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Search: id:A090030
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| A090030 |
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Triangle read by rows: T(n,k) = number of distinct lines through the origin in the n-dimensional cubic lattice of side length k with one corner at the origin. |
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13
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| 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 5, 7, 0, 0, 1, 9, 19, 15, 0, 0, 1, 13, 49, 65, 31, 0, 0, 1, 21, 91, 225, 211, 63, 0, 0, 1, 25, 175, 529, 961, 665, 127, 0, 0, 1, 37, 253, 1185, 2851, 3969, 2059, 255, 0, 0, 1, 45, 415, 2065, 7471, 14833, 16129, 6305, 511, 0, 0, 1, 57, 571, 3745
(list; table; graph; listen)
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OFFSET
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0,9
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COMMENT
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Equivalently, number of lattice points where the GCD of all coordinates = 1.
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FORMULA
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With A(n, k) = A090225(n, k), T(n, k) =(k+1)^n - 1 - the sum for 0 < i < k of Floor[k/i-1]*A(n, i)
T(n, k) = Sum(moebius(i)*((floor((n-k)/i)+1)^k-1), i=1..n-k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 03 2004
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EXAMPLE
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T(n,1) = 2^n-1 because there are 2^n-1 lattice points other than the corner, all of which make distinct lines. T(n,2) = 3^n - 2^n because if the given corner is the origin, all the points with coordinates in {0,1} make lines that are redundant with a point containing a coordinate 2.
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MATHEMATICA
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aux[n_, k_] := If[k==0, 0, (k+1)^n-k^n-Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]]-1}]]; lines[n_, k_] := (k+1)^n-Sum[Floor[k/i-1]*aux[n, i], {i, 1, Floor[k/2]}]-1; lines[n, k]
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CROSSREFS
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Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 give T(n, k) for k = 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 give T(n, k) for n=2, 3, 4, 5, 6, 7 respectively. A090225 counts only points with at least one coordinate = k.
Sequence in context: A035696 A035630 A126723 this_sequence A080159 A060514 A096936
Adjacent sequences: A090027 A090028 A090029 this_sequence A090031 A090032 A090033
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KEYWORD
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nonn,tabl
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AUTHOR
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Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Nov 24 2003
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