A226205 -id:A226205 - cp's OEIS frontend

A379048 Irregular triangular array: row n is the linear recurrence signature of F(i)^n - F(i-1)^n, where F = A000045 (Fibonacci numbers).

1, 1, 2, 2, -1, 3, 6, -3, -1, 4, 19, 4, -1, 8, 40, -60, -40, 8, 1, 13, 104, -260, -260, 104, 13, -1, 21, 273, -1092, -1820, 1092, 273, -21, -1, 33, 747, -3894, -16270, -3894, 747, 33, -1, 55, 1870, -19635, -85085, 136136, 85085, -19635, -1870, 55, 1, 89

Offset: 1

Clark Kimberling, Dec 16 2024

(column 1) = (1, 2, 3, 4, 8, 13, 21, 33, 55, 89, 144, 232, ...) is linearly recurrent with signature (1,1,0,1,-1,-1).
(column 2) = (1, 2, 6, 19, 40, 104, 273, 747, 1870, 4895, 12816, 33784, ...) is linearly recurrent with signature (3, -1, 0, 8, -24, 8, 0, -8, 24, -8, 0, 1, -3, 1).
Row n is also the linear recurrence signature of L(i)^n - L(i-1)^n, where L = A000032 (Lucas numbers).

Cf. A000032, A000045, A226205, A346513.

z = 25; w[i_] := Fibonacci[i];
t[p_] := Table[w[i]^p - w[i - 1]^p, {i, 1, z}];
Column[Table[FindLinearRecurrence[t[p]], {p, 1, 12}]]  (* array *)
Flatten[Table[FindLinearRecurrence[t[p]], {p, 1, 12}]]    (* sequence *)

First 10 rows:
   1
   2      -1
   6      -3       -1
  19       4       -1
  40     -60      -40        8        1
 104    -260     -260      104       13       -1
 273   -1092    -1820     1092      273      -21      -1
 747   -3894   -16270    -3894      747       33      -1
1870  -19635   -85085   136136    85085   -19635   -1870    55   1
4895  -83215  -582505  1514513  1514513  -582505  -83215  4895  89  -1

cp's OEIS Frontend

A379048 Irregular triangular array: row n is the linear recurrence signature of F(i)^n - F(i-1)^n, where F = A000045 (Fibonacci numbers).

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