cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A205868 The least number j such that 8 divides s(k)-s(j), where k(n)=A205867(n) and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

4, 4, 6, 2, 1, 5, 1, 10, 1, 10, 12, 2, 8, 3, 4, 6, 7, 5, 11, 4, 6, 7, 16, 4, 6, 7, 16, 18, 2, 8, 14, 9, 1, 10, 12, 13, 5, 11, 17, 1, 10, 12, 13, 22, 1, 10, 12, 13, 22, 24
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205867.)

Crossrefs

Cf. A204890, A205867, A205840.

Programs

  • Mathematica
    (See the program at A205867.)

A205869 s(k)-s(j), where the pairs (k,j) are given by A205867 and A205868, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

8, 16, 8, 32, 88, 136, 232, 144, 376, 288, 144, 608, 576, 984, 1592, 1584, 1576, 2576, 2440, 4176, 4168, 4160, 2584, 6760, 6752, 6744, 5168, 2584, 10944, 10912, 10336, 17656, 28656, 28568, 28424, 28280, 46360, 46224, 43784, 75024, 74936
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205867.)

Crossrefs

Cf. A204890, A205867, A205840.

Programs

  • Mathematica
    (See the program at A205867.)

A205870 [s(k)-s(j)]/8, where the pairs (k,j) are given by A205867 and A205868, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 2, 1, 4, 11, 17, 29, 18, 47, 36, 18, 76, 72, 123, 199, 198, 197, 322, 305, 522, 521, 520, 323, 845, 844, 843, 646, 323, 1368, 1364, 1292, 2207, 3582, 3571, 3553, 3535, 5795, 5778, 5473, 9378, 9367, 9349, 9331, 5796, 15174, 15163, 15145, 15127
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(6)-s(4) = 13-5 = 8 = 8*1
s(7)-s(4) = 21-5 = 16 = 8*2
s(7)-s(6) = 21-13 = 8 = 8*1
s(8)-s(2) = 34-2 = 32 = 8*4
s(10)-s(1) = 89-1 = 88 = 8*11
s(11)-s(5) = 144-8 = 136 =8*17

Crossrefs

Cf. A204892, A205867, A205869.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
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 = 8; t = d[c]    (* A205866 *)
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}]   (* A205867 *)
Table[j[n], {n, 1, z2}]     (* A205868 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205869 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205870 *)

A205840 [s(k)-s(j)]/2, where the pairs (k,j) are given by A205837 and A205838.

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

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

Examples

			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

Crossrefs

Cf. A204890, A205587, A205839.

Programs

  • 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 *)

A205866 Positions of multiples of 8 in A204922 (differences of Fibonacci numbers).

Original entry on oeis.org

14, 19, 21, 23, 37, 50, 56, 65, 67, 76, 78, 80, 86, 94, 109, 111, 112, 125, 131, 140, 142, 143, 152, 157, 159, 160, 169, 171, 173, 179, 185, 199, 211, 220, 222, 223, 236, 242, 248, 254, 263, 265, 266, 275, 277, 286, 288, 289, 298, 300, 302, 308, 314
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			In A204922=(1,2,1,4,3,2,7,6,5,3,12,11,...), multiples of 8 are in positions 14,19,21,...  See the example at A205867.

Links

Crossrefs

Cf. A204890, A205867, A205840.

Programs

  • Maple
    R:= select(t -> combinat:-fibonacci(t[1])-combinat:-fibonacci(t[2]) mod 8 =0, [seq(seq([i,j],i=0..11),j=0..11)]):
Res:= NULL:
for kk from 3 to 50 do
   km:= kk mod 12;
   n0:= 1 + (kk-2)*(kk-3)/2;
   js:= [$n0 ..(n0+kk-3)];
   Res:= Res, op( select(t -> member([kk, t+2-n0] mod 12,R), js));
od:
Res; # Robert Israel, Jul 27 2020
  • Mathematica
    (See the program at A205867.)
Showing 1-5 of 5 results.

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