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A225918 a(n) = least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n)-1) for n > 1, where f(n) = 1/(n+3) and a(1) = 1. 6
1, 9, 32, 98, 287, 828, 2377, 6812, 19510, 55866, 159958, 457987, 1311283, 3754381, 10749290, 30776629, 88117519, 252291984 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Suppose that f(n) is a sequence of positive real numbers for which the series f(1) + f(2) + ... diverges.  Put a(1) = f(1) and a(n) = least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n)-1) for n > 1.  Conjecture: a(n) is linearly recurrent for the choices of f(n) shown here:

f(n) ...... a(n)................ recurrence coefficients

1/n ....... A003462: 1,4,13,.... (4,-3)

1/(n+1) ... A134931: 1,6,21,.... (4,-3)

1/(n+2) ... A116952: 1,8,29,.... (4,-3)

1/(n+3) ... A225918: 1,9,32,.... (3,0,-1,0,-1)

1/(n+4) ... A225919: 1,11,40,... (4,-4,3,-2)

1/(n+5) ... A225920: 1,13,48,... (3,1,-4,1)

1/(n+6) ... A225921: 1,14,50,... ?

1/(n+7) ... A225922: 1,16,48,... (2,3,-1,-3)

Assuming linear recurrence, it appears that lim a(n+1)/a(n) is the greatest root, R, of the characteristic polynomial of the recurrence, and that lim( 1/(a(n-1)+1) + ... + 1/a(n) ) = log R.

LINKS

Table of n, a(n) for n=1..18.

EXAMPLE

a(1) = 1 by decree; a(2) = 9 because

1/4 + ... + 1/8 < 1 < 1/4 + ... + 1/9, so that a(3) = 32 because

1/10 + ... + 1/31 < 1/4 + ... + 1/9 < 1/10 + ... + 1/32.

Successive values of a(n) yield a chain:

1 < 1/4 + ... + 1/9 < 1/10 + ... + 1/32 < 1/33 + ... + 1/98 < ...

Abbreviating this chain as b(1) = 1 < b(2) < b(3) < b(4) < ... < R = 2.8631..., it appears that lim b(n) = log R = 1.0519... .

MATHEMATICA

nn = 11; f[n_] := 1/(n + 3); a[1] = 1; g[n_] := g[n] = Sum[f[k], {k, 1, n}]; s = 0; a[2] = NestWhile[# + 1 &, 2, ! (s += f[#]) >= a[1] &]; s = 0; a[3] = NestWhile[# + 1 &, a[2] + 1, ! (s += f[#]) >= g[a[2]] - f[1] &]; Do[s = 0; a[z] = NestWhile[# + 1 &, a[z - 1] + 1, ! (s += f[#]) >= g[a[z - 1]] - g[a[z - 2]] &], {z, 4, nn}]; m = Map[a, Range[nn]] (* Peter J. C. Moses, May 13 2013 *)

CROSSREFS

Cf. A003462 (conjectured, for f(n) = 1/n);

A134931 (conjectured, for f(n) = 1/(n+1));

A116952 (conjectured, for f(n) = 1/(n+2)).

Sequence in context: A027620 A152619 A051662 * A231999 A229444 A196016

Adjacent sequences:  A225915 A225916 A225917 * A225919 A225920 A225921

KEYWORD

nonn,more

AUTHOR

Clark Kimberling, May 21 2013

EXTENSIONS

a(12) - a(18) from Robert G. Wilson v, May 22 2013

STATUS

approved

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Last modified June 28 03:08 EDT 2017. Contains 288813 sequences.