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A293411
a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4.
5
1, 2, 3, 4, 7, 12, 19, 30, 49, 80, 129, 208, 337, 546, 883, 1428, 2311, 3740, 6051, 9790, 15841, 25632, 41473, 67104, 108577, 175682, 284259, 459940, 744199, 1204140, 1948339, 3152478, 5100817, 8253296, 13354113, 21607408, 34961521, 56568930, 91530451
OFFSET
0,2
COMMENTS
a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that this sequence has the growth rate of the Fibonacci numbers (A000045).
FORMULA
a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4.
G.f.: (-1 - x - x^2)/(-1 + x + x^3 + x^4).
From Greg Dresden, Aug 25 2021: (Start)
a(2*n) = a(2*n - 1) + a(2*n - 2),
a(2*n) = 2*F(n+1)^2 - (-1)^n = A061646(n+1),
a(2*n+1) = 2*F(n+1)*F(n+2) = A079472(n+2), for F(n) the Fibonacci numbers A000045. (End)
MATHEMATICA
LinearRecurrence[{1, 0, 1, 1}, {1, 2, 3, 4}, 100]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 25 2017
STATUS
approved