A291660 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=2, a(1)=3, a(2)=5, a(3)=7, a sequence related to Lucas numbers.

2, 3, 5, 7, 11, 18, 30, 49, 79, 127, 205, 332, 538, 871, 1409, 2279, 3687, 5966, 9654, 15621, 25275, 40895, 66169, 107064, 173234, 280299, 453533, 733831, 1187363, 1921194, 3108558, 5029753, 8138311, 13168063, 21306373, 34474436, 55780810, 90255247, 146036057, 236291303

Offset

0,1

Comments

The array of successive differences begins:

2, 3, 5, 7, 11, 18, 30, 49, 79, 127, ... = a(n)

1, 2, 2, 4, 7, 12, 19, 30, 48, 78, ... = b(n)

1, 0, 2, 3, 5, 7, 11, 18, 30, 49, ... = a(n-2)

-1, 2, 1, 2, 2, 4, 7, 12, 19, 30, ... = b(n-2)

3, -1, 1, 0, 2, 3, 5, 7, 11, 18, ... = a(n-4)

...

Main diagonal is 2,2,2,... = A007395.

Adding a(n) and first column with alternating signs, one gets two autosequences: 2*Lucas numbers A000032 (2, 1, 3, 4, 7, 11, 18, ...) or 2*A286350 (0, 2, 2, 3, 4, 7, 12, ...) according to signs.

Links

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

OEIS Wiki, Autosequence

Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 1).

Formula

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

a(3n) = A097924(n).

a(3n) + a(3n+1) = a(3n+2).

a(n) = (1/15)*2^(-n-1)*((30-9*sqrt(5))*(1-sqrt(5))^n + (1+sqrt(5))^n*(30 + 9*sqrt(5)) + 5*2^(n+1)*sqrt(3)*sin(n*Pi/3)).

Mathematica

LinearRecurrence[{2, -1, 0, 1}, {2, 3, 5, 7}, 40]

Prog

(GAP)
L:=[2, 3, 5, 7];; for i in [5..10^3] do L[i]:=2*L[i-1]-L[i-2]+L[i-4]; od; L;  #  Muniru A Asiru, Sep 02 2017

Crossrefs

Cf. A000032, A097924, A286350.

Sequence in context: A018146 A293637 A284250 * A069749 A081889 A078139

Adjacent sequences: A291657 A291658 A291659 * A291661 A291662 A291663

Keyword

nonn

Author

Jean-François Alcover and Paul Curtz, Aug 31 2017

Status

approved