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.

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A205877 Numbers k for which 10 divides s(k)-s(j) for some j

Original entry on oeis.org

6, 7, 9, 11, 12, 12, 15, 16, 16, 17, 17, 18, 18, 19, 19, 21, 21, 21, 22, 22, 22, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 27, 28, 29, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 33, 34, 34
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, A205878, A205880.

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 *)

A050939 Numbers that are not the sum of consecutive Fibonacci numbers.

Original entry on oeis.org

9, 14, 15, 17, 22, 23, 24, 25, 27, 28, 30, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 51, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 85, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101
Offset: 1

Views

Author

N. J. A. Sloane, Jan 02 2000

Keywords

Comments

From Clark Kimberling, Dec 16 2009: (Start)
(1) This is the ordered sequence of positive numbers that are not the difference between two Fibonacci numbers; see A007298 for a proof.
(2) Let s=(1,2,1,4,2,1,7,4,2,1,12,7,4,2,1,...) be the lengths of runs of consecutive numbers missing from A050939. Is s=A104582? (End)

Crossrefs

Cf. A007298, A204922, A204924.

Programs

  • Mathematica
    (See A204924, which generates an ordered list of differences of Fibonacci numbers, as in A204922.)

A205850 [s(k)-s(j)]/4, where the pairs (k,j) are given by A205847 and A205848, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 3, 2, 5, 4, 2, 8, 13, 22, 21, 19, 17, 34, 58, 57, 55, 53, 36, 94, 93, 91, 89, 72, 36, 152, 144, 246, 233, 399, 398, 396, 394, 377, 341, 305, 644, 610, 1045, 1044, 1042, 1040, 1023, 987, 951, 646, 1691, 1690, 1688, 1686, 1669, 1633, 1597, 1292, 646
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(4)-s(1) = 5-1 = 4 = 4*1
s(6)-s(1) = 13-1 = 12= 4*3
s(6)-s(4) = 13-5 = 8 = 4*2
s(7)-s(1) = 21-1 = 20 = 4*5
s(7)-s(4) = 21-5 = 16 = 4*4
s(7)-s(6) = 21-13 = 8 = 4*2

Crossrefs

Cf. A204892, A205845, A205849.

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 = 4; t = d[c]    (* A205846 *)
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}]    (* A205847 *)
Table[j[n], {n, 1, z2}]    (* A205848 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205849 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205850 *)

A204928 s(k(n)) - s(j(n)), where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 18, 10, 11, 12, 13, 42, 225, 16, 34, 18, 19, 20, 21, 88, 230, 144, 50, 26, 54, 84, 29, 2550, 31, 32, 33, 34, 17710, 144, 555, 76, 4173, 2440, 123, 42, 86, 88, 225, 230, 47, 144, 343, 50, 2550, 52, 53, 54, 55, 2576, 228, 232, 121304
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

The ordering of the pairs (s(k(n)), s(j(n))) is given by A204922. For a guide to related sequences, see A204892.

Links

Crossrefs

Cf. A204900, A204892, A204929.

Programs

  • Mathematica
    (See the program at A204924.)

A205837 Numbers k for which 2 divides s(k)-s(j) for some j

Original entry on oeis.org

3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(3)-s(1) = 3-1 = 2
s(4)-s(1) = 5-1 = 4
s(4)-s(3) = 5-3 = 2
s(5)-s(2) = 8-2 = 6
s(6)-s(1) = 13-1 = 12
s(6)-s(3) = 13-3 = 10

Crossrefs

Cf. A204892, A205840, A205558.

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 *)

A205855 [s(k)-s(j)]/5, where the pairs (k,j) are given by A205852 and A205853, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 2, 1, 4, 10, 11, 22, 11, 46, 45, 44, 75, 121, 111, 197, 122, 319, 244, 122, 510, 499, 488, 836, 832, 1352, 1342, 1231, 2189, 2185, 1353, 3542, 3538, 2706, 1353, 5731, 5656, 5534, 5412, 9273, 9272, 9271, 9227, 15004, 14994, 14883, 13652
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(5)-s(3) = 8-3 = 5 = 5*1
s(6)-s(3) = 13-3 = 10 = 5*2
s(6)-s(5) = 13-8 = 5 = 5*1
s(7)-s(1) = 21-1 = 20 = 5*4
s(9)-s(4) = 55-5 = 50 = 5*10
s(10)-s(8) = 89-34 = 55 =5*11

Crossrefs

Cf. A204892, A205852, A205854.

Programs

  • 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 = 5; t = d[c]    (* A205851 *)
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}]   (* A205852 *)
Table[j[n], {n, 1, z2}]   (* A205853 *)
Table[s[k[n]]-s[j[n]], {n,1,z2}] (* A205854 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205855 *)

A205860 [s(k)-s(j)]/6, where the pairs (k,j) are given by A205857 and A205858, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 2, 3, 9, 7, 14, 38, 24, 62, 48, 24, 96, 164, 161, 266, 264, 257, 425, 329, 696, 682, 658, 634, 1127, 1124, 963, 1824, 1823, 2951, 2937, 2913, 2889, 2255, 4776, 4774, 4767, 4510, 7704, 12504, 12502, 12495, 12238, 7728, 20232, 20230, 20223
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(5)-s(2) = 8-2 = 6 = 6*1
s(6)-s(1) = 13-1 = 12 = 6*2
s(7)-s(3) = 21-3 = 18 = 6*3
s(9)-s(1) = 55-1 = 54 = 6*9
s(9)-s(6) = 55-13 = 42 = 6*7
s(10)-s(4) = 89-5 = 84 =6*14

Crossrefs

Cf. A204892, A205857, A205859.

Programs

  • 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 = 6; t = d[c]    (* A205856 *)
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}]     (* A205857 *)
Table[j[n], {n, 1, z2}]     (* A205858 *)
Table[s[k[n]]-s[j[n]], {n, 1, z2}]    (* A205859 *)
Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}]  (* A205860 *)

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.

Original entry on oeis.org

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
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(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

Crossrefs

Cf. A204892, A205862, A205864.

Programs

  • 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 *)

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 *)

A205875 [s(k)-s(j)]/9, where the pairs (k,j) are given by A205872 and A205873, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

2, 6, 9, 25, 16, 41, 32, 16, 64, 176, 287, 281, 464, 642, 1216, 1967, 1958, 1942, 1926, 3184, 3178, 2897, 5136, 8336, 8330, 8049, 5152, 13488, 13482, 13201, 10304, 5152, 21824, 20608, 35312, 35310, 57136, 56672, 92448, 92439, 92423, 92407
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(7)-s(3) = 21-3 = 18 = 9*2
s(9)-s(1) = 55-1 = 54 = 9*6
s(10)-s(5) = 89-8 = 81 = 9*9
s(12)-s(5) = 233-8 = 225 = 9*25
s(12)-s(10) = 233-89 = 144 = 9*16
s(13)-s(5) = 377-8 = 369 =9*41

Crossrefs

Cf. A204892, A205872, A205874.

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 = 9; t = d[c]     (* A205871 *)
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}]       (* A205872 *)
Table[j[n], {n, 1, z2}]         (* A205873 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]   (* A205874 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205875 *)
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