|
%I #39 Nov 11 2021 09:52:46
%S 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,
%T 31,36,49,43,39,37,37,39,43,49,64,57,52,49,48,49,52,57,64,81,73,67,63,
%U 61,61,63,67,73,81,100,91,84,79,76,75,76,79,84,91,100,121,111,103,97
%N Array read by antidiagonals, A(n,k) = n^2 + n*k + k^2.
%C Norm of elements in planar hexagonal lattice A_2.
%C Only numbers which appear in A003136 (Loeschian numbers) can appear in this array. - _Peter Luschny_, Nov 10 2021
%H Michael De Vlieger, <a href="/A073254/b073254.txt">Table of n, a(n) for n = 0..11475</a> (triangular rows 0 <= n <= 150, flattened)
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>.
%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F From _Peter Luschny_, Oct 26 2011: (Start)
%F Let m = 2, for the cases m = 3, 4, and 5 see the cross-references.
%F T(n,k) = k^2 - k*n + n^2 = A(n-k,k).
%F 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.
%F T(n,0) = T(n,n) = n^m = n^2 = A000290(n).
%F T(2n,n) = (m+1)*n^m = 3*n^2 = A033428(n).
%F T(2n+1,n+1) = (n+1)^(m+1) - n^(m+1) = (n+1)^3 - n^3 = A003215(n).
%F Sum_{k=0..n} T(n,k) = (5*n^3 + 6*n^2 + n)/6 = A033994(n).
%F T(n+1, k+1)*binomial(n, k)^3/(k+1)^2 = A194595(n,k). (End)
%e Triangle T(n, k) starts:
%e [0] 0
%e [1] 1, 1
%e [2] 4, 3, 4
%e [3] 9, 7, 7, 9
%e [4] 16, 13, 12, 13, 16
%e [5] 25, 21, 19, 19, 21, 25
%e [6] 36, 31, 28, 27, 28, 31, 36
%e [7] 49, 43, 39, 37, 37, 39, 43, 49
%p # Using the triangle formula:
%p A073254 := (n,k) -> k^2 - k*n + n^2: # _Peter Luschny_, Oct 26 2011
%t (* Using the array formula: *)
%t A[n_, k_] := n^2 + n k + k^2;
%t Table[A[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o (PARI) {A(n, k) = n^2 + n*k + k^2}
%Y A033994 gives antidiagonal sums.
%Y Cf. A003136, A000290, A033428, A003215, A033994, A194595.
%Y Cf. A004016, A057427, A003056, A132111.
%Y Cf. A198063 (m=3), A198064 (m=4), A198065 (m=5).
%K nonn,tabl,easy
%O 0,4
%A _Michael Somos_, Jul 23 2002
%E Edited by _Peter Luschny_, Nov 10 2021
|
