A010919 -id:A010919 - cp's OEIS frontend

A020746 Pisot sequence T(3,7), a(n) = floor(a(n-1)^2/a(n-2)).

3, 7, 16, 36, 81, 182, 408, 914, 2047, 4584, 10265, 22986, 51471, 115255, 258081, 577899, 1294040, 2897633, 6488421, 14528964, 32533461, 72849384, 163125366, 365272615, 817923579, 1831505986, 4101133972, 9183316890, 20563412382, 46045882316, 103106587509

Offset: 0

David W. Wilson

nonn

Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Pisot sequences which satisfy no linear recurrences, Acta Arith. 32 (1) (1977) 89-98
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
D. W. Boyd, On linear recurrence relations satisfied by Pisot sequences, Acta Arithm. 47 (1) (1986) 13
D. W. Boyd, Pisot sequences which satisfy no linear recurrences. II, Acta Arithm. 48 (1987) 191
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, in Advances in Number Theory (Kingston ON, 1991), pp. 333-340, Oxford Univ. Press, New York, 1993; with updates from 1996 and 1999.
D. G. Cantor, On families of Pisot E-sequences, Ann. Sci. Ecole Nat. Sup. 9 (2) (1976) 283-308

See A008776 for definitions of Pisot sequences.

Cf. A010919, A010925.

Iv:=[3,7]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..40]]; // Bruno Berselli, Feb 04 2016

RecurrenceTable[{a[0] == 3, a[1] == 7, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 04 2016 *)
nxt[{a_,b_}]:={b,Floor[b^2/a]}; NestList[nxt,{3,7},30][[All,1]] (* Harvey P. Dale, Oct 11 2020 *)

pisotT(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]));
  a
}
pisotT(50, 3, 7) \\ Colin Barker, Jul 29 2016

Conjectured g.f.: (-x^5+x^4-x^3+x^2-2*x+3)/((1-x)*(1-2*x-x^3-x^5)). - Ralf Stephan, May 12 2004

I believe that David Boyd has proved that this g.f. is correct. - N. J. A. Sloane, Aug 11 2016

A022029 a(n) = 3*a(n-1) + a(n-2) - a(n-3) - a(n-5).

Original entry on oeis.org

4, 13, 42, 135, 433, 1388, 4449, 14260, 45706, 146496, 469546, 1504979, 4823727, 15460908, 49554976, 158832563, 509086778, 1631714194, 5229935889, 16762880107, 53728029453, 172207945799, 551957272549

Offset: 0

R. K. Guy

nonn

Sequence agrees with A010919 for n <= 35 only.

Index entries for linear recurrences with constant coefficients, signature (3, 1, -1, 0, -1).

Cf. A010919.

G.f.: (4+x-x^2-x^4)/(1-3x-x^2+x^3+x^5). - R. J. Mathar, Oct 25 2008

A275904 Order of homogeneous linear recurrence satisfied by the Pisot sequence T(n, n^2-n+1).

Original entry on oeis.org

1, 2, 6, 36, 2048

Offset: 1

N. J. A. Sloane, Aug 11 2016

Degree of denominator of minimal g.f. for T(n, n^2-n+1).

Conjecture: a(6) = 6852224. The conjectured generating function for T(6,31) is A(x)/(1+x - x*A(x)) where A(x) = 6 + x - x^2 - x^4 - x^22 - x^1130 - x^6852224 (and as usual there is a common factor of (1+x) in numerator and denominator). - David Boyd, Aug 12 2016.

			T(1,1) is the all-ones sequence, with g.f. 1/(1-x).
T(2,3) is 2,3,4,5,6,... with g.f. (2-x)/(1-2*x+x^2).
T(3,7) is A020746, with a linear recurrence of order 6.
T(4,13) is A010919, with a linear recurrence of order 36.
T(5,21) is A010925, with a linear recurrence of order 2048.

David Boyd, Email communication to N. J. A. Sloane, Aug 06 2016

D. W. Boyd, Pisot sequences which satisfy no linear recurrences, Acta Arith. 32 (1) (1977) 89-98
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
D. W. Boyd, On linear recurrence relations satisfied by Pisot sequences, Acta Arithm. 47 (1) (1986) 13
D. W. Boyd, Pisot sequences which satisfy no linear recurrences. II, Acta Arithm. 48 (1987) 191
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, in Advances in Number Theory (Kingston ON, 1991), pp. 333-340, Oxford Univ. Press, New York, 1993; with updates from 1996 and 1999.

Cf. A008776, A020746, A010919, A010925.

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)

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A020746 Pisot sequence T(3,7), a(n) = floor(a(n-1)^2/a(n-2)).

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A022029 a(n) = 3*a(n-1) + a(n-2) - a(n-3) - a(n-5).

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A275904 Order of homogeneous linear recurrence satisfied by the Pisot sequence T(n, n^2-n+1).

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A278692 Pisot sequence T(4,14).

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