

A073254


Table of n^2 + n*k + k^2 by antidiagonals.


7



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
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OFFSET

0,4


COMMENTS

Norm of elements in planar hexagonal lattice A_2.


LINKS

Table of n, a(n) for n=0..69.


FORMULA

T(n,k) = k^2kn+n^2.
T(n,k) = Sum{j=0..m} Sum{i=0..m} (1)^(j+i)*C(i,j)*n^j*k^(mj) for m = 2.
T(n,0) = T(n,n) = n^m = n^2 = A000290(n).
T(2n,n) = (m+1)n^m = 3n^2 = A033428(n).
T(2n+1,n+1) = (n+1)^(m+1)n^(m+1) = (n+1)^3n^3 = A003215(n).
Sum{k=0..n} T(n,k) = (5n^3+6n^2+n)/6 = A033994(n).
T(n+1,k+1)C(n,k)^3/(k+1)^2 = A194595(n,k).
 Peter Luschny, Oct 26 2011


EXAMPLE

[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

A073254 := (n, k) > k^2k*n+n^2: # Peter Luschny, Oct 26 2011


MATHEMATICA

T[n_, k_] := n^2 + n k + k^2;
Table[T[nk, k], {n, 0, 11}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Jun 22 2018 *)


PROG

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


CROSSREFS

A033994 gives antidiagonal sums. Cf. A004016, A057427, A003056, A198063, A198064, A198065.
Sequence in context: A132984 A277528 A118701 * A094177 A249453 A244954
Adjacent sequences: A073251 A073252 A073253 * A073255 A073256 A073257


KEYWORD

nonn,tabl,easy


AUTHOR

Michael Somos, Jul 23 2002


STATUS

approved



