085787 (k=7), etc.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A001399(n-6) = A069905(n-3) = A211540(n-1) counts 3-part strict partitions.
A011782 counts compositions of any length.
A337461 counts pairwise coprime triples, with unordered version A307719.
Cf. A000212, A001840, A007304, A156040, A220377, A337483, A337604.
Cf. A099174, A130534, A132440, A238363.

Programs

  • Haskell
    a000217 n = a000217_list !! n
a000217_list = scanl1 (+) [0..] -- Reinhard Zumkeller, Sep 23 2011
  • J
    a000217=: *-:@>: NB. Stephen Makdisi, May 02 2018
  • Magma
    [n*(n+1)/2: n in [0..60]]; // Bruno Berselli, Jul 11 2014
  • Magma
    [n: n in [0..1500] | IsSquare(8*n+1)]; // Juri-Stepan Gerasimov, Apr 09 2016
  • Maple
    A000217 := proc(n) n*(n+1)/2; end;
istriangular:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then return true else return false; end if; end proc; # N. J. A. Sloane, May 25 2008
ZL := [S, {S=Prod(B, B, B), B=Set(Z, 1 <= card)}, unlabeled]:
seq(combstruct[count](ZL, size=n), n=2..55); # Zerinvary Lajos, Mar 24 2007
isA000217 := proc(n)
    issqr(1+8*n) ;
end proc: # R. J. Mathar, Nov 29 2015 [This is the recipe Leonhard Euler proposes in chapter VII of his "Vollständige Anleitung zur Algebra", 1765. Peter Luschny, Sep 02 2022]
  • Mathematica
    Array[ #*(# - 1)/2 &, 54] (* Zerinvary Lajos, Jul 10 2009 *)
FoldList[#1 + #2 &, 0, Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[Range[0,70]] (* Harvey P. Dale, Sep 09 2012 *)
CoefficientList[Series[x / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 30 2014 *)
(* For Mathematica 10.4+ *) Table[PolygonalNumber[n], {n, 0, 53}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
LinearRecurrence[{3, -3, 1}, {0, 1, 3}, 54] (* Robert G. Wilson v, Dec 04 2016 *)
(* The following Mathematica program, courtesy of Steven J. Miller, is useful for testing if a sequence is Benford. To test a different sequence only one line needs to be changed. This strongly suggests that the triangular numbers are not Benford, since the second and third columns of the output disagree. - N. J. A. Sloane, Feb 12 2017 *)
fd[x_] := Floor[10^Mod[Log[10, x], 1]]
benfordtest[num_] := Module[{},
   For[d = 1, d <= 9, d++, digit[d] = 0];
   For[n = 1, n <= num, n++,
    {
     d = fd[n(n+1)/2];
     If[d != 0, digit[d] = digit[d] + 1];
     }];
   For[d = 1, d <= 9, d++, digit[d] = 1.0 digit[d]/num];
   For[d = 1, d <= 9, d++,
    Print[d, " ", 100.0 digit[d], " ", 100.0 Log[10, (d + 1)/d]]];
   ];
benfordtest[20000]
Table[Length[Join@@Permutations/@IntegerPartitions[n,{3}]],{n,0,15}] (* Gus Wiseman, Oct 28 2020 *)
  • PARI
    A000217(n) = n * (n + 1) / 2;
  • PARI
    is_A000217(n)=n*2==(1+n=sqrtint(2*n))*n \\ M. F. Hasler, May 24 2012
  • PARI
    is(n)=ispolygonal(n,3) \\ Charles R Greathouse IV, Feb 28 2014
  • PARI
    list(lim)=my(v=List(),n,t); while((t=n*n++/2)<=lim,listput(v,t)); Vec(v) \\ Charles R Greathouse IV, Jun 18 2021
  • Python
    for n in range(0,60): print(n*(n+1)//2, end=', ') # Stefano Spezia, Dec 06 2018
  • Python
    # Intended to compute the initial segment of the sequence, not
# isolated terms. If in the iteration the line "x, y = x + y + 1, y + 1"
# is replaced by "x, y = x + y + k, y + k" then the figurate numbers are obtained,
# for k = 0 (natural A001477), k = 1 (triangular), k = 2 (squares), k = 3 (pentagonal), k = 4 (hexagonal), k = 5 (heptagonal), k = 6 (octagonal), etc.
def aList():
    x, y = 1, 1
    yield 0
    while True:
        yield x
        x, y = x + y + 1, y + 1
A000217 = aList()
print([next(A000217) for i in range(54)]) # Peter Luschny, Aug 03 2019
  • SageMath
    [n*(n+1)/2 for n in (0..60)] # Bruno Berselli, Jul 11 2014
  • Scala
    (1 to 53).scanLeft(0)( + ) // Horstmann (2012), p. 171
  • Scheme
    (define (A000217 n) (/ (* n (+ n 1)) 2)) ;; Antti Karttunen, Jul 08 2017

