cp's OEIS Frontend

Decreasing integers m to 0 followed by decreasing integers m+1 to 0, etc.

The point with coordinates (x = A025581(n), y = A002262(n)) sweeps out the first quadrant by upwards antidiagonals. N. J. A. Sloane, Jul 17 2018

The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 0, k >= 0) by antidiagonals upwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002

Riordan array (x/(1-x)^2, x). - Philippe Deléham, Feb 18 2012

a(n,k) = (A214604(n,k) - A214661(n,k)) / 2. - Reinhard Zumkeller, Jul 25 2012

Sequence B is called a reverse reluctant sequence of sequence A if B is a triangular array read by rows such that row number k lists the first k terms of the sequence A in reverse order. This sequence is the reverse reluctant sequence of sequence 0,1,2,3,..., the nonnegative integers A001477. - Boris Putievskiy, Dec 13 2012

A problem posed by François Viète (Vieta) in his book Zeteticorum liber quinque (1593), liber 2, problem 19 (quoted in the Alten et al. reference, on p. 292) is to find for a rectangle (a >= b >= 1) with given a^3 - b^3, name it C, and a*b, name it F, the difference a-b, name it x. This is a simple exercise which Viète found remarkable. It reduces to a standard cubic equation for x, namely x^3 + 3*F*x = C. Proof: Use the square of the diagonal d^2 = a^2 + b^2. Then (i) C = a^3 - b^3 = (a - b)*(a^2 + b^2 + a*b) = x*(d^2 + F). (ii) use the trivial relation d^2 = (a-b)^2 + 2*a*b = x^2 + 2*F, to eliminate d^2 in (i). End of the Proof. Here for positive integers a = n and b = k: (T(n, k)^2 + 3*A079904(n, k))*T(n, k) = A257238(n, k) (also true for n = k = 0). - Wolfdieter Lang, May 12 2015

See a comment on A051162 on the cubic equation for S = a+b in terms of Cplus = a^3 + b^3 and D = a - b. This equation leads to a - b = sqrt((4*Cplus -S^3)/(3*S)). - Wolfdieter Lang, May 15 2015

The entries correspond to the first of the 2 coordinates of the Cantor Pairs, specifically x=w-(CPKey-(w^2+w)/2), where w=floor((sqrt(8*CPKey+1)-1)/2) and CPKey=Cantor Pair key (A001477). The second of the coordinate pairs is A002262. - Bill McEachen, Sep 12 2015

Or, triangle X(n,m) = T(n-m+1,m) read by rows, in which row n gives the numbers n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n.

Radius of incircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 16 2001

A permutation of A061017. - Matthew Vandermast, Feb 28 2003

In the proof of countability of rational numbers they are arranged in a square array. a(n) = p*q where p/q is the corresponding rational number as read from the array. - Amarnath Murthy, May 29 2003

Permanent of upper right n X n corner is A000442. - Marc LeBrun, Dec 11 2003

Row 12 gives total number of partridges, turtle doves, ... and drummers drumming that you have received at the end of the Twelve Days of Christmas song. - Alonso del Arte, Jun 17 2005

Consider a particle with spin S (a half-integer) and 2S+1 quantum states |m>, m = -S,-S+1,...,S-1,S. Then the matrix element = sqrt((S+m+1)(S-m)) of the spin-raising operator is the square-root of the triangular (tabl) element T(r,o) of this sequence in row r = 2S, and at offset o=2(S+m). T(r,o) is also the intensity || of the transition between the states |m> and |m+1>. For example, the five transitions between the 6 states of a spin S=5/2 particle have relative intensities 5,8,9,8,5. The total intensity of all spin 5/2 transitions (relative to spin 1/2) is 35, which is the tetrahedral number A000292(5). - Stanislav Sykora, May 26 2012

Sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k) = n. See A098359. - Charlie Marion, Apr 22 2013

T(n, k) is also the (k-1)-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n. - Stefano Spezia, Jul 12 2019

From Eric Lengyel, Jun 28 2023: (Start)

X(n, m+1) is the number of degrees of freedom that an m-dimensional flat geometry (point, line, plane, etc.) has when embedded in an n-dimensional Euclidean space.

