A186158 Array associated with "the Mysterious B Sequence", by antidiagonals.

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

Offset: 1

Author

Clark Kimberling, Feb 15 2011

Keywords

• nonn
• tabl

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(2*k-1))^2, V=a(2*k), W=4*b(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+2*d,...) yields b(k)>0 for every k.

Peter Kosinar found a necessary and sufficient condition to be 0c, 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.

Examples

			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).
		

Links

• Clark Kimberling, Unsolved Problems and Rewards
• Clark Kimberling, Partial sums of generating functions as polynomial sequences, The Fibonacci Quarterly 48 (2010) 327-334. (See Theorem 1.)
• Peter Kosinar, On The Mysterious B Sequence

Programs

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

Formula

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