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A350003
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Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest lucky number L(k) such that all n-th differences of (L(k), ..., L(k+n+m)) are zero, where L is A000959; T(n,m) = 0 if no such number exists.
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2
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OFFSET
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2,1
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COMMENTS
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Equivalently, T(n,m) is the smallest lucky number L(k) such that there is a polynomial f of degree at most n-1 such that f(j) = L(j) for k <= j <= k+n+m.
T(n,m) = A000959(k), where k is the smallest positive integer such that A350001(n,k+j) = 0 for 0 <= j <= m.
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LINKS
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FORMULA
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T(n,m) <= T(n-1,m+1).
T(n,m) <= T(n, m+1).
Sum_{j=0..n} (-1)^j*binomial(n,j)*A000959(k+i+j) = 0 for 0 <= i <= m, where A000959(k) = T(n,m).
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EXAMPLE
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Array begins:
n\m| 0 1 2 3
---+-----------------------------------
2 | 37 87 87 72979
3 | 31 87 17781 196089
4 | 87 1263 196089 63955483
5 | 31 3687 17622975 ?
6 | 517 390015 ? ?
7 | 1797 1797 ? ?
8 | 1797 2432367 ? ?
9 | 267 9157647 ? ?
10 | 483 1683501 ? ?
For n = 4 and m = 1, the first six (n+m+1) consecutive lucky numbers for which all fourth (n-th) differences are 0 are (1263, 1275, 1281, 1285, 1291, 1303), so T(4,1) = 1263. The successive differences are (12, 6, 4, 6, 12), (-6, -2, ,2, 6), (4, 4, 4), and (0, 0).
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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