Formula

G.f.: x/(1-x)^3. - Simon Plouffe in his 1992 dissertation
E.g.f.: exp(x)*(x+x^2/2).
a(n) = a(-1-n).
a(n) + a(n-1)*a(n+1) = a(n)^2. - Terrel Trotter, Jr., Apr 08 2002
a(n) = (-1)^n*Sum_{k=1..n} (-1)^k*k^2. - Benoit Cloitre, Aug 29 2002
a(n+1) = ((n+2)/n)*a(n), Sum_{n>=1} 1/a(n) = 2. - Jon Perry, Jul 13 2003
For n > 0, a(n) = A001109(n) - Sum_{k=0..n-1} (2*k+1)*A001652(n-1-k); e.g., 10 = 204 - (1*119 + 3*20 + 5*3 + 7*0). - Charlie Marion, Jul 18 2003
With interpolated zeros, this is n*(n+2)*(1+(-1)^n)/16. - Benoit Cloitre, Aug 19 2003
a(n+1) is the determinant of the n X n symmetric Pascal matrix M_(i, j) = binomial(i+j+1, i). - Benoit Cloitre, Aug 19 2003
a(n) = ((n+1)^3 - n^3 - 1)/6. - Xavier Acloque, Oct 24 2003
a(n) = a(n-1) + (1 + sqrt(1 + 8*a(n-1)))/2. This recursive relation is inverted when taking the negative branch of the square root, i.e., a(n) is transformed into a(n-1) rather than a(n+1). - Carl R. White, Nov 04 2003
a(n) = Sum_{k=1..n} phi(k)*floor(n/k) = Sum_{k=1..n} A000010(k)*A010766(n, k) (R. Dedekind). - Vladeta Jovovic, Feb 05 2004
a(n) + a(n+1) = (n+1)^2. - N. J. A. Sloane, Feb 19 2004
a(n) = a(n-2) + 2*n - 1. - Paul Barry, Jul 17 2004
a(n) = sqrt(Sum_{i=1..n} Sum_{j=1..n} (i*j)) = sqrt(A000537(n)). - Alexander Adamchuk, Oct 24 2004
a(n) = sqrt(sqrt(Sum_{i=1..n} Sum_{j=1..n} (i*j)^3)) = (Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (i*j*k)^3)^(1/6). - Alexander Adamchuk, Oct 26 2004
a(n) == 1 (mod n+2) if n is odd and a(n) == n/2+2 (mod n+2) if n is even. - Jon Perry, Dec 16 2004
a(0) = 0, a(1) = 1, a(n) = 2*a(n-1) - a(n-2) + 1. - Miklos Kristof, Mar 09 2005
a(n) = a(n-1) + n. - Zak Seidov, Mar 06 2005
a(n) = A108299(n+3,4) = -A108299(n+4,5). - Reinhard Zumkeller, Jun 01 2005
a(n) = A111808(n,2) for n > 1. - Reinhard Zumkeller, Aug 17 2005
a(n)*a(n+1) = A006011(n+1) = (n+1)^2*(n^2+2)/4 = 3*A002415(n+1) = 1/2*a(n^2+2*n). a(n-1)*a(n) = (1/2)*a(n^2-1). - Alexander Adamchuk, Apr 13 2006 [Corrected and edited by Charlie Marion, Nov 26 2010]
a(n) = floor((2*n+1)^2/8). - Paul Barry, May 29 2006
For positive n, we have a(8*a(n))/a(n) = 4*(2*n+1)^2 = (4*n+2)^2, i.e., a(A033996(n))/a(n) = 4*A016754(n) = (A016825(n))^2 = A016826(n). - Lekraj Beedassy, Jul 29 2006
a(n)^2 + a(n+1)^2 = a((n+1)^2) [R B Nelsen, Math Mag 70 (2) (1997), p. 130]. - R. J. Mathar, Nov 22 2006
a(n) = A126890(n,0). - Reinhard Zumkeller, Dec 30 2006
a(n)*a(n+k)+a(n+1)*a(n+1+k) = a((n+1)*(n+1+k)). Generalizes previous formula dated Nov 22 2006 [and comments by J. M. Bergot dated May 22 2012]. - Charlie Marion, Feb 04 2011
(sqrt(8*a(n)+1)-1)/2 = n. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 26 2007
a(n) = A023896(n) + A067392(n). - Lekraj Beedassy, Mar 02 2007
Sum_{k=0..n} a(k)*A039599(n,k) = A002457(n-1), for n >= 1. - Philippe Deléham, Jun 10 2007
8*a(n)^3 + a(n)^2 = Y(n)^2, where Y(n) = n*(n+1)*(2*n+1)/2 = 3*A000330(n). - Mohamed Bouhamida, Nov 06 2007 [Edited by Derek Orr, May 05 2015]
A general formula for polygonal numbers is P(k,n) = (k-2)*(n-1)n/2 + n = n + (k-2)*A000217(n-1), for n >= 1, k >= 3. - Omar E. Pol, Apr 28 2008 and Mar 31 2013
a(3*n) = A081266(n), a(4*n) = A033585(n), a(5*n) = A144312(n), a(6*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = A022264(n) - A049450(n). - Reinhard Zumkeller, Oct 09 2008
If we define f(n,i,a) = Sum_{j=0..k-1} (binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j)), then a(n) = -f(n,n-1,1), for n >= 1. - Milan Janjic, Dec 20 2008
4*a(x) + 4*a(y) + 1 = (x+y+1)^2 + (x-y)^2. - Vladimir Shevelev, Jan 21 2009
a(n) = A000124(n-1) + n-1 for n >= 2. a(n) = A000124(n) - 1. - Jaroslav Krizek, Jun 16 2009
An exponential generating function for the inverse of this sequence is given by Sum_{m>=0} ((Pochhammer(1, m)*Pochhammer(1, m))*x^m/(Pochhammer(3, m)*factorial(m))) = ((2-2*x)*log(1-x)+2*x)/x^2, the n-th derivative of which has a closed form which must be evaluated by taking the limit as x->0. A000217(n+1) = (lim_{x->0} d^n/dx^n (((2-2*x)*log(1-x)+2*x)/x^2))^-1 = (lim_{x->0} (2*Gamma(n)*(-1/x)^n*(n*(x/(-1+x))^n*(-x+1+n)*LerchPhi(x/(-1+x), 1, n) + (-1+x)*(n+1)*(x/(-1+x))^n + n*(log(1-x)+log(-1/(-1+x)))*(-x+1+n))/x^2))^-1. - Stephen Crowley, Jun 28 2009
a(n) = A034856(n+1) - A005408(n) = A005843(n) + A000124(n) - A005408(n). - Jaroslav Krizek, Sep 05 2009
a(A006894(n)) = a(A072638(n-1)+1) = A072638(n) = A006894(n+1)-1 for n >= 1. For n=4, a(11) = 66. - Jaroslav Krizek, Sep 12 2009
With offset 1, a(n) = floor(n^3/(n+1))/2. - Gary Detlefs, Feb 14 2010
a(n) = 4*a(floor(n/2)) + (-1)^(n+1)*floor((n+1)/2). - Bruno Berselli, May 23 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=1. - Mark Dols, Aug 20 2010
From Charlie Marion, Oct 15 2010: (Start)
a(n) + 2*a(n-1) + a(n-2) = n^2 + (n-1)^2; and
a(n) + 3*a(n-1) + 3*a(n-2) + a(n-3) = n^2 + 2*(n-1)^2 + (n-2)^2.
In general, for n >= m > 2, Sum_{k=0..m} binomial(m,m-k)*a(n-k) = Sum_{k=0..m-1} binomial(m-1,m-1-k)*(n-k)^2.
a(n) - 2*a(n-1) + a(n-2) = 1, a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 0 and a(n) - 4*a(n-1) + 6*a(n-2) - 4*(a-3) + a(n-4) = 0.
In general, for n >= m > 2, Sum_{k=0..m} (-1)^k*binomial(m,m-k)*a(n-k) = 0.
(End)
a(n) = sqrt(A000537(n)). - Zak Seidov, Dec 07 2010
For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} 4*(sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A110654(n)*A008619(n). - Reinhard Zumkeller, Aug 24 2011
