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A112543 Numerators of fractions n/m in array by antidiagonals. 3
1, 2, 1, 3, 1, 1, 4, 3, 2, 1, 5, 2, 1, 1, 1, 6, 5, 4, 3, 2, 1, 7, 3, 5, 1, 3, 1, 1, 8, 7, 2, 5, 4, 1, 2, 1, 9, 4, 7, 3, 1, 2, 3, 1, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 5, 3, 2, 7, 1, 5, 1, 1, 1, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 6, 11, 5, 9, 4, 1, 3, 5, 2, 3, 1, 1, 14, 13, 4, 11, 2, 3, 8, 7, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column m has period m. - Clark Kimberling, Jul 04 2013

Read as a triangle with antidiagonals for rows, T(n,k) gives the number of distinct locations at which two points the same distance from a center rotating around that center in the same direction at speeds n+1 and k will coincide. - Thomas Anton, Nov 23 2018

LINKS

Clark Kimberling, Antidiagonals n = 1..60, flattened

Eric Weisstein's World of Mathematics, Exponential Integral

FORMULA

For n/m: g.f.: x/(1-x)*log(1/(1-y)), e.g.f.: x*e^x*(Ei(y)-log(y)+gamma) = x*e^x*integral_t=0^y (e^t-1)dt.

n/gcd(m,n). - Clark Kimberling, Jul 04 2013

EXAMPLE

a(2,4)=1/2 because 2/4 = 1/2.

Northwest corner:

1 2 3 4 5 6 7

1 1 3 2 5 3 7

1 2 1 4 5 2 7

1 1 3 1 5 3 7

1 2 3 4 1 1 7

1 1 1 2 5 1 7

1 2 3 4 5 6 1

MATHEMATICA

d[m_, n_] := n/GCD[m, n]; z = 12;

TableForm[Table[d[m, n], {m, 1, z}, {n, 1, z}] ] (*array*)

Flatten[Table[d[k, m + 1 - k], {m, 1, z}, {k, 1, m}]] (*sequence*)

(* Clark Kimberling, Jul 04 2013 *)

PROG

(PARI) t1(n)=binomial(floor(3/2+sqrt(2*n)), 2)-n+1 t2(n)=n-binomial(floor(1/2+sqrt(2*n)), 2) vector(100, n, t1(n)/gcd(t1(n), t2(n)))

CROSSREFS

Denominators in A112544. Reduced version of A004736/A002260.

Sequence in context: A011973 A115139 A198788 * A099478 A133913 A209485

Adjacent sequences:  A112540 A112541 A112542 * A112544 A112545 A112546

KEYWORD

easy,frac,nonn,tabl

AUTHOR

Franklin T. Adams-Watters, Sep 11 2005

EXTENSIONS

Keyword tabl added by Franklin T. Adams-Watters, Sep 02 2009

STATUS

approved

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Last modified November 22 18:55 EST 2019. Contains 329410 sequences. (Running on oeis4.)