A275628 Pisot sequence E(31,51), a(n)=[a(n-1)^2/a(n-2)+1/2].

31, 51, 84, 138, 227, 373, 613, 1007, 1654, 2717, 4463, 7331, 12042, 19780, 32490, 53367, 87659, 143986, 236507, 388479, 638103, 1048127, 1721619, 2827875, 4644975, 7629684, 12532269, 20585095, 33812403, 55539146, 91226783, 149846127, 246132342, 404288926, 664071752, 1090782516

Offset

0,1

Links

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, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

Formula

It is known (Boyd, 1977) that this sequence does not satisfy a linear recurrence. - N. J. A. Sloane, Aug 07 2016

Maple

A[0]:= 31:
A[1]:= 51:
for n from 2 to 100 do
  A[n]:= floor(A[n-1]^2/A[n-2]+1/2)
od:
seq(A[n], n=0..100); # Robert Israel, Aug 18 2016

Mathematica

nxt[{a_, b_}]:={b, Floor[b^2/a+1/2]}; NestList[nxt, {31, 51}, 40][[All, 1]] (* Harvey P. Dale, Nov 02 2020 *)

Prog

(PARI) pisotE(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]+1/2));
  a
}
pisotE(50, 31, 51) \\ Colin Barker, Aug 08 2016
(Python)
a, b = 31, 51
A275628_list = [a, b]
for i in range(1000):
    c, d = divmod(b**2, a)
    a, b = b, c + (0 if 2*d < a else 1)
    A275628_list.append(b) # Chai Wah Wu, Aug 08 2016

Crossrefs

Cf. A008776, A010902.

Sequence in context: A248904 A068779 A068473 * A176507 A163321 A182380

Adjacent sequences: A275625 A275626 A275627 * A275629 A275630 A275631

Keyword

nonn

Author

N. J. A. Sloane, Aug 07 2016

Status

approved