a(2*k-1) = A000384(k), a(2*k) = A014105(k), k > 0. - Omar E. Pol, Sep 13 2011
a(n) = A026741(n)*A026741(n+1). - Charles R Greathouse IV, Apr 01 2012
a(n) + a(a(n)) + 1 = a(a(n)+1). - J. M. Bergot, Apr 27 2012
a(n) = -s(n+1,n), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n)*a(n+1) = a(Sum_{m=1..n} A005408(m))/2, for n >= 1. For example, if n=8, then a(8)*a(9) = a(80)/2 = 1620. - Ivan N. Ianakiev, May 27 2012
a(n) = A002378(n)/2 = (A001318(n) + A085787(n))/2. - Omar E. Pol, Jan 11 2013
G.f.: x * (1 + 3x + 6x^2 + ...) = x * Product_{j>=0} (1+x^(2^j))^3 = x * A(x) * A(x^2) * A(x^4) * ..., where A(x) = (1 + 3x + 3x^2 + x^3). - Gary W. Adamson, Jun 26 2012
G.f.: G(0) where G(k) = 1 + (2*k+3)*x/(2*k+1 - x*(k+2)*(2*k+1)/(x*(k+2) + (k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
a(n) = A002088(n) + A063985(n). - Reinhard Zumkeller, Jan 21 2013
G.f.: x + 3*x^2/(Q(0)-3*x) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) + a(n+1) + a(n+2) + a(n+3) + n = a(2*n+4). - Ivan N. Ianakiev, Mar 16 2013
a(n) + a(n+1) + ... + a(n+8) + 6*n = a(3*n+15). - Charlie Marion, Mar 18 2013
a(n) + a(n+1) + ... + a(n+20) + 2*n^2 + 57*n = a(5*n+55). - Charlie Marion, Mar 18 2013
3*a(n) + a(n-1) = a(2*n), for n > 0. - Ivan N. Ianakiev, Apr 05 2013
In general, a(k*n) = (2*k-1)*a(n) + a((k-1)*n-1). - Charlie Marion, Apr 20 2015
Also, a(k*n) = a(k)*a(n) + a(k-1)*a(n-1). - Robert Israel, Apr 20 2015
a(n+1) = det(binomial(i+2,j+1), 1 <= i,j <= n). - Mircea Merca, Apr 06 2013
a(n) = floor(n/2) + ceiling(n^2/2) = n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = floor((n+1)/(exp(2/(n+1))-1)). - Richard R. Forberg, Jun 22 2013
Sum_{n>=1} a(n)/n! = 3*exp(1)/2 by the e.g.f. Also see A067764 regarding ratios calculated this way for binomial coefficients in general. - Richard R. Forberg, Jul 15 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2) - 2 = 0.7725887... . - Richard R. Forberg, Aug 11 2014
2/(Sum_{n>=m} 1/a(n)) = m, for m > 0. - Richard R. Forberg, Aug 12 2014
A228474(a(n))=n; A248952(a(n))=0; A248953(a(n))=a(n); A248961(a(n))=A000330(n). - Reinhard Zumkeller, Oct 20 2014
a(a(n)-1) + a(a(n+2)-1) + 1 = A000124(n+1)^2. - Charlie Marion, Nov 04 2014
a(n) = 2*A000292(n) - A000330(n). - Luciano Ancora, Mar 14 2015
a(n) = A007494(n-1) + A099392(n) for n > 0. - Bui Quang Tuan, Mar 27 2015
Sum_{k=0..n} k*a(k+1) = a(A000096(n+1)). - Charlie Marion, Jul 15 2015
Let O(n) be the oblong number n(n+1) = A002378(n) and S(n) the square number n^2 = A000290(n). Then a(n) + a(n+2k) = O(n+k) + S(k) and a(n) + a(n+2k+1) = S(n+k+1) + O(k). - Charlie Marion, Jul 16 2015
A generalization of the Nov 22 2006 formula, a(n)^2 + a(n+1)^2 = a((n+1)^2), follows. Let T(k,n) = a(n) + k. Then for all k, T(k,n)^2 + T(k,n+1)^2 = T(k,(n+1)^2 + 2*k) - 2*k. - Charlie Marion, Dec 10 2015
a(n)^2 + a(n+1)^2 = a(a(n) + a(n+1)). Deducible from N. J. A. Sloane's a(n) + a(n+1) = (n+1)^2 and R. B. Nelson's a(n)^2 + a(n+1)^2 = a((n+1)^2). - Ben Paul Thurston, Dec 28 2015
Dirichlet g.f.: (zeta(s-2) + zeta(s-1))/2. - Ilya Gutkovskiy, Jun 26 2016
a(n)^2 - a(n-1)^2 = n^3. - Miquel Cerda, Jun 29 2016
a(n) = A080851(0,n-1). - R. J. Mathar, Jul 28 2016
a(n) = A000290(n-1) - A034856(n-4). - Peter M. Chema, Sep 25 2016
a(n)^2 + a(n+3)^2 + 19 = a(n^2 + 4*n + 10). - Charlie Marion, Nov 23 2016
2*a(n)^2 + a(n) = a(n^2+n). - Charlie Marion, Nov 29 2016
G.f.: x/(1-x)^3 = (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...), where r(x) = (1 + x + x^2)^3 = (1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6). - Gary W. Adamson, Dec 03 2016
a(n) = sum of the elements of inverse of matrix Q(n), where Q(n) has elements q_i,j = 1/(1-4*(i-j)^2). So if e = appropriately sized vector consisting of 1's, then a(n) = e'.Q(n)^-1.e. - Michael Yukish, Mar 20 2017
a(n) = Sum_{k=1..n} ((2*k-1)!!*(2*n-2*k-1)!!)/((2*k-2)!!*(2*n-2*k)!!). - Michael Yukish, Mar 20 2017
Sum_{i=0..k-1} a(n+i) = (3*k*n^2 + 3*n*k^2 + k^3 - k)/6. - Christopher Hohl, Feb 23 2019
a(n) = A060544(n + 1) - A016754(n). - Ralf Steiner, Nov 09 2019
a(n) == 0 (mod n) iff n is odd (see De Koninck reference). - Bernard Schott, Jan 10 2020
8*a(k)*a(n) + ((a(k)-1)*n + a(k))^2 = ((a(k)+1)*n + a(k))^2. This formula reduces to the well-known formula, 8*a(n) + 1 = (2*n+1)^2, when k = 1. - Charlie Marion, Jul 23 2020
a(k)*a(n) = Sum_{i = 0..k-1} (-1)^i*a((k-i)*(n-i)). - Charlie Marion, Dec 04 2020
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(7)*Pi/2)/(2*Pi).
Product_{n>=2} (1 - 1/a(n)) = 1/3. (End)
a(n) = Sum_{k=1..2*n-1} (-1)^(k+1)*a(k)*a(2*n-k). For example, for n = 4, 1*28 - 3*21 + 6*15 - 10*10 + 15*6 - 21*3 + 28*1 = 10. - Charlie Marion, Mar 23 2022
2*a(n) = A000384(n) - n^2 + 2*n. In general, if P(k,n) = the n-th k-gonal number, then (j+1)*a(n) = P(5 + j, n) - n^2 + (j+1)*n. More generally, (j+1)*P(k,n) = P(2*k + (k-2)*(j-1),n) - n^2 + (j+1)*n. - Charlie Marion, Mar 14 2023
a(n) = A109613(n) * A004526(n+1). - Torlach Rush, Nov 10 2023
a(n) = (1/6)* Sum_{k = 0..3*n} (-1)^(n+k+1) * k*(k + 1) * binomial(3*n+k, 2*k). - Peter Bala, Nov 03 2024
From Peter Bala, Jul 05 2025: (Start)
The following series telescope: for k >= 0,
Sum_{n >= 1} a(n)*a(n+2)*...*a(n+2*k)/(a(n+1)*a(n+3)*...*a(n+2*k+3)) = 1/(2*k + 3);
Sum_{n >= 1} a(n+1)*a(n+3)*...*a(n+2*k+1)/(a(n)*a(n+2)*...*a(n+2*k+2)) = 2/(2*k + 3) * Sum_{i = 1..2*k+3} 1/i. (End)

