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A259577 Sum of numbers in the n-th antidiagonal of the reciprocity array of 1. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..500

FORMULA

a(n) = sum{sum{floor((n*k + x)/m), k=0..m-1, m=1..n}, where x = 1.

a(n) = n^3 / 4 + O(n^2). - Charles R Greathouse IV, Mar 22 2017

MATHEMATICA

f[n_] := Sum[Floor[(n*k + 1)/m], {m, n}, {k, 0, m - 1}]; Array[f, 50]

PROG

(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);

main(size)=return(vector(size, n, a(n))) \\ Anders Hellström, Jul 06 2015

(PARI) a(n)=sum(m=1, n, sum(k=0, m-1, (n*k+1)\m)) \\ Charles R Greathouse IV, Mar 22 2017

CROSSREFS

Cf. A259572, A259574, A259575.

Sequence in context: A048094 A181522 A031872 * A172348 A254821 A192953

Adjacent sequences:  A259574 A259575 A259576 * A259578 A259579 A259580

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 01 2015

STATUS

approved

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Last modified June 5 09:32 EDT 2020. Contains 334829 sequences. (Running on oeis4.)