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
Keywords
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, 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
Programs
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Magma
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
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Mathematica
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 *) -
PARI
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
Formula
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