A098353 -id:A098353 - cp's OEIS frontend

A141387 Triangle read by rows: T(n,m) = n + 2m(n - m) (0 <= m <= n).

0, 1, 1, 2, 4, 2, 3, 7, 7, 3, 4, 10, 12, 10, 4, 5, 13, 17, 17, 13, 5, 6, 16, 22, 24, 22, 16, 6, 7, 19, 27, 31, 31, 27, 19, 7, 8, 22, 32, 38, 40, 38, 32, 22, 8, 9, 25, 37, 45, 49, 49, 45, 37, 25, 9, 10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10

Offset: 0

Roger L. Bagula, Aug 03 2008

Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k)=T(2j,j+m) satisfy:(1/4)T(2j,j+m) =  = . Matrices for J_1^2 and J_2^2 are sparse. These diagonal equalities and the off-diagonal equalities of A268759 determine the only nonzero entries. Comments on A268759 provide a conjecture for the clear interpretation of these numbers in the context of binomial coefficients and other geometrical sequences. - Bradley Klee, Feb 20 2016
This sequence appears in the probability of the coin tossing "Gambler's Ruin". Call the probability of winning a coin toss = p, and the probability of losing the toss is 1-p = q, and call z = q/p. A gambler starts with $1, and tosses for $1 stakes till he has $0 (ruin) or has $n (wins). The average time T_win_lose(n) of a game (win OR lose) is a well-known function of z and n. The probability of the gambler winning P_win(n) is also known, and is equal to (1-z)/(1-z^n). T_win(n) defined as the average time it takes the gambler to win such a game is not so well known (I have not found it in the literature). I calculated T_win(n) and found it to be T_win(n) = P_win(n) * Sum_{m=0..n} T(n,m) * z^m. - Steve Newman, Oct 24 2016
As a square array A(n,m), gives the odd number's index of the product of n-th and m-th odd number. See formula. - Rainer Rosenthal, Sep 07 2022

			As a triangle:
  { 0},
  { 1,  1},
  { 2,  4,  2},
  { 3,  7,  7,  3},
  { 4, 10, 12, 10,  4},
  { 5, 13, 17, 17, 13,  5},
  { 6, 16, 22, 24, 22, 16,  6},
  { 7, 19, 27, 31, 31, 27, 19,  7},
  { 8, 22, 32, 38, 40, 38, 32, 22,  8},
  { 9, 25, 37, 45, 49, 49, 45, 37, 25,  9},
  {10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10}
From _Peter Munn_, Sep 28 2022: (Start)
Square array A(n,m) starts:
  0,  1,  2,  3,  4,  5,  6,  7, ...
  1,  4,  7, 10, 13, 16, 19, 22, ...
  2,  7, 12, 17, 22, 27, 32, 37, ...
  3, 10, 17, 24, 31, 38, 45, 52, ...
  4, 13, 22, 31, 40, 49, 58, 67, ...
  5, 16, 27, 38, 49, 60, 71, 82, ...
  6, 19, 32, 45, 58, 71, 84, 97, ...
...
(End)

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Michael De Vlieger, Table of n, a(n) for n = 0..10010 (rows 0 <= n <= 70, flattened)
Tomislav Došlić, On the Laplacian Szeged Spectrum of Paths, Iranian J. Math. Chem. (2020) Vol. 11, No. 1, 57-63.
W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum, Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

[0, 0] together with the row sums give A007290.

Cf. A003991, A268759, A094053, A098353, A114327.

T[n_, m_] = n + 2* m *(-m + n);
a = Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[a]
(* second program: *)
Flatten[ Table[2 j + 2 j^2 - 2 m^2, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Feb 20 2016 *)

{T(n, m) = if( m<0 || nMichael Somos, May 28 2017

T(n,m) = n + 2*m*(n-m).

Square array A(n,m) = 2*n*m + n + m, read by antidiagonals, satisfying 2*A(n,m) + 1 = (2*n+1)*(2*m+1) = A005408(n)*A005408(m) = A098353(n+1,m+1). - Rainer Rosenthal, Oct 01 2022

Edited by N. J. A. Sloane, Feb 21 2016

A098352 Multiplication table of the even numbers read by antidiagonals.

Original entry on oeis.org

4, 8, 8, 12, 16, 12, 16, 24, 24, 16, 20, 32, 36, 32, 20, 24, 40, 48, 48, 40, 24, 28, 48, 60, 64, 60, 48, 28, 32, 56, 72, 80, 80, 72, 56, 32, 36, 64, 84, 96, 100, 96, 84, 64, 36, 40, 72, 96, 112, 120, 120, 112, 96, 72, 40, 44, 80, 108, 128, 140, 144, 140, 128, 108, 80, 44

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

			   4   8  12  16  20  24  28  32
   8  16  24  32  40  48  56  64
  12  24  36  48  60  72  84  96
  16  32  48  64  80  96 112 128
  20  40  60  80 100 120 140 160
  24  48  72  96 120 144 168 192
  28  56  84 112 140 168 196 224
  32  64  96 128 160 192 224 256

G. C. Greubel, Antidiagonals n = 1..100, flattened

Cf. A003991, A098353.

Flat(List([1..12], n-> List([1..n], k-> 4*k*(n-k+1) ))); # G. C. Greubel, Aug 16 2019

[4*k*(n-k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 16 2019

seq(seq(4*k*(n-k+1), k = 1..n), n = 1..12); # G. C. Greubel, Aug 16 2019

Table[4*k*(n-k+1), {n,12}, {k,n}]//Flatten (* G. C. Greubel, Aug 16 2019 *)

T(n,k) = 4*k*(n-k+1); \\ G. C. Greubel, Aug 16 2019

[[4*k*(n-k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Aug 16 2019

T(n,k) = 4*A003991(n,k). - R. J. Mathar, Dec 08 2015

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A141387 Triangle read by rows: T(n,m) = n + 2m(n - m) (0 <= m <= n).

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A098352 Multiplication table of the even numbers read by antidiagonals.

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cp's OEIS Frontend

A141387 Triangle read by rows: T(n,m) = n + 2*m*(n - m) (0 <= m <= n).

Views

Author

Keywords

Comments

Examples

References

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Programs

Mathematica

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Extensions

A098352 Multiplication table of the even numbers read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A141387 Triangle read by rows: T(n,m) = n + 2m(n - m) (0 <= m <= n).