login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A295683
a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 2, a(1) = 1, a(2) = 0, a(3) = 1.
2
2, 1, 0, 1, 4, 5, 6, 11, 20, 31, 48, 79, 130, 209, 336, 545, 884, 1429, 2310, 3739, 6052, 9791, 15840, 25631, 41474, 67105, 108576, 175681, 284260, 459941, 744198, 1204139, 1948340, 3152479, 5100816, 8253295, 13354114, 21607409, 34961520, 56568929, 91530452
OFFSET
0,1
COMMENTS
a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).
FORMULA
G.f.: (2 - x - x^2 - x^3)/(1 - x - x^3 - x^4).
a(n) = (2/5) * A000045(n) + (4/5) * A000045(n-1) + (6/5) * A056594(n) + (3/5) * A056594(n-1) for n >= 1. - Robert Israel, Apr 03 2019
MAPLE
f:= gfun:-rectoproc(a(n) = a(n-1) + a(n-3) + a(n-4), a(0) = 2, a(1) = 1, a(2) = 0, a(3) = 1}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Apr 03 2019
MATHEMATICA
LinearRecurrence[{1, 0, 1, 1}, {2, 1, 0, 1}, 45]
PROG
(PARI) my(x='x+O('x^45)); Vec((-2 + x + x^2 + x^3)/(-1 + x + x^3 + x^4)) \\ Georg Fischer, Apr 03 2019
(Magma) I:=[2, 1, 0, 1]; [n le 4 select I[n] else Self(n-1) +Self(n-3) +Self(n-4): n in [1..45]]; // G. C. Greubel, Apr 03 2019
(Sage) ((2-x-x^2-x^3)/(1-x-x^3-x^4)).series(x, 45).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 29 2017
EXTENSIONS
a(0) = 2 amended by Georg Fischer, Apr 03 2019
STATUS
approved