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A205840 [s(k)-s(j)]/2, where the pairs (k,j) are given by A205837 and A205838. 45
1, 2, 1, 3, 6, 5, 4, 10, 9, 8, 4, 16, 13, 27, 26, 25, 21, 17, 44, 43, 42, 38, 34, 17, 71, 68, 55, 116, 115, 114, 110, 106, 89, 72, 188, 187, 186, 182, 178, 161, 144, 72, 304, 301, 288, 233, 493, 492, 491, 487, 483, 466, 449, 377, 305, 798, 797, 796, 792, 788 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let s(n)=F(n+1), where F=A000045 (Fibonacci numbers), so that s=(1,2,3,5,8,13,21,...).  If c is a positive integer, there are infinitely many pairs (k,j) such that c divides s(k)-s(j).  The set of differences s(k)-s(j) is ordered as a sequence at A204922.  Guide to related sequences:

c....k..........j..........s(k)-s(j)....[s(k)-s(j)]/c

2....A205837....A205838....A205839......A205840

3....A205842....A205843....A205844......A205845

4....A205847....A205848....A205849......A205850

5....A205852....A205853....A205854......A205855

6....A205857....A205858....A205859......A205860

7....A205862....A205863....A205864......A205865

8....A205867....A205868....A205869......A205870

9....A205872....A205873....A205874......A205875

10...A205877....A205878....A205879......A205880

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

The first six terms match these differences:

s(3)-s(1) = 3-1 = 2 = 2*1

s(4)-s(1) = 5-1 = 4 = 2*2

s(4)-s(3) = 5-3 = 2 = 2*1

s(5)-s(2) = 8-2 = 6 = 2*3

s(6)-s(1) = 13-1 = 12 = 2*6

s(6)-s(3) = 13-3 = 10 = 2*5

MATHEMATICA

s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;

f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];

Table[s[n], {n, 1, 30}]

u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

Table[u[m], {m, 1, z1}]   (* A204922 *)

v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]

c = 2; t = d[c]    (* A205556 *)

k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]

j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2

Table[k[n], {n, 1, z2}]    (* A205837 *)

Table[j[n], {n, 1, z2}]    (* A205838 *)

Table[s[k[n]] - s[j[n]], {n, 1, z2}](* A205839 *)

Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] * A205840 *)

CROSSREFS

Cf. A204890, A205587, A205839.

Sequence in context: A317788 A120576 A063707 * A171084 A332318 A118287

Adjacent sequences:  A205837 A205838 A205839 * A205841 A205842 A205843

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 01 2012

STATUS

approved

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Last modified May 28 16:37 EDT 2022. Contains 354119 sequences. (Running on oeis4.)