A010925 -id:A010925 - cp's OEIS frontend

A010919 Pisot sequence T(4,13), a(n) = floor(a(n-1)^2/a(n-2)).

4, 13, 42, 135, 433, 1388, 4449, 14260, 45706, 146496, 469546, 1504979, 4823727, 15460908, 49554976, 158832563, 509086778, 1631714194, 5229935889, 16762880107, 53728029453, 172207945799, 551957272549, 1769121798104, 5670351840955, 18174492018967

Offset: 0

Simon Plouffe and N. J. A. Sloane

nonn

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

Cf. A022029, A010925.

a[0] = 4; a[1] = 13; a[n_] := a[n] = Floor[a[n-1]^2/a[n-2]]; Array[a, 30, 0] (* Jean-François Alcover, Dec 14 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, 4, 13) \\ Colin Barker, Jul 29 2016

Appears to satisfy the g.f. (4+x-x^2-x^4-x^36)/(1-3*x-x^2+x^3+x^5+x^37), where there is a common factor of 1+x that can be canceled, so the sequence appears to satisfy a linear recurrence of order 36. I believe that David Boyd has proved that the sequence does indeed satisfy this recurrence. - N. J. A. Sloane, Aug 11 2016

A010920 Pisot sequence T(3,13), a(n) = floor( a(n-1)^2/a(n-2) ).

Original entry on oeis.org

3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072

Offset: 0

Simon Plouffe

Vincenzo Librandi, 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.

Cf. A010925, A010903.

I:=[3,13]; [n le 2 select I[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..25]]; // Bruno Berselli, Sep 03 2013

RecurrenceTable[{a[0] == 3, a[1] == 13, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 25}] (* Bruno Berselli, Sep 03 2013 *)

Empirical G.f.: (3-2*x)/(1-5*x+3*x^2). - Colin Barker, Feb 21 2012
Empirical: a(n) = 5*a(n-1)-3*a(n-2) with n>1, a(0)=3 and a(1)=13. - Vincenzo Librandi, Apr 17 2012
The empirical g.f. and recurrence above hold for n<=6000. - Bruno Berselli, Sep 03 2013
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

A020746 Pisot sequence T(3,7), a(n) = floor(a(n-1)^2/a(n-2)).

Original entry on oeis.org

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

David W. Wilson

nonn

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

See A008776 for definitions of Pisot sequences.

Cf. A010919, A010925.

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

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 *)

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

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

A010901 Pisot sequences E(4,7), P(4,7).

Original entry on oeis.org

4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786, 26931732, 47261895, 82938844, 145547525, 255418101, 448227521, 786584466

Offset: 0

Simon Plouffe

nonn

Essentially the same as A005251: a(n) = A005251(n+5).

See A008776 for definitions of Pisot sequences.

Colin Barker, Table of n, a(n) for n = 0..1000
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).
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 1).

Cf. A005251, A008776, A010925.

LinearRecurrence[{2, -1, 1}, {4, 7, 12}, 35] (* Jean-François Alcover, Oct 05 2018 *)

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, 7) \\ Colin Barker, Jul 27 2016

a(n) = 2a(n-1) - a(n-2) + a(n-3) for n>=3. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016

Edited by N. J. A. Sloane, Jul 26 2016

A010905 Pisot sequence E(4,15): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=15.

Original entry on oeis.org

4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244

Offset: 0

Simon Plouffe

nonn

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

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.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A010925, A001353, A079935, A195503.

/* By definition: */ [n le 2 select 11*n-7 else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..22]]; // Bruno Berselli, Apr 16 2012

a[0] = 4; a[1] = 15; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2] + 1/2]; Table[a[n], {n, 0, 24}] (* Michael De Vlieger, Jul 27 2016 *)

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, 15) \\ Colin Barker, Jul 27 2016

@cached_function
def A010905(n):
    if n==0: return 4
    elif n==1: return 15
    else: return 4*A010905(n-1) - A010905(n-2)
[A010905(n) for n in range(30)] # G. C. Greubel, Dec 13 2018

a(n) = 4*a(n-1) - a(n-2) for n>=2. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016
This was conjectured by Colin Barker, Apr 16 2012, and implies the G.f.: (4-x)/(1-4*x+x^2) and the formula a(n) = ((1+sqrt(3))^(2*n+4)-(1-sqrt(3))^(2*n+4))/(2^(n+3)*sqrt(3)).
Partial sums of A079935. - Erin Pearse, Dec 13 2018

Edited by N. J. A. Sloane, Jul 26 2016 and Sep 09 2016

A019992 a(n) = 4*a(n-1) + a(n-2) - a(n-3) - a(n-5).

