OFFSET
0,4
COMMENTS
Norm of elements in planar hexagonal lattice A_2.
Only numbers which appear in A003136 (Loeschian numbers) can appear in this array. - Peter Luschny, Nov 10 2021
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (triangular rows 0 <= n <= 150, flattened)
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
FORMULA
From Peter Luschny, Oct 26 2011: (Start)
Let m = 2, for the cases m = 3, 4, and 5 see the cross-references.
T(n,k) = k^2 - k*n + n^2 = A(n-k,k).
T(n,k) = Sum_{j=0..m} Sum_{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j) for m = 2.
T(n,0) = T(n,n) = n^m = n^2 = A000290(n).
T(2n,n) = (m+1)*n^m = 3*n^2 = A033428(n).
T(2n+1,n+1) = (n+1)^(m+1) - n^(m+1) = (n+1)^3 - n^3 = A003215(n).
Sum_{k=0..n} T(n,k) = (5*n^3 + 6*n^2 + n)/6 = A033994(n).
T(n+1, k+1)*binomial(n, k)^3/(k+1)^2 = A194595(n,k). (End)
EXAMPLE
Triangle T(n, k) starts:
[0] 0
[1] 1, 1
[2] 4, 3, 4
[3] 9, 7, 7, 9
[4] 16, 13, 12, 13, 16
[5] 25, 21, 19, 19, 21, 25
[6] 36, 31, 28, 27, 28, 31, 36
[7] 49, 43, 39, 37, 37, 39, 43, 49
MAPLE
# Using the triangle formula:
A073254 := (n, k) -> k^2 - k*n + n^2: # Peter Luschny, Oct 26 2011
MATHEMATICA
(* Using the array formula: *)
A[n_, k_] := n^2 + n k + k^2;
Table[A[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
PROG
(PARI) {A(n, k) = n^2 + n*k + k^2}
CROSSREFS
A033994 gives antidiagonal sums.
KEYWORD
AUTHOR
Michael Somos, Jul 23 2002
EXTENSIONS
Edited by Peter Luschny, Nov 10 2021
STATUS
approved