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.

A205878 The least number j such that 10 divides s(k)-s(j), where k(n)=A205877(n) and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

3, 1, 4, 8, 3, 6, 13, 13, 15, 8, 11, 1, 7, 4, 9, 1, 7, 18, 13, 15, 16, 5, 4, 9, 19, 3, 6, 12, 5, 23, 1, 7, 18, 21, 10, 14, 10, 28, 10, 28, 30, 5, 23, 26, 13, 15, 16, 22, 4, 9
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205877.)

Crossrefs

Cf. A204890, A205877, A205840.

Programs

  • Mathematica
    (See the program at A205877.)

A205879 s(k)-s(j), where the pairs (k,j) are given by A205877 and A205878, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

10, 20, 50, 110, 230, 220, 610, 1220, 610, 2550, 2440, 4180, 4160, 6760, 6710, 17710, 17690, 13530, 28280, 27670, 27060, 46360, 75020, 74970, 68260, 121390, 121380, 121160, 196410, 150050, 317810, 317790, 313630, 300100, 514140
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205877.)

Crossrefs

Cf. A204890, A205877, A205840.

Programs

  • Mathematica
    (See the program at A205877.)

A205880 [s(k)-s(j)]/10, where the pairs (k,j) are given by A205877 and A205878, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 2, 5, 11, 23, 22, 61, 122, 61, 255, 244, 418, 416, 676, 671, 1771, 1769, 1353, 2828, 2767, 2706, 4636, 7502, 7497, 6826, 12139, 12138, 12116, 19641, 15005, 31781, 31779, 31363, 30010, 51414, 83143, 134618, 83204, 217822, 166408, 83204
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first three terms match these differences:
s(6)-s(3) = 13-3 = 10 = 10*1
s(7)-s(1) = 21-1 = 20 = 10*2
s(9)-s(4) = 55-5 = 50 = 10*5

Crossrefs

Cf. A204892, A205877, A205879.

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 = 10; t = d[c]    (* A205876 *)
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}]   (* A205877 *)
Table[j[n], {n, 1, z2}]   (* A205878 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205879 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205880 *)

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 *)

A205876 Positions of multiples of 10 in A204922 (differences of Fibonacci numbers).

Original entry on oeis.org

13, 16, 32, 53, 58, 61, 104, 118, 120, 128, 131, 137, 143, 157, 162, 191, 197, 208, 223, 225, 226, 236, 257, 262, 272, 279, 282, 288, 305, 323, 326, 332, 343, 346, 361, 392, 416, 434, 445, 463, 465, 470, 488, 491, 509, 511, 512, 518, 532, 537, 547
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 10 are in positions 13,16,32,...  See the example at A205877.

Crossrefs

Cf. A204890, A205877, A205840.

Programs

  • Mathematica
    (See the program at A205877.)
Showing 1-5 of 5 results.

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