Original entry on oeis.org

5, 21, 88, 368, 1538, 6427, 26857, 112229, 468978, 1959746, 8189306, 34221135, 143001871, 597570335, 2497102330, 10434788478, 43604464772, 182212543365, 761422279419, 3181800093939, 13295975323332, 55560674643076, 232174661258332, 970201922073653, 4054239874815929, 16941690784755705, 70795240417122020

Offset: 0

R. K. Guy

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,1,-1,0,-1).

This is different from A010925. See the comments in that sequence.

I:=[5, 21, 88, 368, 1538]; [n le 5 select I[n] else 4*Self(n-1)+Self(n-2)-Self(n-3)-Self(n-5): n in [1..30]]; // Vincenzo Librandi, Apr 20 2012

CoefficientList[Series[(5+x-x^2-x^4)/(1-4*x-x^2+x^3+x^5),{x,0,30}],x] (* Vincenzo Librandi, Apr 20 2012 *)
LinearRecurrence[{4,1,-1,0,-1},{5,21,88,368,1538},30] (* Harvey P. Dale, May 03 2020 *)

G.f.: (5+x-x^2-x^4)/(1-4*x-x^2+x^3+x^5). - Colin Barker, Feb 21 2012

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004

A020748 Pisot sequence T(4,10), a(n) = floor(a(n-1)^2/a(n-2)).

Original entry on oeis.org

4, 10, 25, 62, 153, 377, 928, 2284, 5621, 13833, 34042, 83774, 206159, 507335, 1248496, 3072412, 7560869, 18606469, 45788478, 112680418, 277294139, 682390435, 1679287948, 4132543288, 10169735361, 25026602289, 61587720810, 151560619806, 372974046999

Offset: 0

David W. Wilson

nonn

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

See A008776 for definitions of Pisot sequences.

RecurrenceTable[{a[0]==4,a[1]==10,a[n]==Floor[a[n-1]^2/a[n-2]]},a,{n,30}] (* Harvey P. Dale, Dec 26 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, 4, 10) \\ Colin Barker, Jul 29 2016

G.f.: (-3x^5+2x^4+x^3-x^2-2x+4)/[(1-x)(1-2x-x^2-2x^5)] (conjectured). - Ralf Stephan, May 12 2004

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

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

A020750 Pisot sequence T(5,9), a(n) = floor(a(n-1)^2/a(n-2)).

Original entry on oeis.org

5, 9, 16, 28, 49, 85, 147, 254, 438, 755, 1301, 2241, 3860, 6648, 11449, 19717, 33955, 58474, 100698, 173411, 298629, 514265, 885608, 1525092, 2626337, 4522773, 7788595, 13412614, 23097646, 39776083, 68497749, 117958865, 203135052, 349815584, 602411753

Offset: 0

David W. Wilson

nonn

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

See A008776 for definitions of Pisot sequences.

Iv:=[5, 9]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..40]]; // Bruno Berselli, Feb 04 2016

RecurrenceTable[{a[0] == 5, a[1] == 9, 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,{5,9},40][[All,1]] (* Harvey P. Dale, Aug 29 2021 *)

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, 9) \\ Colin Barker, Jul 29 2016

a(n) = 2*a(n-1) - 2*a(n-4) + a(n-5) (holds at least up to n = 40000 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

Deleted unproved recurrence and program based on it. - N. J. A. Sloane, Feb 04 2016

A048582 Pisot sequence L(4,9).

Original entry on oeis.org

4, 9, 21, 49, 115, 270, 634, 1489, 3498, 8218, 19307, 45359, 106565, 250361, 588192, 1381884, 3246565, 7627402, 17919636, 42099965, 98908653, 232373629, 545933059, 1282602102, 3013314774, 7079409829, 16632196530, 39075285666, 91802543767, 215678705823

Offset: 0

David W. Wilson

nonn

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

See A008776 for definitions of Pisot sequences.

a[n_] := a[n] = Switch[n, 0, 4, 1, 9, _, Ceiling[a[n-1]^2/a[n-2]]];
a /@ Range[0, 29] (* Jean-François Alcover, Oct 22 2019 *)

pisotL(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]));
  a
}
pisotL(50, 4, 9) \\ Colin Barker, Aug 07 2016

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) + a(n-4) - 2*a(n-5) + a(n-6) - a(n-7) (conjectured). Recurrence is satisfied for at least 760000 terms. - Chai Wah Wu, Jul 25 2016

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

A010919 Pisot sequence T(4,13), a(n) = floor(a(n-1)^2/a(n-2)).

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A010920 Pisot sequence T(3,13), a(n) = floor( a(n-1)^2/a(n-2) ).

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A020746 Pisot sequence T(3,7), a(n) = floor(a(n-1)^2/a(n-2)).

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A010901 Pisot sequences E(4,7), P(4,7).

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A010905 Pisot sequence E(4,15): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=15.

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A019992 a(n) = 4*a(n-1) + a(n-2) - a(n-3) - a(n-5).

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A020748 Pisot sequence T(4,10), a(n) = floor(a(n-1)^2/a(n-2)).

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