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A259577
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Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.
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3
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1, 2, 6, 13, 26, 44, 72, 108, 156, 215, 290, 381, 486, 610, 758, 924, 1112, 1329, 1566, 1839, 2134, 2456, 2816, 3220, 3640, 4099, 4608, 5153, 5726, 6368, 7020, 7744, 8504, 9305, 10180, 11103, 12042, 13060, 14146, 15296, 16460, 17739, 19026, 20421, 21876
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OFFSET
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1,2
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COMMENTS
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The "reciprocity law" that Sum{[(n*k+x)/m] : k = 0..m} = Sum{[(m*k+x)/n] : k = 0..n} where x is a real number and m and n are positive integers, is proved in Section 3.5 of Concrete Mathematics (see References). See A259572 for a guide to related sequences.
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REFERENCES
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R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
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LINKS
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FORMULA
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a(n) = sum{sum{floor((n*k + x)/m), k=0..m-1, m=1..n}, where x = 1.
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MATHEMATICA
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f[n_] := Sum[Floor[(n*k + 1)/m], {m, n}, {k, 0, m - 1}]; Array[f, 50]
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PROG
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(PARI) a(n)=x=1; r=0; for(m=1, n, for(k=0, m-1, r=r+floor((n*k+x)/m))); return(r);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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