A294364 Linear recurrence with signature (1,1,-1,1,1), where the first terms are powers of 2 (1,2,4,8,16).

1, 2, 4, 8, 16, 23, 37, 56, 94, 152, 250, 401, 649, 1046, 1696, 2744, 4444, 7187, 11629, 18812, 30442, 49256, 79702, 128957, 208657, 337610, 546268, 883880, 1430152, 2314031, 3744181, 6058208, 9802390, 15860600, 25662994, 41523593, 67186585, 108710174, 175896760, 284606936

Offset

0,2

Comments

The interest of this sequence mainly lies in the peculiarities of its array of successive differences, which begins:

1, 2, 4, 8, 16, 23, 37, 56, 94, ...

1, 2, 4, 8, 7, 14, 19, 38, 58, ...

1, 2, 4, -1, 7, 5, 19, 20, 40, ...

1, 2, -5, 8, -2, 14, 1, 20, 13, ...

1, -7, 13, -10, 16, -13, 19, -7, 31, ...

-8, 20, -23, 26, -29, 32, -26, 38, -23, ...

28, -43, 49, -55, 61, -58, 64, -61, 67, ...

The main diagonal is A000079 (powers of 2).

The first upper subdiagonal is A254076.

The second upper subdiagonal (4, 8, 7, 14, 19, 38, ...) is not in the OEIS.

The third upper subdiagonal is A185346 (2^n-9).

Every row, once computed mod 9, is 6-periodic, repeating (1, 2, 4, 8, 7, 5) (A153130).

Links

Table of n, a(n) for n=0..39.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,1).

Formula

G.f.: (1+x+x^2+3*x^3+5*x^4) / (1-x-x^2+x^3-x^4-x^5).

a(n) = (9/2)*fibonacci(n) + (-1)^n - sqrt(3)*sin(n*Pi/3).

a(n) ~ (9/2)*fibonacci(n).

Mathematica

LinearRecurrence[{1, 1, -1, 1, 1}, {1, 2, 4, 8, 16}, 40]

Crossrefs

Cf. A000045, A000079, A153130, A185346, A254076.

Sequence in context: A062729 A004620 A018618 * A108566 A057615 A018416

Adjacent sequences: A294361 A294362 A294363 * A294365 A294366 A294367

Keyword

nonn,easy

Author

Jean-François Alcover and Paul Curtz, Oct 29 2017

Status

approved