A238379 -id:A238379 - cp's OEIS frontend

A097835 First differences of Chebyshev polynomials S(n,27) = A097781(n) with Diophantine property.

1, 26, 701, 18901, 509626, 13741001, 370497401, 9989688826, 269351100901, 7262490035501, 195817879857626, 5279820266120401, 142359329305393201, 3838422070979496026, 103495036587140999501, 2790527565781827490501

Offset: 0

Wolfdieter Lang, Sep 10 2004

(5*b(n))^2 - 29*a(n)^2 = -4 with b(n)=A097834(n) give all positive solutions of this Pell equation.

			All positive solutions of Pell equation x^2 - 29*y^2 = -4 are (5=5*1,1), (140=5*28,26), (3775=5*755,701), (101785=5*20357,18901), ...

Indranil Ghosh, Table of n, a(n) for n = 0..697
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (27,-1).

Cf. similar sequences listed in A238379.

a:=[1,26];; for n in [3..30] do a[n]:=27*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-27*x+x^2) )); // G. C. Greubel, Jan 12 2019

LinearRecurrence[{27,-1},{1,26},30] (* Harvey P. Dale, May 31 2013 *)

my(x='x+O('x^30)); Vec((1-x)/(1-27*x+x^2)) \\ G. C. Greubel, Jan 12 2019

((1-x)/(1-27*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019

a(n) = ((-1)^n)*S(2*n, 5*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-27*x+x^2).
a(n) = S(n, 27) - S(n-1, 27) = T(2*n+1, sqrt(29)/2)/(sqrt(29)/2), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
a(n) = 27*a(n-1) - a(n-2), a(0)=1, a(1)=26. - Philippe Deléham, Nov 18 2008

A153111 Solutions of the Pell-like equation 1 + 6AA = 7BB, with A, B integers.

Original entry on oeis.org

1, 25, 649, 16849, 437425, 11356201, 294823801, 7654062625, 198710804449, 5158826853049, 133930787374825, 3477041644892401, 90269151979827601, 2343520909830625225, 60841274503616428249, 1579529616184196509249, 41006928746285492812225

Offset: 1

Ctibor O. Zizka, Dec 18 2008

B is of the form B(i) = 26*B(i-1) - B(i-2) for B(0) = 1, B(1) = 25 (this sequence).
A is of the form A(i) = 26*A(i-1) - A(i-2) for A(0) = 1, A(1) = 27.
In general a Pell-like equation of the form 1 + X*A*A = (X + 1)*B*B has the solution A(i) = (4*X + 2)*A(i-1) - A(i-2), for A(0) = 1 and A(1) = (4*X + 3), and B(i) = (4*X + 2)*B(i-1) - B(i-2) for B(0) = 1 and B(1) = (4*X + 1).
Examples in the OEIS:
  X = 1 gives A002315 for A(i) and A001653 for B(i);
  X = 2 gives A054320 for A(i) and A072256 for B(i);
  X = 3 gives A028230 for A(i) and A001570 for B(i);
  X = 4 gives A049629 for A(i) and A007805 for B(i);
  X = 5 gives A133283 for A(i) and A157014 for B(i);
  X = 6 gives A157461 for A(i) and this sequence for B(i).
Positive values of x (or y) satisfying x^2 - 26*x*y + y^2 + 24 = 0. - Colin Barker, Feb 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Luigi Cimmino, Algebraic relations for recursive sequences, arXiv:math/0510417 [math.NT], 2005-2008.
Jeroen Demeyer, Diophantine sets of polynomials over number fields, arXiv:0807.1970 [math.NT], 2008.
Franz Lemmermeyer, Conics - a Poor Man's Elliptic Curves, arXiv:math/0311306 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (26,-1).

Cf. A002315, A001653, A054320, A072256, A001078, A028230, A001570, A049629, A007805, A133283, A140480.

Cf. similar sequences listed in A238379.

