A020745 -id:A020745 - cp's OEIS frontend

A207100 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with each no smaller than the sum of its two previous neighbors modulo (k+1).

2, 3, 3, 4, 6, 5, 5, 10, 12, 8, 6, 15, 26, 26, 12, 7, 21, 45, 68, 55, 18, 8, 28, 75, 140, 176, 115, 27, 9, 36, 112, 274, 441, 458, 239, 40, 10, 45, 164, 462, 989, 1382, 1193, 498, 59, 11, 55, 225, 760, 1904, 3579, 4322, 3103, 1038, 87, 12, 66, 305, 1158, 3504, 7868

Offset: 1

R. H. Hardin Feb 15 2012

Table starts
..2....3.....4......5......6.......7.......8........9.......10........11
..3....6....10.....15.....21......28......36.......45.......55........66
..5...12....26.....45.....75.....112.....164......225......305.......396
..8...26....68....140....274.....462.....760.....1158.....1720......2431
.12...55...176....441....989....1904....3504.....5925.....9652.....14850
.18..115...458...1382...3579....7868...16224....30390....54294.....90959
.27..239..1193...4322..12964...32531...75114...155922...305362....557095
.40..498..3103..13511..46952..134517..347794...800088..1717686...3412442
.59.1038..8069..42238.170076..556259.1610482..4105829..9663330..20904257
.87.2162.20982.132051.616065.2300219.7457403.21069969.54364034.128056753

			Some solutions for n=5 k=3
..2....2....0....0....0....1....0....3....2....2....3....1....0....0....2....0
..2....2....1....0....0....1....3....3....3....2....3....3....0....3....3....1
..1....2....3....0....3....3....3....3....1....2....2....1....2....3....2....3
..3....3....3....2....3....2....2....2....3....0....3....3....3....2....3....0
..1....1....3....2....3....1....3....3....2....3....3....3....3....2....2....3

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A020745(n-2)
Row 2 is A000217(n+1)
Row 3 is A199771(n+1)

A018917 Define the generalized Pisot sequence T(a(0),a(1)) by: a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n). This is T(3,5).

Original entry on oeis.org

3, 5, 8, 12, 17, 24, 33, 45, 61, 82, 110, 147, 196, 261, 347, 461, 612, 812, 1077, 1428, 1893, 2509, 3325, 4406, 5838, 7735, 10248, 13577, 17987, 23829, 31568, 41820, 55401, 73392, 97225, 128797, 170621, 226026, 299422, 396651, 525452, 696077, 922107

Offset: 0

R. K. Guy

nonn

Not to be confused with the Pisot T(3,5) sequence, which is A020745. - R. J. Mathar, Feb 13 2016

Is 1 followed by this sequence equal to A167385? - Bruno Berselli, Feb 17 2016

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
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.
Index entries for Pisot sequences

Tiv:=[3,5]; [n le 2 select Tiv[n] else Ceiling(Self(n-1)^2/Self(n-2))-1: n in [1..50]]; // Bruno Berselli, Feb 17 2016

RecurrenceTable[{a[1] == 3, a[2] == 5, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 50}] (* Bruno Berselli, Feb 17 2016 *)

T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
T(3, 5, 60) \\ Colin Barker, Feb 14 2016

Conjecture: a(n)=a(n-1)+a(n-2)-a(n-4). G.f.: (3+2*x-x^3)/(1-x)/(1-x^2-x^3). [Colin Barker, Feb 16 2012]

Conjecture: a(n) = a(n-1) + A000931(n+8). - Reinhard Zumkeller, Dec 30 2012

A020749 Pisot sequence T(5,8), a(n) = floor(a(n-1)^2/a(n-2)).

Original entry on oeis.org

5, 8, 12, 18, 27, 40, 59, 87, 128, 188, 276, 405, 594, 871, 1277, 1872, 2744, 4022, 5895, 8640, 12663, 18559, 27200, 39864, 58424, 85625, 125490, 183915, 269541, 395032, 578948, 848490, 1243523, 1822472, 2670963, 3914487, 5736960, 8407924, 12322412, 18059373

Offset: 0

David W. Wilson

nonn

Colin Barker, Table of n, a(n) for n = 0..1000

Subsequence of A020745.

See A008776 for definitions of Pisot sequences.

Txy:=[5,8]; [n le 2 select Txy[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..40]]; // Bruno Berselli, Feb 05 2016

RecurrenceTable[{a[0] == 5, a[1] == 8, a[n] == Floor[a[n - 1]^2/a[n - 2] ]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 05 2016 *)

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, 5, 8) \\ Colin Barker, Jul 29 2016

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).

Note the warning in A010925 from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004: [A010925] and other examples show that it is essential to reject conjectured generating functions for Pisot sequences until a proof or reference is provided. - N. J. A. Sloane, Jul 26 2016

cp's OEIS Frontend

A207100 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with each no smaller than the sum of its two previous neighbors modulo (k+1).

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A018917 Define the generalized Pisot sequence T(a(0),a(1)) by: a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n). This is T(3,5).

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A020749 Pisot sequence T(5,8), a(n) = floor(a(n-1)^2/a(n-2)).

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Magma

Mathematica

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