A259577 - OEIS

A259577

Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.

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

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