I:=[1,25]; [n le 2 select I[n] else 26*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 22 2014

CoefficientList[Series[(1 - x)/(x^2 - 26 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 22 2014 *)
LinearRecurrence[{26, -1}, {1, 25}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(-x*(x-1)/(x^2-26*x+1) + O(x^100)) \\ Colin Barker, Feb 20 2014

a(n) = 26*a(n-1) - a(n-2). - Colin Barker, Feb 20 2014
G.f.: -x*(x - 1) / (x^2 - 26*x + 1). - Colin Barker, Feb 20 2014
a(n) = (1/14)*(7 - sqrt(42))*(1 + (13 + 2*sqrt(42))^(2*n - 1))/(13 + 2*sqrt(42))^(n - 1). - Bruno Berselli, Feb 25 2014
E.g.f.: (1/7)*(7*cosh(2*sqrt(42)*x) - sqrt(42)*sinh(2*sqrt(42)*x))*exp(13*x) - 1. - Franck Maminirina Ramaharo, Jan 07 2019

More terms from Philippe Deléham, Sep 19 2009; corrected by N. J. A. Sloane, Sep 20 2009

Additional term from Colin Barker, Feb 20 2014

A159668 Expansion of (1 - x)/(1 - 28*x + x^2).

Original entry on oeis.org

1, 27, 755, 21113, 590409, 16510339, 461699083, 12911063985, 361048092497, 10096435525931, 282339146633571, 7895399670214057, 220788851619360025, 6174192445671866643, 172656599627192905979, 4828210597115729500769, 135017240119613233115553

Offset: 0

Paul Weisenhorn, Apr 19 2009

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j) + 1 = a(j)*a(j) and 15*n(j) + 1 = b(j)*b(j) with positive integer numbers.

Positive values of x (or y) satisfying x^2 - 28*x*y + y^2 + 26 = 0. - Colin Barker, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..199
Index entries for linear recurrences with constant coefficients, signature (28,-1).

Cf. A097311, A157456, A159669, A159673.

Cf. similar sequences listed in A238379.

[n le 2 select 27^(n-1) else 28*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 25 2014

for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
# Second program
seq(simplify(ChebyshevU(n,14) -ChebyshevU(n-1,14)), n=0..40); # G. C. Greubel, Sep 26 2022

CoefficientList[Series[(1-x)/(1-28x+x^2), {x,0,40}], x] (* Vincenzo Librandi, Feb 25 2014 *)
LinearRecurrence[{28,-1},{1,27},40] (* Harvey P. Dale, Apr 09 2014 *)

Vec((-x+1)/(x^2-28*x+1) + O(x^100)) \\ Colin Barker, Feb 23 2014

def A159668(n): return chebyshev_U(n,14) - chebyshev_U(n-1,14)
[A159668(n) for n in range(40)] # G. C. Greubel, Sep 26 2022

G.f.: (1 - x)/(1 - 28*x + x^2).
The a(j) recurrence is a(0)=1, a(1)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (this sequence).
The b(j) recurrence is b(0)=1, b(1)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (A159669).
The n(j) recurrence is n(0) = n(1) = 0, n(2) = 56, n(t+3) = 783*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 56, 43848, 34289136, ... (A159673).
a(n) = (1/30)*(15-sqrt(195))*(1+(14+sqrt(195))^(2*n+1))/(14+sqrt(195))^n. - Bruno Berselli, Feb 25 2014
a(n) = 28*a(n-1) - a(n-2), a(0)=1, a(1)=27. - Harvey P. Dale, Apr 09 2014
a(n) = A097311(n) - A097311(n-1). - G. C. Greubel, Sep 26 2022

More terms from Colin Barker, Feb 23 2014

New name and offset changed to 0 from Joerg Arndt, Feb 23 2014

A159664 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 11n(j) + 1 = a(j)a(j) and 13n(j) + 1 = b(j)b(j), with positive integer numbers.