Extensions

Edited by Derek Orr, May 05 2015

A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Sometimes called "triprimes" or "3-almost primes".
See also A001358 for product of two primes (sometimes called semiprimes).
If you graph a(n)/n for n up to 10000 (and probably quite a bit higher), it appears to be converging to something near 3.9. In fact the limit is infinite. - Franklin T. Adams-Watters, Sep 20 2006
Meng shows that for any sufficiently large odd integer n, the equation n = a + b + c has solutions where each of a, b, c is 3-almost prime. The number of such solutions is (log log n)^6/(16 (log n)^3)*n^2*s(n)*(1 + O(1/log log n)), where s(n) = Sum_{q >= 1} Sum_{a = 1..q, (a, q) = 1} exp(i*2*Pi*n*a/q)*mu(n)/phi(n)^3 > 1/2. - Jonathan Vos Post, Sep 16 2005, corrected & rewritten by M. F. Hasler, Apr 24 2019
Also, a(n) are the numbers such that exactly half of their divisors are composite. For the numbers in which exactly half of the divisors are prime, see A167171. - Ivan Neretin, Jan 12 2016

Examples

			From _Gus Wiseman_, Nov 04 2020: (Start)
Also Heinz numbers of integer partitions into three parts, counted by A001399(n-3) = A069905(n) with ordered version A000217, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence of terms together with their prime indices begins:
      8: {1,1,1}     70: {1,3,4}     130: {1,3,6}
     12: {1,1,2}     75: {2,3,3}     138: {1,2,9}
     18: {1,2,2}     76: {1,1,8}     147: {2,4,4}
     20: {1,1,3}     78: {1,2,6}     148: {1,1,12}
     27: {2,2,2}     92: {1,1,9}     153: {2,2,7}
     28: {1,1,4}     98: {1,4,4}     154: {1,4,5}
     30: {1,2,3}     99: {2,2,5}     164: {1,1,13}
     42: {1,2,4}    102: {1,2,7}     165: {2,3,5}
     44: {1,1,5}    105: {2,3,4}     170: {1,3,7}
     45: {2,2,3}    110: {1,3,5}     171: {2,2,8}
     50: {1,3,3}    114: {1,2,8}     172: {1,1,14}
     52: {1,1,6}    116: {1,1,10}    174: {1,2,10}
     63: {2,2,4}    117: {2,2,6}     175: {3,3,4}
     66: {1,2,5}    124: {1,1,11}    182: {1,4,6}
     68: {1,1,7}    125: {3,3,3}     186: {1,2,11}
(End)

References

  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974). See p. 211.

Links

Crossrefs

Cf. A000040, A001358 (biprimes), A014613 (quadruprimes), A033942, A086062, A098238, A123072, A123073, A101605 (characteristic function).
Cf. A109251 (number of 3-almost primes <= 10^n).
Subsequence of A145784. - Reinhard Zumkeller, Oct 19 2008
Cf. A007304 is the squarefree case.
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), this sequence (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Cf. A253721 (final digits).
A014311 is a different ranking of ordered triples, with strict case A337453.
A046316 is the restriction to odds, with strict case A307534.
A075818 is the restriction to evens, with strict case A075819.
A285508 is the nonsquarefree case.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
Cf. A000212, A000217, A046389, A140106, A307719, A321773.

Programs

  • Haskell
    a014612 n = a014612_list !! (n-1)
a014612_list = filter ((== 3) . a001222) [1..] -- Reinhard Zumkeller, Apr 02 2012
  • Maple
    with(numtheory); A014612:=n->`if`(bigomega(n)=3, n, NULL); seq(A014612(n), n=1..250) # Wesley Ivan Hurt, Feb 05 2014
  • Mathematica
    threeAlmostPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@244, threeAlmostPrimeQ[ # ] &] (* Robert G. Wilson v, Jan 04 2006 *)
NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; NestList[NextkAlmostPrime[#, 3] &, 2^3, 56] (* Robert G. Wilson v, Jan 27 2013 *)
Select[Range[244], PrimeOmega[#] == 3 &] (* Jayanta Basu, Jul 01 2013 *)
  • PARI
    isA014612(n)=bigomega(n)==3 \\ Charles R Greathouse IV, May 07 2011
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2,lim\4, forprime(q=2,min(lim\(2*p),p), t=p*q; forprime(r=2,min(lim\t,q),listput(v,t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 04 2013
  • Python
    from sympy import factorint
def ok(n): f = factorint(n); return sum(f[p] for p in f) == 3
print(list(filter(ok, range(245)))) # Michael S. Branicky, Aug 12 2021
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A014612(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024
  • Scala
    def primeFactors(number: Int, list: List[Int] = List())
                                                      : List[Int] = {
  for (n <- 2 to number if (number % n == 0)) {
    return primeFactors(number / n, list :+ n)
  }
  list
}
(1 to 250).filter(primeFactors().size == 3) // _Alonso del Arte, Nov 04 2020, based on algorithm by Victor Farcic (vfarcic)

