A020744 Pisot sequences P(8,10), T(8,10).

8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Conjecturally, even sums of four primes. - Charles R Greathouse IV, Feb 16 2012

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Subsequence of A005843, A020739. See A008776 for definitions of Pisot sequences.

Cf. A020705, A028563.

Programs

Mathematica

LinearRecurrence[{2,-1},{8,10},70] (* Harvey P. Dale, Jul 19 2015 *)
P[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Ceiling[a[n - 1]^2/a[n - 2] - 1/2]; Table[a[n], {n, 0, z}]]; P[8, 10, 65] (* or *)
T[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2]]; Table[a[n], {n, 0, z}]]; T[8, 10, 65] (* Michael De Vlieger, Aug 08 2016 *)

PARI

a(n)=2*n+8 \\ Charles R Greathouse IV, Feb 16 2012

PARI

pisotP(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
  a
}
pisotP(50, 8, 10) \\ Colin Barker, Aug 08 2016

Formula

a(n) = 2*n + 8.

a(n) = 2*a(n-1) - a(n-2).

From Elmo R. Oliveira, Oct 30 2024: (Start)

G.f.: 2*(4 - 3*x)/(1 - x)^2.

E.g.f.: 2*(4 + x)*exp(x).

a(n) = 2*A020705(n) = A028563(n+1) - A028563(n). (End)