OFFSET
0,4
COMMENTS
From Clark Kimberling, May 31 2011: (Start)
If we arrange A000027 as an array with northwest corner
1 2 4 7 17 ...
3 5 8 12 18 ...
6 9 13 18 24 ...
10 14 19 25 32 ...
diagonals can be numbered as follows depending on their distance to the main diagonal:
Diag 0: 1, 5, 13, 25, ...
Diag 1: 2, 8, 18, 32, ...
Diag -1: 3, 9, 19, 33, ...,
then a(n) in the flattened array is the number of the diagonal that contains n+1. (End)
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in Jordan-Schwinger form (cf. Harter, Klee, Schwinger). Triangle terms T(n,k) = T(2j,j-m) satisfy: (1/2) T(2j,j-m) = <j,m|J_3|j,m> = m. Matrix J_3 is diagonal, so this equality determines the only nonzero entries. - Bradley Klee, Jan 29 2016
For the characteristic polynomial of the n X n matrix M_n (Det(x*1_n - M_n)) with elements M_n(i, j) = i-j see the Michael Somos, Nov 14 2002, comment on A002415. - Wolfdieter Lang, Feb 05 2018
The entries of the n-th antidiagonal, T(n,1), T(n-1,2), ... , T(1,n), are the eigenvalues of the Hamming graph H(2,n-1) (or hypercube Q(n-1)). - Miquel A. Fiol, May 21 2024
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
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.
FORMULA
G.f. for the table: Sum_{n, m>=0} T(n,m)*x^n*y^n = (x-y)/((1-x)^2*(1-y)^2).
E.g.f. for the table: Sum_{n, m>=0} T(n,m)x^n/n!*y^m/m! = (x-y)*e^{x+y}.
a(n+1) = A004736(n) - A002260(n) or a(n+1) = ((t*t+3*t+4)/2-n) - (n-t*(t+1)/2), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012
G.f. as sequence: -(1+x)/(1-x)^2 + (Sum_{j>=0} (2*j+1)*x^(j*(j+1)/2) / (1-x). The sum is related to Jacobi theta functions. - Robert Israel, Jan 29 2016
Triangle t(n, k) = n - 2*k, for n >= 0, k = 0..n. (see the Maple program). - Wolfdieter Lang, Feb 05 2018
EXAMPLE
From Wolfdieter Lang, Feb 05 2018: (Start)
The table T(n, m) begins:
n\m 0 1 2 3 4 5 ...
0: 0 -1 -2 -3 -4 -5 ...
1: 1 0 -1 -2 -3 -4 ...
2: 2 1 0 -1 -2 -3 ...
3: 3 2 1 0 -1 -2 ...
4: 4 3 2 1 0 -1 ...
5: 5 4 3 2 1 0 ...
...
The triangle t(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 0
1: 1 -1
2: 2 0 -2
3: 3 1 -1 -3
4: 4 2 0 -2 -4
5: 5 3 1 -1 -3 -5
6: 6 4 2 0 -2 -4 -6
7: 7 5 3 1 -1 -3 -5 -7
8: 8 6 4 2 0 -2 -4 -6 -8
9: 9 7 5 3 1 -1 -3 -5 -7 -9
10: 10 8 6 4 2 0 -2 -4 -6 -8 -10
... Reformatted and corrected. (End)
MAPLE
seq(seq(i-2*j, j=0..i), i=0..30); # Robert Israel, Jan 29 2016
MATHEMATICA
max = 12; a025581 = NestList[Prepend[#, First[#]+1]&, {0}, max]; a002262 = Table[Range[0, n], {n, 0, max}]; a114327 = a025581 - a002262 // Flatten (* Jean-François Alcover, Jan 04 2016 *)
Flatten[Table[-2 m, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)
PROG
(Haskell)
a114327 n k = a114327_tabl !! n !! k
a114327_row n = a114327_tabl !! n
a114327_tabl = zipWith (zipWith (-)) a025581_tabl a002262_tabl
-- Reinhard Zumkeller, Aug 09 2014
(PARI) T(n, m) = n-m \\ Charles R Greathouse IV, Feb 07 2017
(Python)
from math import isqrt
def A114327(n): return ((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)))**2+1-k # Chai Wah Wu, Nov 09 2024
CROSSREFS
KEYWORD
AUTHOR
Franklin T. Adams-Watters, Feb 06 2006
EXTENSIONS
Formula improved by Reinhard Zumkeller, Aug 09 2014
STATUS
approved