Formula

Product p_i^e_i with Sum e_i = 3.
a(n) ~ 2n log n / (log log n)^2 as n -> infinity [Landau, p. 211].
Tau(a(n)) = 2 * (omega(a(n)) + 1) = 2*A083399(a(n)), where tau = A000005 and omega = A001221. - Wesley Ivan Hurt, Jun 28 2013
a(n) = A078840(3,n). - R. J. Mathar, Jan 30 2019

Extensions

More terms from Patrick De Geest, Jun 15 1998

A007304 Sphenic numbers: products of 3 distinct primes.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426, 429, 430, 434, 435, 438
Offset: 1

Views

Author

Simon Plouffe

Keywords

Comments

Note the distinctions between this and "n has exactly three prime factors" (A014612) or "n has exactly three distinct prime factors." (A033992). The word "sphenic" also means "shaped like a wedge" [American Heritage Dictionary] as in dentation with "sphenic molars." - Jonathan Vos Post, Sep 11 2005
Also the volume of a sphenic brick. A sphenic brick is a rectangular parallelepiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. Example: The distinct prime triple (3,5,7) produces a 3x5x7 unit brick which has volume 105 cubic units. 3-D analog of 2-D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3-D A107768 Golden 3-almost primes = Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007
Sum(n>=1, 1/a(n)^s) = (1/6)*(P(s)^3 - P(3*s) - 3*(P(s)*P(2*s)-P(3*s))), where P is prime zeta function. - Enrique Pérez Herrero, Jun 28 2012
Also numbers n with A001222(n)=3 and A001221(n)=3. - Enrique Pérez Herrero, Jun 28 2012
n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - Peter Dolland, Apr 11 2020

Examples

			From _Gus Wiseman_, Nov 05 2020: (Start)
Also Heinz numbers of strict integer partitions into three parts, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are counted by A001399(n-6) = A069905(n-3), with ordered version A001399(n-6)*6. The sequence of terms together with their prime indices begins:
     30: {1,2,3}     182: {1,4,6}     286: {1,5,6}
     42: {1,2,4}     186: {1,2,11}    290: {1,3,10}
     66: {1,2,5}     190: {1,3,8}     310: {1,3,11}
     70: {1,3,4}     195: {2,3,6}     318: {1,2,16}
     78: {1,2,6}     222: {1,2,12}    322: {1,4,9}
    102: {1,2,7}     230: {1,3,9}     345: {2,3,9}
    105: {2,3,4}     231: {2,4,5}     354: {1,2,17}
    110: {1,3,5}     238: {1,4,7}     357: {2,4,7}
    114: {1,2,8}     246: {1,2,13}    366: {1,2,18}
    130: {1,3,6}     255: {2,3,7}     370: {1,3,12}
    138: {1,2,9}     258: {1,2,14}    374: {1,5,7}
    154: {1,4,5}     266: {1,4,8}     385: {3,4,5}
    165: {2,3,5}     273: {2,4,6}     399: {2,4,8}
    170: {1,3,7}     282: {1,2,15}    402: {1,2,19}
    174: {1,2,10}    285: {2,3,8}     406: {1,4,10}
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • "Sphenic", The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Mifflin Company, 2000.

Links

Crossrefs

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A002033, A010051, A020639, A037074, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A107768, A179643, A179695.
Cf. A162143 (a(n)^2).
For the following, NNS means "not necessarily strict".
A014612 is the NNS version.
A046389 is the restriction to odds (NNS: A046316).
A075819 is the restriction to evens (NNS: A075818).
A239656 gives first differences.
A285508 lists terms of A014612 that are not squarefree.
A307534 is the case where all prime indices are odd (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338557 is the case where all prime indices are even (NNS: A338556).
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers.
A008289 counts strict partitions by sum and length.
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
Cf. A000212, A000217, A001840, A101271, A284825, A321773, A337599, A337605.

Programs

  • Haskell
    a007304 n = a007304_list !! (n-1)
a007304_list = filter f [1..] where
f u = p < q && q < w && a010051 w == 1 where
p = a020639 u; v = div u p; q = a020639 v; w = div v q
-- Reinhard Zumkeller, Mar 23 2014
  • Maple
    with(numtheory): a:=proc(n) if bigomega(n)=3 and nops(factorset(n))=3 then n else fi end: seq(a(n),n=1..450); # Emeric Deutsch
A007304 := proc(n)
    option remember;
    local a;
    if n =1 then
        30;
    else
        for a from procname(n-1)+1 do
            if bigomega(a)=3 and nops(factorset(a))=3 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Dec 06 2016
is_a := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A007304List := upto -> select(is_a, [seq(1..upto)]):  # Peter Luschny, Apr 14 2025
  • Mathematica
    Union[Flatten[Table[Prime[n]*Prime[m]*Prime[k], {k, 20}, {n, k+1, 20}, {m, n+1, 20}]]]
Take[ Sort@ Flatten@ Table[ Prime@i Prime@j Prime@k, {i, 3, 21}, {j, 2, i - 1}, {k, j - 1}], 53] (* Robert G. Wilson v *)
With[{upto=500},Sort[Select[Times@@@Subsets[Prime[Range[Ceiling[upto/6]]],{3}],#<=upto&]]] (* Harvey P. Dale, Jan 08 2015 *)
Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==3&] (* Gus Wiseman, Nov 05 2020 *)
  • PARI
    for(n=1,1e4,if(bigomega(n)==3 && omega(n)==3,print1(n", "))) \\ Charles R Greathouse IV, Jun 10 2011
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2,(lim)^(1/3),forprime(q=p+1,sqrt(lim\p),t=p*q;forprime(r=q+1,lim\t,listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); Set(v) \\ Charles R Greathouse IV, Jan 21 2025
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A007304(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024
  • SageMath
    def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