Original entry on oeis.org

1, 23, 551, 13201, 316273, 7577351, 181540151, 4349386273, 104203730401, 2496540143351, 59812759710023, 1433009692897201, 34332419869822801, 822545067182850023, 19706749192518577751, 472139435553263016001, 11311639704085793806273, 271007213462505788334551

Offset: 1

Paul Weisenhorn, Apr 19 2009

Positive values of x (or y) satisfying x^2 - 24*x*y + y^2 + 22 = 0. - Colin Barker, Feb 19 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (24,-1).

Cf. A077423, A157456, A159661, A159665.

Cf. similar sequences listed in A238379.

[n le 2 select 23^(n-1) else 24*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 21 2014

for a from 1 by 2 to 100000 do b:=sqrt((13*a*a-2)/11): if (trunc(b)=b) then
n:=(a*a-1)/11: La:=[La),a]:Lb:=[op(Lb),b]: Ln:=[op(Ln),n]: end if: end do:
# Second program
seq(simplify(ChebyshevU(n-1,12) - ChebyshevU(n-2,12)), n=1..30); # G. C. Greubel, Sep 27 2022

CoefficientList[Series[(1-x)/(1-24x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 21 2014 *)
LinearRecurrence[{24,-1}, {1,23}, 30] (* G. C. Greubel, Sep 27 2022 *)

Vec(x*(1-x)/(1-24*x+x^2) + O(x^100)) \\ Colin Barker, Feb 19 2014

a(n) = round((12+sqrt(143))^(-n)*(13+sqrt(143)-(-13+sqrt(143))*(12+sqrt(143))^(2*n))/26) \\ Colin Barker, Jul 25 2016

def A159664(n): return chebyshev_U(n-1,12) - chebyshev_U(n-2,12)
[A159664(n) for n in range(1,30)] # G. C. Greubel, Sep 27 2022

The a(j) recurrence is a(1)=1, a(2)=23, a(t+2) = 24*a(t+1) - a(t) resulting in terms 1, 23, 551, 13201, ... (this sequence).
The b(j) recurrence is b(1)=1, b(2)=25, b(t+2) = 24*b(t+1) - b(t) resulting in terms 1, 25, 599, 14351, ... (A159661).
The n(j) recurrence is n(0)=n(1)=1, n(2)=48, n(t+3) = 575*(n(t+2) - n(t+1)) + n(t) resulting in terms 0, 0, 48, 27600, 15842400, ... (A159665).
G.f.: x*(1-x)/(1 - 24*x + x^2). - Colin Barker, Feb 19 2014
a(n) = (12+sqrt(143))^(-n)*(13+sqrt(143)-(-13+sqrt(143))*(12+sqrt(143))^(2*n))/26. - Colin Barker, Jul 25 2016
a(n) = A077423(n-1) - A077423(n-2). - G. C. Greubel, Sep 27 2022

More terms from Colin Barker, Feb 19 2014

A159674 Expansion of (1 - x)/(1 - 32*x + x^2).

Original entry on oeis.org

1, 31, 991, 31681, 1012801, 32377951, 1035081631, 33090234241, 1057852414081, 33818187016351, 1081124132109151, 34562154040476481, 1104907805163138241, 35322487611179947231, 1129214695752595173151, 36099547776471865593601, 1154056314151347103822081

Offset: 0

Paul Weisenhorn, Apr 19 2009

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 15*n(j) + 1 = a(j)*a(j) and 17*n(j) + 1 = b(j)*b(j) with positive integers.

Positive values of x (or y) satisfying x^2 - 32*x*y + y^2 + 30 = 0. - Colin Barker, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (32,-1).

Cf. A029548, A157456, A159675, A159677.

Cf. similar sequences listed in A238379.

