A205864 -id:A205864

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A205840

[s(k)-s(j)]/2, where the pairs (k,j) are given by A205837 and A205838.

+10
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.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 01 2012

STATUS

approved

A205862

Numbers k for which 7 divides s(k)-s(j) for some j<k; each k occurs once for each such j; s(k) denotes the (k+1)-st Fibonacci number.

+10
6

5, 8, 9, 9, 10, 12, 13, 13, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 27, 28, 28, 28, 29, 29, 29, 29, 29, 29, 29

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For a guide to related sequences, see A205840.

LINKS

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

EXAMPLE

The first six terms match these differences:

s(5)-s(1) = 8-1 = 7 = 7*1

s(8)-s(6) = 34-13 = 21 = 7*3

s(9)-s(6) = 55-13 = 42 = 7*6

s(9)-s(8) = 55-34 = 21 = 7*3

s(10)-s(4) = 89-5 = 84 = 7*12

s(13)-s(6) = 377-13 = 364 =7*52

MATHEMATICA

s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; 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 = 7; t = d[c] (* A205861 *)

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}] (* A205862 *)

Table[j[n], {n, 1, z2}] (* A205863 *)

Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205864 *)

Table[(s[k[n]]-s[j[n]])/c, {n, 1, z2}] (* A205865 *)

CROSSREFS

Cf. A204892, A205863, A205865.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 02 2012

STATUS

approved

A205865

[s(k)-s(j)]/7, where the pairs (k,j) are given by A205862 and A205863, and s(k) denotes the (k+1)-st Fibonacci number.

+10
3

1, 3, 6, 3, 12, 33, 52, 49, 46, 87, 86, 138, 228, 227, 141, 369, 368, 282, 141, 597, 564, 966, 1563, 1551, 2530, 2529, 2443, 2302, 2161, 4092, 4089, 4086, 4040, 6621, 6483, 10716, 10713, 10710, 10664, 6624, 17340, 17337, 17334, 17288, 13248

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For a guide to related sequences, see A205840.

LINKS

Table of n, a(n) for n=1..45.

EXAMPLE

The first six terms match these differences:

s(5)-s(1) = 8-1 = 7 = 7*1

s(8)-s(6) = 34-13 = 21 = 7*3

s(9)-s(6) = 55-13 = 42 = 7*6

s(9)-s(8) = 55-34 = 21 = 7*3

s(10)-s(4) = 89-5 = 84 = 7*12

s(13)-s(6) = 377-13 = 364 =7*52

MATHEMATICA

s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; 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 = 7; t = d[c] (* A205861 *)

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}] (* A205862 *)

Table[j[n], {n, 1, z2}] (* A205863 *)

Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205864 *)

Table[(s[k[n]]-s[j[n]])/c, {n, 1, z2}] (* A205865 *)

CROSSREFS

Cf. A204892, A205862, A205864.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 02 2012

STATUS

approved

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