Formula

A008683(a(n)) = -1.
A000005(a(n)) = 8. - R. J. Mathar, Aug 14 2009
A002033(a(n)-1) = 13. - Juri-Stepan Gerasimov, Oct 07 2009, R. J. Mathar, Oct 14 2009
A178254(a(n)) = 36. - Reinhard Zumkeller, May 24 2010
A050326(a(n)) = 5, subsequence of A225228. - Reinhard Zumkeller, May 03 2013
a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015

Extensions

More terms from Robert G. Wilson v, Jan 04 2006
Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009

A101268 Number of compositions of n into pairwise relatively prime parts.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 101, 160, 254, 403, 635, 984, 1492, 2225, 3281, 4814, 7044, 10271, 14889, 21416, 30586, 43401, 61205, 85748, 119296, 164835, 226423, 309664, 422302, 574827, 781237, 1060182, 1436368, 1942589, 2622079, 3531152, 4742316, 6348411
Offset: 0

Views

Author

Vladeta Jovovic, Dec 18 2004

Keywords

Comments

Here a singleton is always considered pairwise relatively prime. Compare to A337462. - Gus Wiseman, Oct 18 2020

Examples

			From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (31)    (23)
             (111)  (112)   (32)
                    (121)   (41)
                    (211)   (113)
                    (1111)  (131)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)
(End)

Links

Crossrefs

Row sums of A282748.
A051424 is the unordered version, with strict case A007360.
A335235 ranks these compositions.
A337461 counts these compositions of length 3, with unordered version A307719 and unordered strict version A220377.
A337462 does not consider a singleton to be coprime unless it is (1), with strict version A337561.
A337562 is the strict case.
A337664 looks only at distinct parts, with non-constant version A337665.
A000740 counts relatively prime compositions, with strict case A332004.
A178472 counts compositions with a common factor.
Cf. A087087, A302569, A305713, A326675, A327516, A328673, A333227, A333228.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]<=1||CoprimeQ@@#&]],{n,0,10}] (* Gus Wiseman, Oct 18 2020 *)

Formula

It seems that no formula is known.

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 14 2017

A327516 Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 22, 26, 32, 37, 42, 50, 59, 69, 80, 91, 101, 115, 133, 152, 170, 190, 210, 235, 265, 300, 334, 366, 398, 441, 484, 541, 597, 648, 703, 770, 848, 935, 1022, 1102, 1184, 1281, 1406, 1534, 1661, 1789, 1916, 2062, 2244, 2435
Offset: 0

Views

Author

Gus Wiseman, Sep 19 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A302696.
Note that the definition excludes partitions with repeated parts other than 1 (cf. A038348, A304709).

Examples

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (52)       (71)
                    (1111)  (311)    (411)     (61)       (431)
                            (2111)   (3111)    (511)      (521)
                            (11111)  (21111)   (3211)     (611)
                                     (111111)  (4111)     (5111)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

Links

Crossrefs

Cf. A038348, A302569, A304709, A304711.
A000837 is the relatively prime instead of pairwise coprime version.
A051424 includes all singletons, with strict case A007360.
A101268 is the ordered version (with singletons).
A302696 ranks these partitions, with complement A335241.
A305713 is the strict case.
A307719 counts these partitions of length 3.
A018783 counts partitions with a common divisor.
A328673 counts pairwise non-coprime partitions.
Cf. A087087, A220377, A326675, A333227, A333228, A335238.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],#=={}||CoprimeQ@@#&]],{n,0,30}]

Formula

For n > 1, a(n) = A051424(n) - 1. - Gus Wiseman, Sep 18 2020

A220377 Number of partitions of n into three distinct and mutually relatively prime parts.

Original entry on oeis.org

1, 0, 2, 1, 3, 1, 6, 1, 7, 3, 7, 3, 14, 3, 15, 6, 14, 6, 25, 6, 22, 10, 25, 9, 42, 8, 34, 15, 37, 15, 53, 13, 48, 22, 53, 17, 78, 17, 65, 30, 63, 24, 99, 24, 88, 35, 84, 30, 126, 34, 103, 45, 103, 38, 166, 35, 124, 57, 128, 51, 184, 44, 150, 67, 172, 52, 218
Offset: 6

Views

Author

Carl Najafi, Dec 13 2012

Keywords

Comments

The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - Gus Wiseman, Oct 14 2020

Examples

			For n=10 we have three such partitions: 1+2+7, 1+4+5 and 2+3+5.
From _Gus Wiseman_, Oct 14 2020: (Start)
The a(6) = 1 through a(20) = 15 triples (empty column indicated by dot, A..H = 10..17):
321  .  431  531  532  731  543  751  743  753  754  971  765  B53  875
        521       541       651       752  951  853  B51  873  B71  974
                  721       732       761  B31  871  D31  954  D51  A73
                            741       851       952       972       A91
                            831       941       B32       981       B54
                            921       A31       B41       A71       B72
                                      B21       D21       B43       B81
                                                          B52       C71
                                                          B61       D43
                                                          C51       D52
                                                          D32       D61
                                                          D41       E51
                                                          E31       F41
                                                          F21       G31
                                                                    H21
(End)

Links

Crossrefs

Cf. A015617, A300815.
A023022 is the 2-part version.
A101271 is the relative prime instead of pairwise coprime version.
A220377*6 is the ordered version.
A305713 counts these partitions of any length, with Heinz numbers A302797.
A307719 is the non-strict version.
A337461 is the non-strict ordered version.
A337563 is the case with no 1's.
A337605 is the pairwise non-coprime instead of pairwise coprime version.
A001399(n-6) counts strict 3-part partitions, with Heinz numbers A007304.
A008284 counts partitions by sum and length, with strict case A008289.
A318717 counts pairwise non-coprime strict partitions.
A326675 ranks pairwise coprime sets.
A327516 counts pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000217, A007360, A023023, A051424, A078374, A087087, A302696, A333227, A337485, A337561.