A029548:= func< n | Evaluate(ChebyshevSecond(n),16) >;
[A029548(n+1) -A029548(n): n in [0..30]]; // G. C. Greubel, Sep 25 2022

for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
n:=(a*a-1)/15: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
# Second program
seq(simplify(ChebyshevU(n, 16) -ChebyshevU(n-1, 16)), n=0..30); # G. C. Greubel, Sep 25 2022

CoefficientList[Series[(1-x)/(1-32*x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{32,-1},{1,31},30] (* Harvey P. Dale, Mar 21 2017 *)

concat([0], Vec((-x+1)/(x^2-32*x+1) + O(x^100))) \\ Colin Barker, Feb 24 2014

def A159674(n): return chebyshev_U(n, 16) - chebyshev_U(n-1, 16)
[A159674(n) for n in range(31)] # G. C. Greubel, Sep 25 2022

The a(j) recurrence is: a(0)=1, a(1)=31, a(t+2) = 32*a(t+1) - a(t) resulting in terms 1, 31, 991, 31681, ... (this sequence).
The b(j) recurrence is: b(0)=1, b(1)=33, b(t+2) = 32*b(t+1) - b(t) resulting in terms 1, 33, 1055, 33727, ... (A159675).
The n(j) recurrence is: n(-1) = n(0) = 0, n(1) = 64, n(t+3) = 1023*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 64, 65472, 66912384, ... (A159677).
a(n) = (1/34)*(17-sqrt(255))*(1+(16+sqrt(255))^(2*n+1))/(16+sqrt(255))^n. - Bruno Berselli, Feb 25 2014
a(n) = ChebyshevU(n, 16) - ChebyshevU(n-1, 16) = A029548(n) - A029548(n-1). - G. C. Greubel, Sep 25 2022

More terms and new name from Colin Barker, Feb 24 2014

Set offset to 0 by Joerg Arndt, Feb 25 2014

A161595 The list of the A values in the common solutions to the 2 equations 15k+1=A^2, 19k+1=B^2.

Original entry on oeis.org

1, 16, 271, 4591, 77776, 1317601, 22321441, 378146896, 6406175791, 108526841551, 1838550130576, 31146825378241, 527657481299521, 8939030356713616, 151435858582831951, 2565470565551429551, 43461563755791470416, 736281113282903567521, 12473317362053569177441

Offset: 1

Paul Weisenhorn, Jun 14 2009

The 2 equations are equivalent to the Pell equation x^2- 285*y^2=1,
with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.
Also: the first differences of A078366.
Positive values of x (or y) satisfying x^2 - 17xy + y^2 + 15 = 0. - Colin Barker, Feb 14 2014

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
Index entries for linear recurrences with constant coefficients, signature (17,-1).

Cf. A078366, A160682, A161599 (sequence of B), A161583 (sequence of k).

Cf. similar sequences listed in A238379.

t:=0: for a from 1 to 1000000 do b:=sqrt((19*a^2-4)/15):
if (trunc(b)=b) then t:=t+1: n:=(a^2-1)/15: print(t,a,b,n): end if: end do:

Rest[CoefficientList[Series[x (1-x)/(1-17x+x^2),{x,0,40}],x]] (* or *) LinearRecurrence[{17,-1},{1,16},20] (* Harvey P. Dale, Oct 12 2012 *)

Vec(x*(1-x)/(1-17*x+x^2) + O(x^100)) \\ Colin Barker, Feb 14 2014

a(t+2) = 17*a(t+1)-a(t).
a(t) = ((285+15*w)*((17+w)/2)^(t-1)+(285-15*w)*((17-w)/2)^(t-1))/570, where w=sqrt(285).
a(t) = ceiling of ((285+15*w)*((17+w)/2)^(t-1))/570.
G.f.: x*(1-x)/(1-17*x+x^2).
a(n) = 17*a(n-1)-a(n-2). - Colin Barker, Feb 14 2014

Edited, extended by R. J. Mathar, Sep 02 2009

A111216 a(n) = 31*a(n-1)-a(n-2).

Original entry on oeis.org

1, 30, 929, 28769, 890910, 27589441, 854381761, 26458245150, 819351217889, 25373429509409, 785756963573790, 24333092441278081, 753540108716046721, 23335410277756170270, 722644178501725231649, 22378634123275726010849, 693015013643045781104670

Offset: 0

N. J. A. Sloane, following a suggestion from R. K. Guy, Oct 26 2005

Take 31 numbers consisting of 29 ones together with any two successive terms from this sequence. This set has the property that the sum of their squares is 31 times their product. (Guy)

Positive values of x (or y) satisfying x^2 - 31xy + y^2 + 29 = 0. - Colin Barker, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (31,-1).

