A010904 Pisot sequence E(4,14): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=14.
4, 14, 49, 172, 604, 2121, 7448, 26154, 91841, 322504, 1132488, 3976785, 13964668, 49037590, 172197809, 604680724, 2123364868, 7456295833, 26183134320, 91943310482, 322863269121, 1133749589840, 3981215131600, 13980224615841, 49092217790004, 172389637059934
Offset: 0
Keywords
References
- Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- 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.
- S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)
Programs
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Magma
I:=[4, 14]; [n le 2 select I[n] else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..25]]; // Bruno Berselli, Sep 03 2013
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Mathematica
RecurrenceTable[{a[0] == 4, a[1] == 14, a[n] == Floor[a[n-1]^2/a[n-2] + 1/2]}, a ,{n, 30}] (* Harvey P. Dale, May 02 2012 *) -
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, 4, 14) \\ Colin Barker, Jul 27 2016
Formula
Theorem: a(0)=4, a(1)=14, a(2)=49; for n>2, a(n) = 4*a(n-1)-2*a(n-2)+a(n-3). Proved using the PtoRv program of Ekhad-Sloane-Zeilberger. (Conjectured by Harvey P. Dale, May 02 2012.) - N. J. A. Sloane, Sep 09 2016