A022031 -id:A022031 - cp's OEIS frontend

A022032 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(5,26).

5, 26, 135, 700, 3629, 18813, 97527, 505582, 2620947, 13587040, 70435478, 365138879, 1892887004, 9812762803, 50869551972, 263708740319, 1367071205166, 7086923541985, 36738748574433, 190454382472052, 987319198674433, 5118281802804775, 26533271760636405, 137548993480193164

Offset: 0

R. K. Guy

nonn

The empirical g.f. / recurrence agrees with the original definition for at least 2000 terms (and a(2000) ~ 10^1430). - M. F. Hasler, Feb 11 2016

Colin Barker, Table of n, a(n) for n = 0..1000

Cf. A022018 - A022025, A022026 - A022031.

(* This empirical recurrence should not be used to extend the data. *) LinearRecurrence[{5, 1, 0, -1, -1, -1, -1}, {5, 26, 135, 700, 3629, 18813, 97527}, 24] (* Jean-François Alcover, Dec 12 2016 *)

a=[5,26];for(n=2,2000, a=concat(a, ceil(a[n]^2/a[n-1])-1));A022032(n)=a[n+1] \\ M. F. Hasler, Feb 11 2016

Empirical g.f.: -(x^6+x^5+x^4+x^3-x-5) / (x^7+x^6+x^5+x^4-x^2-5*x+1). - Colin Barker, Sep 18 2015

a(n+1) = ceiling(a(n)^2/a(n-1))-1 for all n > 0. - M. F. Hasler, Feb 11 2016

Edited by M. F. Hasler, Feb 11 2016

A278692 Pisot sequence T(4,14).

Original entry on oeis.org

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

Ilya Gutkovskiy, Nov 28 2016

Index entries for Pisot sequences

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)).

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

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

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

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)

cp's OEIS Frontend

A022032 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(5,26).

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