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A306276 a(0) = a(1) = a(2) = a(3) = 1; thereafter a(n) = a(n-2) + a(n-3) + a(n-4). 1
1, 1, 1, 1, 3, 3, 5, 7, 11, 15, 23, 33, 49, 71, 105, 153, 225, 329, 483, 707, 1037, 1519, 2227, 3263, 4783, 7009, 10273, 15055, 22065, 32337, 47393, 69457, 101795, 149187, 218645, 320439, 469627, 688271, 1008711, 1478337, 2166609, 3175319, 4653657, 6820265 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The characteristic equation of this sequence is x^4 = x^2 + x + 1. The characteristic equation of A000930 is x^3 = x^2 + 1 [1], which can be rewritten as x^4 = x^3 + x [2]. By substituting the value of x^3 from equation [1] in equation [2], we get x^4 = (x^2 + 1) + x, which is the characteristic equation for this sequence. Hence the ratio a(n+1)/a(n) has the same limit as the A000930 sequence does, about 1.465571231.

LINKS

Table of n, a(n) for n=0..43.

FORMULA

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

a(n) = a(n-2) + a(n-3) + a(n-4) for n >= 4, a(n) = 1 for n < 4.

Lim_{n->infinity} a(n+1)/a(n) = A092526.

MATHEMATICA

Nest[Append[#, Total@ #[[-4 ;; -2]] ] &, {1, 1, 1, 1}, 40] (* or *)

CoefficientList[Series[(x^3 - x - 1)/(x^4 + x^3 + x^2 - 1), {x, 0, 43}], x] (* Michael De Vlieger, Feb 09 2019 *)

CROSSREFS

Cf. A000930, A092526.

Sequence in context: A141867 A163646 A323529 * A001588 A107029 A240180

Adjacent sequences:  A306273 A306274 A306275 * A306277 A306278 A306279

KEYWORD

nonn

AUTHOR

Joseph Damico, Feb 02 2019

STATUS

approved

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Last modified November 13 17:34 EST 2019. Contains 329106 sequences. (Running on oeis4.)