Cf. A049685.

Cf. similar sequences listed in A238379.

I:=[1,30]; [n le 2 select I[n] else 31*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 26 2014

CoefficientList[Series[(1 - x)/(1 - 31 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)

Vec((1-x)/(1-31*x+x^2) + O(x^100)) \\ Colin Barker, Feb 24 2014

G.f.: (1-x)/(1-31*x+x^2). [Philippe Deléham, Nov 18 2008]

a(n) = A200442(n) - A200442(n-1). - R. J. Mathar, Feb 13 2016

A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 39, 1559, 62321, 2491281, 99588919, 3981065479, 159143030241, 6361740144161, 254310462736199, 10166056769303799, 406387960309415761, 16245352355607326641, 649407706263983649879, 25960062898203738668519, 1037753108221885563090881

Offset: 0

Ilya Gutkovskiy, Feb 18 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2) with n>1 and b(0)=1, b(1)=1, is (1 - (k - 1)*x)/(1 - k*x +x^2). This recurrence gives the closed form b(n) = (2^( -n - 1)*((k - 2)*(k - sqrt(k^2 - 4))^n + sqrt(k^2 - 4)*(k - sqrt(k^2 - 4))^n - (k - 2)*(sqrt(k^2 - 4) + k)^n + sqrt(k^2 - 4)*(sqrt(k^2 - 4) + k)^n))/sqrt(k^2 - 4).

Index entries for linear recurrences with constant coefficients, signature (40,-1).

Cf. A001519, A001835, A001653, A049685, A070997, A070998, A072256, A078922, A160682, A007805, A075839, A157014, A159664, A159668, A157877, A238379, A097315.

[n le 2 select 1 else 40*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 19 2016

Table[Cosh[n Log[20 + Sqrt[399]]] - Sqrt[19/21] Sinh[n Log[20 + Sqrt[399]]], {n, 0, 17}]
Table[(2^(-n - 2) (38 (40 - 2 Sqrt[399])^n + 2 Sqrt[399] (40 - 2 Sqrt[399])^n - 38 (40 + 2 Sqrt[399])^n + 2 Sqrt[399] (40 + 2 Sqrt[399])^n))/Sqrt[399], {n, 0, 17}]
LinearRecurrence[{40, -1}, {1, 1}, 17]

G.f.: (1 - 39*x)/(1 - 40*x + x^2).
a(n) = cosh(n*log(20 + sqrt(399))) - sqrt(19/21)*sinh(n*log(20 + sqrt(399))).
a(n) = (2^(-n - 2)*(38*(40 - 2*sqrt(399))^n + 2*sqrt(399)*(40 - 2*sqrt(399))^n - 38*(40 + 2*sqrt(399))^n + 2*sqrt(399)*(40 + 2*sqrt(399))^n))/sqrt(399).
Sum_{n>=0} 1/a(n) = 2.0262989201139499769986...

cp's OEIS Frontend

A097835 First differences of Chebyshev polynomials S(n,27) = A097781(n) with Diophantine property.

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A153111 Solutions of the Pell-like equation 1 + 6*A*A = 7*B*B, with A, B integers.

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A159668 Expansion of (1 - x)/(1 - 28*x + x^2).

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A159664 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 11*n(j) + 1 = a(j)*a(j) and 13*n(j) + 1 = b(j)*b(j), with positive integer numbers.

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A159674 Expansion of (1 - x)/(1 - 32*x + x^2).

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A161595 The list of the A values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

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A153111 Solutions of the Pell-like equation 1 + 6AA = 7BB, with A, B integers.

A159664 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 11n(j) + 1 = a(j)a(j) and 13n(j) + 1 = b(j)b(j), with positive integer numbers.

A161595 The list of the A values in the common solutions to the 2 equations 15k+1=A^2, 19k+1=B^2.