Programs

  • Mathematica
    Table[Length@Select[ IntegerPartitions[ n, {3}], #[[1]] != #[[2]] != #[[3]] && GCD[#[[1]], #[[2]]] == 1 && GCD[#[[1]], #[[3]]] == 1 && GCD[#[[2]], #[[3]]] == 1 &], {n, 6, 100}]
Table[Count[IntegerPartitions[n,{3}],?(CoprimeQ@@#&&Length[ Union[#]] == 3&)],{n,6,100}] (* _Harvey P. Dale, May 22 2020 *)
  • PARI
    a(n)=my(P=partitions(n));sum(i=1,#P,#P[i]==3&&P[i][1]Charles R Greathouse IV, Dec 14 2012

Formula

a(n > 2) = A307719(n) - 1. - Gus Wiseman, Oct 15 2020

A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 9, 3, 15, 9, 21, 9, 39, 9, 45, 21, 45, 21, 87, 21, 93, 39, 87, 39, 153, 39, 135, 63, 153, 57, 255, 51, 207, 93, 225, 93, 321, 81, 291, 135, 321, 105, 471, 105, 393, 183, 381, 147, 597, 147, 531, 213, 507, 183, 759, 207, 621, 273, 621, 231
Offset: 0

Views

Author

Gus Wiseman, Sep 02 2020

Keywords

Examples

			The a(3) = 1 through a(9) = 9 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
           (1,2,1)  (1,3,1)  (1,2,3)  (1,5,1)  (1,2,5)  (1,3,5)
           (2,1,1)  (3,1,1)  (1,3,2)  (5,1,1)  (1,3,4)  (1,5,3)
                             (1,4,1)           (1,4,3)  (1,7,1)
                             (2,1,3)           (1,5,2)  (3,1,5)
                             (2,3,1)           (1,6,1)  (3,5,1)
                             (3,1,2)           (2,1,5)  (5,1,3)
                             (3,2,1)           (2,5,1)  (5,3,1)
                             (4,1,1)           (3,1,4)  (7,1,1)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

Links

Crossrefs

A000212 counts the unimodal instead of coprime version.
A220377*6 is the strict case.
A307719 is the unordered version.
A337462 counts these compositions of any length.
A337563 counts the case of partitions with no 1's.
A337603 only requires the *distinct* parts to be pairwise coprime.
A337604 is the intersecting instead of coprime version.
A014612 ranks 3-part partitions.
A302696 ranks pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000217, A001399, A001840, A014311, A101268, A284825, A337562, A326675, A333227, A337601, A337602.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,30}]

A337561 Number of pairwise coprime strict compositions of n, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 8, 6, 16, 12, 22, 40, 40, 66, 48, 74, 74, 154, 210, 228, 242, 240, 286, 394, 806, 536, 840, 654, 1146, 1618, 2036, 2550, 2212, 2006, 2662, 4578, 4170, 7122, 4842, 6012, 6214, 11638, 13560, 16488, 14738, 15444, 16528, 25006, 41002, 32802
Offset: 0

Views

Author

Gus Wiseman, Sep 18 2020

Keywords

Examples

			The a(1) = 1 through a(9) = 12 compositions (empty column shown as dot):
   (1)  .  (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)    (1,8)
           (2,1)  (3,1)  (2,3)  (5,1)    (2,5)  (3,5)    (2,7)
                         (3,2)  (1,2,3)  (3,4)  (5,3)    (4,5)
                         (4,1)  (1,3,2)  (4,3)  (7,1)    (5,4)
                                (2,1,3)  (5,2)  (1,2,5)  (7,2)
                                (2,3,1)  (6,1)  (1,3,4)  (8,1)
                                (3,1,2)         (1,4,3)  (1,3,5)
                                (3,2,1)         (1,5,2)  (1,5,3)
                                                (2,1,5)  (3,1,5)
                                                (2,5,1)  (3,5,1)
                                                (3,1,4)  (5,1,3)
                                                (3,4,1)  (5,3,1)
                                                (4,1,3)
                                                (4,3,1)
                                                (5,1,2)
                                                (5,2,1)

Links

Crossrefs

A072706 counts unimodal strict compositions.
A220377*6 counts these compositions of length 3.
A305713 is the unordered version.
A337462 is the not necessarily strict version.
A000740 counts relatively prime compositions, with strict case A332004.
A051424 counts pairwise coprime or singleton partitions.
A101268 considers all singletons to be coprime, with strict case A337562.
A178472 counts compositions with a common factor > 1.
A327516 counts pairwise coprime partitions, with strict case A305713.
A328673 counts pairwise non-coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A007359, A007360, A087087, A216652, A220377, A302569, A307719, A326675, A333227, A337461.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||UnsameQ@@#&&CoprimeQ@@#&]],{n,0,10}]

Formula

a(n) = A337562(n) - 1 for n > 1.

A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290
Offset: 0

Views

Author

Clark Kimberling, Apr 15 2012

Keywords

Comments

For a guide to related sequences, see A211422.
Also the number of partitions of n+1 into three parts, where each part > 1. - Peter Woodward, May 25 2015
a(n) is also equal to the number of partitions of n+4 into three distinct parts, where each part > 1. - Giovanni Resta, May 26 2015
Number of different distributions of n+1 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Dec 31 2015
After the first three terms, partial sums of A008615. - Robert Israel, Dec 31 2015
For n >= 2, also the number of partitions of n - 2 into 3 parts. The Heinz numbers of these partitions are given by A014612. - Gus Wiseman, Oct 11 2020

Examples

			a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015
a(11) = 5 Number of different distributions of 11 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
a(1) = a(2) = a(3) = a(4) = a(5) = 0, since with fewer than 6 identical balls there is no such distribution with 3 boxes that holds for 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...
From _Gus Wiseman_, Oct 11 2020: (Start)
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three parts > 1 [Woodward] are the following (A = 10, B = 11, C = 12). The ordered version is A000217(n - 4) and the Heinz numbers are A046316.
  222  322  332  333  433  443  444  544  554  555  655
            422  432  442  533  543  553  644  654  664
                 522  532  542  552  643  653  663  754
                      622  632  633  652  662  744  763
                           722  642  733  743  753  772
                                732  742  752  762  844
                                822  832  833  843  853
                                     922  842  852  862
                                          932  933  943
                                          A22  942  952
                                               A32  A33
                                               B22  A42
                                                    B32
                                                    C22
The a(5) = 1 through a(15) = 14 partitions of n + 4 into three distinct parts > 1 [Resta] are the following (A = 10, B = 11, C = 12, D = 13, E = 14). The ordered version is A211540*6 and the Heinz numbers are A046389.
  432  532  542  543  643  653  654  754  764  765  865
            632  642  652  743  753  763  854  864  874
                 732  742  752  762  853  863  873  964
                      832  842  843  862  872  954  973
                           932  852  943  953  963  982
                                942  952  962  972  A54
                                A32  A42  A43  A53  A63
                                     B32  A52  A62  A72
                                          B42  B43  B53
                                          C32  B52  B62
                                               C42  C43
                                               D32  C52
                                                    D42
                                                    E32
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three distinct parts [Uslu and Becenen] are the following (A = 10, B = 11, C = 12, D = 13). The ordered version is A211540(n)*6 and the Heinz numbers are A007304.
  321  421  431  432  532  542  543  643  653  654  754
            521  531  541  632  642  652  743  753  763
                 621  631  641  651  742  752  762  853
                      721  731  732  751  761  843  862
                           821  741  832  842  852  871
                                831  841  851  861  943
                                921  931  932  942  952
                                     A21  941  951  961
                                          A31  A32  A42
                                          B21  A41  A51
                                               B31  B32
                                               C21  B41
                                                    C31
                                                    D21
(End)

