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A225918 a(n) is the 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. 7
1, 9, 32, 98, 287, 828, 2377, 6812, 19510, 55866, 159958, 457987, 1311283, 3754381, 10749290, 30776629, 88117519, 252291984, 722344942 (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) = 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,... ?

1/(n+6) ... A225921: 1,14,50,... (3,0,-1,-3,2,2,-1,-1)

1/(n+7) ... A225922: 1,16,48,... ?

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

LINKS

Table of n, a(n) for n=1..19.

EXAMPLE

a(1) = 1 by decree; a(2) = 9 because 1/5 + ... + 1/11 < 1 < 1/5 + ... + 1/(9+3), so that a(3) = 32 because 1/13 + ... + 1/34 < 1/5 + ... + 1/12 < 1/13 + ... + 1/(32+3).

Successive values of a(n) yield a chain: 1 < 1/(1+4) + ... + 1/(9+3) < 1/(9+4) + ... + 1/(32+3) < 1/(32+4) + ... + 1/(98+3) < ...

Abbreviating this chain as b(1) = 1 < b(2) < b(3) < b(4) < ... < R = 2.8631..., it appears that lim_{n->infinity} 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 *)

PROG

(PARI) lista(nn) = {default(realprecision, 100); my(k=5, r=1, s); print1(1); for(n=2, nn, s=0; while((s+=1./k)<r, k++); r=s; print1(", ", (k++)-4)); } \\ Jinyuan Wang, Jun 14 2020

CROSSREFS

Cf. A003462, A134931, A116952.

Cf. A225919, A225920, A225921, A225922.

Sequence in context: A152619 A051662 A326247 * A231999 A297298 A229444

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

a(19) from Jinyuan Wang, Jun 14 2020

STATUS

approved

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Last modified July 8 22:01 EDT 2020. Contains 335537 sequences. (Running on oeis4.)