X(n+1, m+1) is the number of degrees of freedom that an m-ball has when embedded in an n-dimensional Euclidean space. (End)

T(n, k) is also the average number of steps it takes a person to fall off a board of length n+k, if the person starts a random walk at k. - Ruediger Jehn, May 12 2025

This is permutation of natural numbers larger than 1.

From Antti Karttunen, Dec 19 2014: (Start)

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.

For navigating in this array:

A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.

A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.

First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).

(End)

The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

Old name: Square array read by antidiagonals: T(i,j) = product prime(k)^(Ei(k) XOR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; XOR is the bitwise operation on binary representation of the exponents.

Analogous to multiplication, with XOR replacing +.

From Peter Munn, Apr 01 2019: (Start)

(1) Defines an abelian group whose underlying set is the positive integers. (2) Every element is self-inverse. (3) For all n and k, A(n,k) is a divisor of n*k. (4) The terms of A050376, sometimes called Fermi-Dirac primes, form a minimal set of generators. In ordered form, it is the lexicographically earliest such set.

The unique factorization of positive integers into products of distinct terms of the group's lexicographically earliest minimal set of generators seems to follow from (1) (2) and (3).

From (1) and (2), every row and every column of the table is a self-inverse permutation of the positive integers. Rows/columns numbered by nonmembers of A050376 are compositions of earlier rows/columns.

It is a subgroup of the equivalent group over the nonzero integers, which has -1 as an additional generator.

As generated by A050376, the subgroup of even length words is A000379. The complementary set of odd length words is A000028.

The subgroup generated by A000040 (the primes) is A005117 (the squarefree numbers).

(End)

Considered as a binary operation, the result is (the squarefree part of the product of its operands) times the square of (the operation's result when applied to the square roots of the square parts of its operands). - Peter Munn, Mar 21 2022

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

This array can be obtained by taking every second column from array A242378, starting from its column 2.

Permutation of natural numbers larger than 1.

The terms on row n are all divisible by n-th prime, A000040(n).

Each column is strictly growing, and the terms in the same column have the same prime signature.

A055396(n) gives the row number of row where n occurs,

and A246277(n) gives its column number, both starting from 1.

From Antti Karttunen, Jan 03 2015: (Start)

A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.

(End)

The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 2, 1 <= k <= n - 1) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002

Smallest integer such that n-1 <= C(a(n),2). - Frank Ruskey, Nov 06 2007

a(n) = inverse (frequency distribution) sequence of A161680. - Jaroslav Krizek, Jun 19 2009

Taken as a triangle t(n, m) with offset 1, i.e., n >= m >= 1, this gives all positive integer limits r = r (a = m, b = A063929(n, m)) of the (a,b)-Fibonacci ratio F(a,b;k+1)/F(a,b;k) for k -> infinity. See the Jan 11 2015 comment on A063929. - Wolfdieter Lang, Jan 12 2015

Square array, T(n,k) = n + k + 2, n > = 0 and k >= 0, read by antidiagonals. Northwest corner:

2, 3, 4, 5, ...

3, 4, 5, 6, ...

4, 5, 6, 7, ...

5, 6, 7, 8, ...

... - Franck Maminirina Ramaharo, Nov 21 2018

a(n) is the pair chromatic number of an empty graph with n vertices. (The pair chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of colors is repeated.) - Allan Bickle, Dec 26 2021

The starting offset of the sequence giving the terms of square array is 2. However, we can tacitly assume that a(1) = 1 when the sequence is used as a permutation of natural numbers. However, term 1 itself is out of the array.

The choice of offset = 2 for the terms starting in rows >= 1 is motivated by the desire to have a permutation of the integers n -> a(n) with a(n) = A(A002260(n-1), A004736(n-1)) for n > 1 and a(1) := 1. However, since this sequence is declared as a "table", offset = 2 would mean that the first *row* (not element) has index 2. I think the sequence should have offset = 1 and the permutation of the integers would be n -> a(n-1) with a(0) := 1 (if a(1) = A(1,1) = 2). Or, the sequence could have offset 0, with an additional row 0 of length 1 with the only element a(0) = A(0,1) = 1, the permutation still being n -> a(n-1) if a(n=0, 1, 2, ...) = (1, 2, 4, ...). This would be in line with considering 1 as the first ludic number, and A(n, 1) = A003309(n+1) for n >= 0. - M. F. Hasler, Nov 12 2024

