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A073254
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Table of n^2+n*k+k^2 by antidiagonals.
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7
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Norm of elements in planar hexagonal lattice A_2.
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FORMULA
| T(n,k) = k^2-kn+n^2.
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 = 3n^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) = (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
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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
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MAPLE
| A073254 := (n, k) -> k^2-k*n+n^2: - Peter Luschny, Oct 26 2011
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PROG
| (PARI) T(n, k)=n^2+n*k+k^2
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CROSSREFS
| A033994 gives antidiagonal sums. Cf. A004016, A057427, A003056, A198063, A198064, A198065.
Sequence in context: A075246 A132984 A118701 * A094177 A060374 A074296
Adjacent sequences: A073251 A073252 A073253 * A073255 A073256 A073257
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KEYWORD
| nonn,tabl,easy
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AUTHOR
| Michael Somos, Jul 23, 2002
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