login
This site is supported by donations to The OEIS Foundation.

 

Logo

The submissions stack has been unacceptably high for several months now. Please voluntarily restrict your submissions and please help with the editing. (We don't want to have to impose further limits.)

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A186158 Array associated with "the Mysterious B Sequence", by antidiagonals. 1
18, 5, 165, 3, 18, 1333, 2, 8, 56, 10353, 2, 5, 18, 165, 78958, 1, 3, 9, 38, 472, 596438, 1, 3, 6, 18, 80, 1333, 4479398, 1, 2, 5, 11, 32, 165, 3727, 33514643, 1, 2, 4, 8, 18, 56, 333, 10353, 250104748, 1, 2, 3, 6, 12, 28, 96, 668, 28635, 1862945616, 1, 2, 3, 5, 9, 18, 45, 165, 1333, 78958 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This array represents the solution of Problem 7 in "Unsolved Problems and Rewards" in Links (below). Problem 7 is restated here:

For any sequence A=(a(0),a(1),...) of positive real numbers, create a sequence B as follows: let b(0)=a(0) and for k>0, let U=[a(2k-1)]^2, V=a(2k), W=4b(k-1), b(k)=V-U/W, and assume for each k that W is not zero.  Determine conditions on c and d for which the arithmetic sequence A=(c,c+d,c+2d,...) yields b(k)>0 for every k.

Peter Kosinar found a necessary and sufficient condition to be 0<d<=c.  He also proved that if d>c, then the sequence B contains one and only one negative number.  The number in row i, column j, is the unique k for which b(k)<0 when c=i and d=i+j.

REFERENCES

C. Kimberling, Partial sums of generating functions as polynomial sequences, The Fibonacci Quarterly 48 (2010) 327-334.  (See Theorem 1.)

LINKS

Table of n, a(n) for n=1..65.

C. Kimberling, Unsolved Problems and Rewards

C. Kimberling, Partial sums of generating functions as polynomial sequences (abstract), The Fibonacci Quarterly 48 (2010).

Peter Kosinar, On The Mysterious B Sequence

FORMULA

Starting with A=(c,c+d,c+2d,...), put b(0)=a(0) and for k>0, put U=[a(2k-1]^2, V=a(2k), W=4b(k-1), b(k)=V-U/W.

For i>=1 and j>=1, put f(i,i+j)=(the index k for which b(k)<0).  Then the array, T, is given by T(i,j)=f(i,i+j).

EXAMPLE

Northwest corner:

18.......5.....3....2...2...1...1...1...1

165......18....9....6...5...4...3...3...2

1333.....56....18...9...6...5...4...3...3

10353....165...38...18..11..8...6...5...4

78958....472...80...32..18..12..9...7...6

596438...1333..165..56..28..18..12..9...8

4479388..3727..333..96..45..26..18..13..10

Column 1 continues with 33514643,250104748,1862945616.

T(1,1)=18 because when (c,d)=(1,2), the only negative number in the sequence B is b(18).

MATHEMATICA

B[0, c_, d_]:=c;

B[k_, c_, d_]:=B[Mod[k, 2], c, d]=c+2d*k-((c+d(-1+2k))^2)/(4B[Mod[k-1, 2], c, d]);

Table[Table[NestWhile[#1+1&, 1, B[#1, c, d]>0&], {d, c+1, c+10}], {c, 1, 5}]//TableForm

(* by Peter J. C. Moses, Feb 08 2011 *)

CROSSREFS

Sequence in context: A040312 A214893 A065909 * A038642 A040311 A221351

Adjacent sequences:  A186155 A186156 A186157 * A186159 A186160 A186161

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 15 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 28 03:16 EDT 2015. Contains 261112 sequences.