A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

Offset

1,4

Comments

T(n,k) = A003991(n-1,k) for 1 <= k < n;

T(n,k) = T(n,n-1-k) for k < n;

T(n,1) = n-1; T(n,n) = 0; T(n,2) = A005843(n-2) for n > 1;

T(n,3) = A008585(n-3) for n>2; T(n,4) = A008586(n-4) for n > 3;

T(n,5) = A008587(n-5) for n>4; T(n,6) = A008588(n-6) for n > 5;

T(n,7) = A008589(n-7) for n>6; T(n,8) = A008590(n-8) for n > 7;

T(n,9) = A008591(n-9) for n>8; T(n,10) = A008592(n-10) for n > 9;

T(n,11) = A008593(n-11) for n>10; T(n,12) = A008594(n-12) for n > 11;

T(n,13) = A008595(n-13) for n>12; T(n,14) = A008596(n-14) for n > 13;

T(n,15) = A008597(n-15) for n>14; T(n,16) = A008598(n-16) for n > 15;

T(n,17) = A008599(n-17) for n>16; T(n,18) = A008600(n-18) for n > 17;

T(n,19) = A008601(n-19) for n>18; T(n,20) = A008602(n-20) for n > 19;

Row sums give A000292; triangle sums give A000332;

All numbers m > 0 occur A000005(m) times;

A002378(n) = T(A005408(n),n+1) = n*(n+1).

k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013

Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) = <j,m|J_1|j,m+1> = <j,m+1|J_1|j,m> = i <j,m|J_2|j,m+1> = -i <j,m+1|J_2|j,m>. Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016

T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

Links

Table of n, a(n) for n=1..79.

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.

Example

From M. F. Hasler, Feb 02 2013: (Start)
Triangle begins:
  0;
  1, 0;
  2, 2, 0;
  3, 4, 3, 0;
  4, 6, 6, 4, 0;
  5, 8, 9, 8, 5, 0;
  (...)
If an additional 0 was added at the beginning, this would become:
  0;
  0, 1;
  0, 2, 2;
  0, 3, 4; 3;
  0, 4, 6, 6, 4;
  0, 5, 8, 9, 8, 5;
  ... (End)

Mathematica

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

Prog

(Magma) /* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016
(PARI) {for(n=1, 13, for(k=1, n, print1(k*(n - k), " "); ); print(); ); } \\ Indranil Ghosh, Mar 12 2017

Crossrefs

J_3: A114327; J_1^2, J_2^2: A141387, A268759.

Cf. A000005, A002378, A003991, A005408.

Cf. A000292 (row sums), A000332 (triangle sums).

T(n,k) for values of k:

A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

Sequence in context: A369524 A339422 A367746 * A294598 A077264 A349344

Adjacent sequences: A094050 A094051 A094052 * A094054 A094055 A094056

Keyword

nonn,tabl

Author

Reinhard Zumkeller, May 31 2004

Status

approved