Complement of A000096 = increasing sequence of positive integers excluding n*(n+3)/2. - Jonathan Vos Post, Aug 13 2005

As a triangle: (1; 3,4; 6,7,8; 10,11,12,13; ...), Row sums = A127736: (1, 7, 21, 46, 85, 141, 217, ...). - Gary W. Adamson, Oct 25 2007

Odd primes are a subsequence except 5, cf. A004139. - Reinhard Zumkeller, Jul 18 2011

A023532(a(n)) = 1. - Reinhard Zumkeller, Dec 04 2012

T(n,k) = ((n+k)^2+n-k)/2 - 1, n,k > 0, read by antidiagonals. - Boris Putievskiy, Jan 14 2013

A023531(a(n)) = 0. - Reinhard Zumkeller, Feb 14 2015

Numbers that are not triangular (nontriangular numbers).

Also definable as follows: a(1)=2; for n>1, a(n) is smallest integer greater than a(n-1) such that the condition "n and a(a(n)) have opposite parities" can always be satisfied. - Benoit Cloitre and Matthew Vandermast, Mar 10 2003

Record values in A256188 that are greater than 1. - Reinhard Zumkeller, Mar 26 2015

From Daniel Forgues, Apr 10 2015: (Start)

With n >= 1, k >= 1:

t(n+k) - k, 1 <= k <= n+k-1, n >= 1;

t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;

where t(n+k) = t(n+k-1) + (n+k) is the (n+k)-th triangular number, while the number of compositions of n+k into 2 parts is C(n+k-1, 2-1) = n+k-1, the number of nontriangular numbers between t(n+k-1) and t(n+k), just right!

Related to Hilbert's Infinite Hotel:

0) All rooms, numbered through the positive integers, are full;

1) An infinite number of trains, each containing an infinite number of passengers, arrives: i.e., a 2-D lattice of pairs of positive integers;

2) Move occupant of room m, m >= 1, to room t(m) = m*(m+1)/2, where t(m) is the m-th triangular number;

3) Assign n-th passenger from k-th train to room t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;

4) Everybody has his or her own room, no room is empty, for m >= 1.

If situation 1 happens again, repeat steps 2 and 3, you're back to 4.

(End)

1711 + 2*a(n)*(58 + a(n)) is prime for n<=21. The terms that do not have this property start 29,32,34,43,47,58,59,60,62,63,65,68,70,73,... - Benedict W. J. Irwin, Nov 22 2016

Also numbers k with the property that in the symmetric representation of sigma(k) both Dyck paths have a central peak or both Dyck paths have a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

Number of compositions of n into one sort of 1's, two sorts of 2's, ..., k sorts of k's, ... - Joerg Arndt, Jun 21 2011

Also the number of spanning trees of a graph formed by joining a single vertex to all vertices of a path on n-1 vertices. - Edward Scheinerman (ers(AT)jhu.edu), Feb 28 2007

Row sums of triangle A128908. - Philippe Deléham, Nov 21 2007

Let P = the partial sum operator, A000012: (1; 1,1; 1,1,1; ...) and A153463 = M, the partial sum & shift operator. It appears that beginning with any randomly taken sequence S(n), iterates of the operations M * S(n), -> M * ANS, -> P * ANS, etc. (or starting with P) will rapidly converge upon a two-sequence limit cycle of (1, 2, 5, 13, 34, ...) and (1, 1, 3, 8, 21, ...). - Gary W. Adamson, Dec 27 2008

Eigensequence of triangle A004736. - Paul Barry, Nov 03 2010

a(n) = the sum of the products of all compositions of n.

Number of nonisomorphic graded posets with 0 and uniform Hasse graph of rank n, with exactly 2 elements of each rank level above 0.(Uniform used in the sense of Retakh, Serconek and Wilson. Graded poset is being used in Stanley's sense that every maximal chain has the same length n.) - David Nacin, Feb 26 2012

a(n) is the top left entry in the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 0, 1] or of the 3 X 3 matrix [1, 1, 1; 1, 1, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014