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

%I A020744 #29 Oct 30 2024 21:22:22
%S A020744 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,
%T A020744 54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,
%U A020744 100,102,104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138
%N A020744 Pisot sequences P(8,10), T(8,10).
%C A020744 Conjecturally, even sums of four primes. - _Charles R Greathouse IV_, Feb 16 2012
%H A020744 Colin Barker, <a href="/A020744/b020744.txt">Table of n, a(n) for n = 0..1000</a>
%H A020744 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A020744 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A020744 a(n) = 2*n + 8.
%F A020744 a(n) = 2*a(n-1) - a(n-2).
%F A020744 From _Elmo R. Oliveira_, Oct 30 2024: (Start)
%F A020744 G.f.: 2*(4 - 3*x)/(1 - x)^2.
%F A020744 E.g.f.: 2*(4 + x)*exp(x).
%F A020744 a(n) = 2*A020705(n) = A028563(n+1) - A028563(n). (End)
%t A020744 LinearRecurrence[{2,-1},{8,10},70] (* _Harvey P. Dale_, Jul 19 2015 *)
%t A020744 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 A020744 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 *)
%o A020744 (PARI) a(n)=2*n+8 \\ _Charles R Greathouse IV_, Feb 16 2012
%o A020744 (PARI) pisotP(nmax, a1, a2) = {
%o A020744   a=vector(nmax); a[1]=a1; a[2]=a2;
%o A020744   for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
%o A020744   a
%o A020744 }
%o A020744 pisotP(50, 8, 10) \\ _Colin Barker_, Aug 08 2016
%Y A020744 Subsequence of A005843, A020739. See A008776 for definitions of Pisot sequences.
%Y A020744 Cf. A020705, A028563.
%K A020744 nonn,easy
%O A020744 0,1
%A A020744 _David W. Wilson_