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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A256658 Rectangular array by antidiagonals: row n consists of numbers k such that F(n+1) is the trace of the minimal alternating Fibonacci representation of k, where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 9, 2, 14, 15, 3, 17, 23, 24, 5, 22, 28, 37, 39, 8, 27, 36, 45, 60, 63, 13, 30, 44, 58, 73, 97, 102, 21, 35, 49, 71, 94, 118, 157, 165, 34, 43, 57, 79, 115, 152, 191, 254, 267, 55, 48, 70, 92, 128, 186, 246, 309, 411, 432, 89, 51, 78, 113, 149, 207, 301
Offset: 1

Views

Author

Clark Kimberling, Apr 08 2015

Keywords

Comments

See A256655 for definitions. This array and the array at A256659 partition the positive integers. The row differences are Fibonacci numbers. The columns satisfy the Fibonacci recurrence x(n) = x(n-1) + x(n-2).

Examples

			Northwest corner:
1    9     14    17    22    27    30    35    43
2    15    23    28    36    44    49    57    70
3    24    37    45    58    71    79    92    113
5    39    69    73    94    115   128   149   183
8    63    97    118   152   186   207   241   296
13   102   157   191   246   301   335   390   479

Crossrefs

Cf. A000045, A256655, A256659.

Programs

  • Mathematica
    b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[18];  r[0] = {0};
 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
t = Table[Last[r[n]], {n, 0, 1000}];  (* A256656 *)
TableForm[Table[Flatten[-1 + Position[t, b[n]]], {n, 2, 8}]]   (* A256658 *)
TableForm[Table[Flatten[-1 + Position[t, -b[n]]], {n, 2, 8}]]  (* A256659 *)

A256659 Rectangular array by antidiagonals: row n consists of numbers k such that -F(n+1) is the trace of the minimal alternating Fibonacci representation of k, where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

4, 7, 6, 12, 11, 10, 20, 19, 18, 16, 25, 32, 31, 29, 26, 33, 40, 52, 50, 47, 42, 38, 53, 65, 84, 81, 76, 68, 41, 61, 86, 105, 136, 131, 123, 110, 46, 66, 99, 139, 170, 220, 212, 199, 178, 54, 74, 107, 160, 225, 275, 356, 343, 322, 288, 59, 87, 120, 173, 259
Offset: 1

Views

Author

Clark Kimberling, Apr 08 2015

Keywords

Comments

See A256655 for definitions. This array and the array at A256658 partition the positive integers. The row differences are Fibonacci numbers. The columns satisfy the Fibonacci recurrence x(n) = x(n-1) + x(n-2).

Examples

			Northwest corner:
4    7    12    20    25    33    38    41    46
6    11   19    32    40    53    61    66    74
10   18   31    52    65    86    99    102   120
16   29   50    84    105   139   160   173   194
26   47   81    136   170   225   259   280   314

Crossrefs

Cf. A000045, A256655, A246658.

Programs

  • Mathematica
    b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[18];  r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
t = Table[Last[r[n]], {n, 0, 1000}];  (* A256656 *)
TableForm[Table[Flatten[-1 + Position[t, b[n]]], {n, 2, 8}]]   (* A256658 *)
TableForm[Table[Flatten[-1 + Position[t, -b[n]]], {n, 2, 8}]]  (* A256659 *)

A256660 Number of terms in the minimal alternating Fibonacci representation of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 2, 1, 3, 3, 2, 3, 2, 2, 2, 1, 3, 3, 3, 4, 2, 3, 3, 2, 3, 2, 2, 2, 1, 3, 3, 3, 4, 3, 4, 4, 2, 3, 3, 3, 4, 2, 3, 3, 2, 3, 2, 2, 2, 1, 3, 3, 3, 4, 3, 4, 4, 3, 5, 4, 4, 4, 2, 3, 3, 3, 4, 3, 4, 4, 2, 3, 3, 3, 4, 2, 3, 3, 2, 3
Offset: 0

Views

Author

Clark Kimberling, Apr 08 2015

Keywords

Comments

See A256655 for definitions.

Links

Crossrefs

Cf. A000045, A256655.

Programs

  • Mathematica
    b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[12];  r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]
t = Table[Length[r[n]], {n, 0, 120}]  (* A256660 *)

A256661 Rectangular array by antidiagonals: row n shows the numbers k such that R(k) consists of n terms, where R(k) is the minimal alternating Fibonacci representation of k.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 7, 14, 25, 8, 10, 15, 38, 64, 13, 11, 17, 40, 98, 169, 21, 12, 22, 41, 103, 258, 441, 34, 16, 23, 46, 104, 271, 674, 1156, 55, 18, 24, 59, 106, 273, 708, 1766, 3025, 89, 19, 27, 61, 119, 274, 713, 1855, 4622, 7921, 144, 20, 28, 62, 153
Offset: 1

Views

Author

Clark Kimberling, Apr 08 2015

Keywords

Comments

See A256655 for definitions. Every positive integer occurs exactly once.
(row 1): A000045 (Fibonacci numbers)
(col 1): A007598 (squared Fibonacci numbers)
(col 2): A127546 (conjectured)

Examples

			Northwest corner:
1     2     3     5     8     13    21
4     6     7     10    11    12    62
9     14    15    17    22    23    24
25    38    40    41    46    59    61
64    98    103   104   106   119   153
169   258   271   273   274   279   313
R(1) = 1, in row 1
R(2) = 2, in row 1
R(3) = 3, in row 1
R(4) = 5 - 1, in row 2
R(9) = 13 - 5 + 1, in row 3
R(25) = 34 - 13 + 5 - 1, in row 4
R(64) = 89 - 34 + 13 - 5 + 1, in row 5

Crossrefs

Cf. A000045, A256655, A007598, A127546.

Programs

  • Mathematica
    b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[23];  r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
u = Table[Length[r[n]], {n, 1, 6000}];
TableForm[Table[Flatten[Position[u, k]], {k, 1, 9}]]
Previous Showing 11-14 of 14 results.

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