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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
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 (list; graph; refs; listen; history; text; internal format)
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
OEIS Wiki, Autosequence
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
Sequence in context: A018146 A293637 A284250 * A069749 A081889 A078139
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 28 15:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)