Links

Crossrefs

Cf. A001399, A069905, A211422.
All of the following pertain to 3-part strict partitions.
- A000009 counts these partitions of any length, with non-strict version A000041.
- A007304 gives the Heinz numbers, with non-strict version A014612.
- A101271 counts the relatively prime case, with non-strict version A023023.
- A220377 counts the pairwise coprime case, with non-strict version A307719.
- A337605 counts the pairwise non-coprime case, with non-strict version A337599.
Cf. A000217, A001840, A156040, A284825, A337453, A337483, A337484, A337563.

Programs

  • Magma
    I:=[0,0,0,0,0,1]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015
  • Maple
    f:= gfun:-rectoproc({a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6),seq(a(i)=0,i=0..4),a(5)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Dec 31 2015
  • Mathematica
    t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211540 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 05 2020 *)
  • PARI
    {a(n) = round( (n-2)^2 / 12 )}; / * Michael Somos, Feb 02 2015 */
  • PARI
    concat(vector(5), Vec(x^5/(1-x-x^2+x^4+x^5-x^6) + O(x^100))) \\ Altug Alkan, Jan 10 2016

Formula

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - Alois P. Heinz, Nov 03 2012
a(n) = 3*k^2-6*k+3 (for n = 6*k-3), 3*k^2-5*k+2 (for n = 6*k-2), 3*k^2-4*k+1 (for n = 6*k-1), 3*k^2-3*k+1 (for n = 6*k), 3*k^2-2*k (for n = 6*k+1), 3*k^2-k (for n = 6*k+2). - Ece Uslu, Esin Becenen, Dec 31 2015
a(n) = A004526(n-2) + a(n-2) for n > 2. - Ece Uslu, Esin Becenen, Dec 31 2015
G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - Robert Israel, Dec 31 2015
a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - Wesley Ivan Hurt, Apr 27 2019
From Gus Wiseman, Oct 11 2020: (Start)
a(n+2) = A069905(n) = A001399(n-3) counts 3-part partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part strict partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part partitions with no 1's.
a(n-4) = A069905(n-6) = A001399(n-9) counts 3-part strict partitions with no 1's.
A000217(n-2) counts 3-part compositions.
a(n-1)*6 = A069905(n-3)*6 = A001399(n-6)*6 counts 3-part strict compositions.
A000217(n-5) counts 3-part compositions with no 1's.
a(n-4)*6 = A069905(n-6)*6 = A001399(n-9)*6 counts 3-part strict compositions with no 1's.
(End)

A284825 Number of partitions of n into 3 parts without common divisors such that every pair of them has common divisors.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 5, 0, 0, 0, 1, 0, 5, 0, 1, 0, 6, 0, 6, 0, 0, 0, 4, 0, 6, 0, 0, 0, 9, 0, 2, 1, 2, 0, 9, 0, 8, 1, 1, 0, 5, 0, 14, 0, 1, 0, 15, 0, 14, 0, 0, 1, 14, 0, 14, 0, 2, 0, 15, 0, 6, 1, 2, 1, 11, 0, 18, 1, 1, 0, 10, 0, 23
Offset: 31

Views

Author

Alois P. Heinz, Apr 03 2017

Keywords

Comments

The Heinz numbers of these partitions are the intersection of A014612 (triples), A289509 (relatively prime), and A337694 (pairwise non-coprime). - Gus Wiseman, Oct 16 2020

Examples

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(41) = 2: [6,14,21], [6,15,20].
From _Gus Wiseman_, Oct 14 2020: (Start)
Selected terms and the corresponding triples:
  a(31)=1: a(41)=2: a(59)=3:  a(77)=4:  a(61)=5:  a(71)=6:
-------------------------------------------------------------
  15,10,6  20,15,6  24,20,15  39,26,12  33,22,6   39,26,6
           21,14,6  24,21,14  42,20,15  40,15,6   45,20,6
                    35,14,10  45,20,12  45,10,6   50,15,6
                              50,15,12  28,21,12  35,21,15
                                        36,15,10  36,20,15
                                                  36,21,14
(End)

Links

Crossrefs

Cf. A082024, A230034, A230035.
A023023 does not require pairwise non-coprimality, with strict case A101271.
A202425 and A328672 count these partitions of any length, ranked by A328868.
A284825*6 is the ordered version.
A307719 is the pairwise coprime instead of non-coprime version.
A337599 does not require relatively primality, with strict case A337605.
A200976 and A328673 count pairwise non-coprime partitions.
A289509 gives Heinz numbers of relatively prime partitions.
A327516 counts pairwise coprime partitions, ranked by A333227.
A337694 gives Heinz numbers of pairwise non-coprime partitions.
Cf. A000217, A000741, A001399, A007304, A014612, A220377, A318716, A328679, A337604, A337666.

Programs

  • Maple
    a:= proc(n) option remember; add(add(`if`(igcd(i, j)>1
      and igcd(i, j, n-i-j)=1 and igcd(i, n-i-j)>1 and
      igcd(j, n-i-j)>1, 1, 0), j=i..(n-i)/2), i=2..n/3)
    end:
seq(a(n), n=31..137);
  • Mathematica
    a[n_] := a[n] = Sum[Sum[If[GCD[i, j] > 1 && GCD[i, j, n - i - j] == 1 && GCD[i, n - i - j] > 1 && GCD[j, n - i - j] > 1, 1, 0], {j, i, (n - i)/2} ], {i, 2, n/3}];
Table[a[n], {n, 31, 137}] (* Jean-François Alcover, Jun 13 2018, from Maple *)
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&&stabQ[#,CoprimeQ]&]],{n,31,100}] (* Gus Wiseman, Oct 14 2020 *)

Formula

a(n) > 0 iff n in { A230035 }.
a(n) = 0 iff n in { A230034 }.
Showing 1-10 of 32 results. Next

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