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A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1. 70
1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 18, 20, 20, 18, 14, 8, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 <m+1|S_+|m> = 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 |<m+1|S_+|m><m|S_-|m+1>| 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

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 46.

LINKS

T. D. Noe, Rows n = 1..100 of triangle, flattened

A. Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9 no. 2 (1997), p. 319-335.

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

FORMULA

Rectangular array: T(n, m)=n*m, i>=1, j>= 1.

Triangle X(n, m) = T(n-m+1, m) = (n-m+1)*m.

Sum_{i=1..n} Sum_{j=1..n} a(n) = A000537(n) [Sum of first n cubes; or n-th triangular number squared.] Determinant of all n X n contiguous subarrays of A003991 is 0. - Gerald McGarvey, Sep 26 2004

G.f. as rectangular array: x * y / [ (1-x)^2 * (1-y)^2 ].

a(n) = i*j, where i=floor((1+sqrt(8n-7))/2), j=n-i*(i-1)/2. - Hieronymus Fischer, Aug 08 2007

As an infinite lower triangular matrix equals A000012 * A002260; where A000012 = (1; 1,1; 1,1,1; ...) and A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Oct 23 2007

As a linear array, the sequence is a(n) = A002260(n)*A004736(n) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

G.f. as linear array: (x - 3*x^2 + Sum_{k >= 0} ((k+2-x-(k+1)*x^2)*x^((k^2+3*k+4)/2)))/(1-x)^3. - Robert Israel, Dec 14 2015

EXAMPLE

The array T starts in row n=1 with columns m>=1 as:

   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15

   2   4   6   8  10  12  14  16  18  20  22  24  26  28  30

   3   6   9  12  15  18  21  24  27  30  33  36  39  42  45

   4   8  12  16  20  24  28  32  36  40  44  48  52  56  60

   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75

   6  12  18  24  30  36  42  48  54  60  66  72  78  84  90

   7  14  21  28  35  42  49  56  63  70  77  84  91  98 105

   8  16  24  32  40  48  56  64  72  80  88  96 104 112 120

   9  18  27  36  45  54  63  72  81  90  99 108 117 126 135

  10  20  30  40  50  60  70  80  90 100 110 120 130 140 150

The triangle X(n, m) begins

   n\m  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...

   1:   1

   2:   2  2

   3:   3  4  3

   4:   4  6  6  4

   5:   5  8  9  8  5

   6:   6 10 12 12 10  6

   7:   7 12 15 16 15 12  7

   8:   8 14 18 20 20 18 14  8

   9:   9 16 21 24 25 24 21 16  9

  10:  10 18 24 28 30 30 28 24 18 10

  11:  11 20 27 32 35 36 35 32 27 20 11

  12:  12 22 30 36 40 42 42 40 36 30 22 12

  13:  13 24 33 40 45 48 49 48 45 40 33 24 13

  14:  14 26 36 44 50 54 56 56 54 50 44 36 26 14

  15:  15 28 39 48 55 60 63 64 63 60 55 48 39 28 15

  ... Formatted by Wolfdieter Lang, Dec 02 2014

MAPLE

seq(seq(i*(n-i), i=1..n-1), n=2..10); # Robert Israel, Dec 14 2015

MATHEMATICA

Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)

PROG

(PARI) A003991(n, k) = if(k<1 || n<1, 0, k*n)

CROSSREFS

Main diagonal gives squares A000290. Antidiagonal sums are tetrahedral numbers A000292. See A004247 for another version.

Cf. A003989, A003990, A003056, A049581, A000442, A027424, A002260, A033638, A075374.

Sequence in context: A205153 A091257 A216622 * A131923 A119457 A241356

Adjacent sequences:  A003988 A003989 A003990 * A003992 A003993 A003994

KEYWORD

tabl,nonn,nice,easy,look

AUTHOR

Marc LeBrun

EXTENSIONS

More terms from Michael Somos

STATUS

approved

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Last modified June 25 00:41 EDT 2017. Contains 288708 sequences.