A073254 Array read by antidiagonals, A(n,k) = n^2 + n*k + k^2.

0, 1, 1, 4, 3, 4, 9, 7, 7, 9, 16, 13, 12, 13, 16, 25, 21, 19, 19, 21, 25, 36, 31, 28, 27, 28, 31, 36, 49, 43, 39, 37, 37, 39, 43, 49, 64, 57, 52, 49, 48, 49, 52, 57, 64, 81, 73, 67, 63, 61, 61, 63, 67, 73, 81, 100, 91, 84, 79, 76, 75, 76, 79, 84, 91, 100, 121, 111, 103, 97

Offset: 0

Author

Michael Somos, Jul 23 2002

Keywords

• nonn
• tabl
• easy

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

Examples

			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
		

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.
• Index entries for sequences related to A2 = hexagonal = triangular lattice

Crossrefs

A033994 gives antidiagonal sums.

Cf. A003136, A000290, A033428, A003215, A033994, A194595.

Cf. A004016, A057427, A003056, A132111.

Cf. A198063 (m=3), A198064 (m=4), A198065 (m=5).

Programs

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 *)

PARI

{A(n, k) = n^2 + n*k + k^2}

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)

Extensions

Edited by Peter Luschny, Nov 10 2021