A278692 Pisot sequence T(4,14).

4, 14, 49, 171, 596, 2077, 7238, 25223, 87897, 306303, 1067403, 3719680, 12962320, 45171020, 157411717, 548547468, 1911575138, 6661446313, 23213770727, 80895217952, 281903201529, 982374694626, 3423373822671, 11929753885009, 41572739387791, 144872448909191, 504850696923520, 1759300875378480

Offset: 0

Author

Ilya Gutkovskiy, Nov 28 2016

Keywords

• nonn
• easy

Links

• Index entries for Pisot sequences

Crossrefs

Cf. A008776 for definitions of Pisot sequences.

Cf. A010919, A019495, A022031.

Cf. A010904 (Pisot sequence E(4,14)), A251221 (seems to be Pisot sequence P(4,14)), A277084 (Pisot sequence L(4,14)).

Programs

Mathematica

RecurrenceTable[{a[0] == 4, a[1] == 14, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 27}]

PARI

first(n)=my(v=vector(n+1)); v[1]=4; v[2]=14; for(i=3,#v, v[i]=v[i-1]^2\v[i-2]); v \\ Charles R Greathouse IV, Nov 28 2016

Python

from itertools import islice
def A278692_gen(): # generator of terms
    a, b = 4, 14
    yield from (a,b)
    while True:
        a, b = b, b**2//a
        yield b
A278692_list = list(islice(A278692_gen(),30)) # Chai Wah Wu, Dec 06 2023

Formula

a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 4, a(1) = 14.

Conjectures: (Start)

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

a(n) = 4*a(n-1) - 2*a(n-2) + a(n-3) - a(n-4). (End)