A078366 -id:A078366 - cp's OEIS frontend

A097831 Partial sums of Chebyshev sequence S(n,17)= U(n,17/2)=A078366(n).

1, 18, 306, 5185, 87840, 1488096, 25209793, 427078386, 7235122770, 122570008705, 2076455025216, 35177165419968, 595935357114241, 10095723905522130, 171031371036761970, 2897437583719431361, 49085407552193571168

Offset: 0

Wolfdieter Lang, Aug 31 2004

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (18,-18,1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

LinearRecurrence[{18,-18,1},{1,18,306},20] (* Harvey P. Dale, Nov 20 2022 *)

a(n) = sum(S(k, 17), k=0..n) with S(k, 17) = U(k, 17/2) = A078366(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-17*x+x^2)) = 1/(1-18*x+18*x^2-x^3).
a(n) = 18*a(n-1)-18*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=18.
a(n) = 17*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 17) - S(n, 17) -1)/15.

A097781 Chebyshev polynomials S(n,27) with Diophantine property.

Original entry on oeis.org

1, 27, 728, 19629, 529255, 14270256, 384767657, 10374456483, 279725557384, 7542215592885, 203360095450511, 5483180361570912, 147842509666964113, 3986264580646460139, 107481301167787459640, 2898008866949614950141

Offset: 0

Wolfdieter Lang, Aug 31 2004

All positive integer solutions of Pell equation b(n)^2 - 725*a(n)^2 = +4 together with b(n)=A090248(n+1), n>=0. Note that D=725=29*5^2 is not squarefree.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 27's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,26}. - Milan Janjic, Jan 26 2015

			(x,y) = (27;1), (727;27), (19602;728), ... give the positive integer solutions to x^2 - 29*(5*y)^2 =+4.

Vincenzo Librandi, Table of n, a(n) for n = 0..700
R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (27,-1).

Cf. A078362, A078366.

I:=[1, 27, 728]; [n le 3 select I[n] else 27*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

with (combinat):seq(fibonacci(2*n, 5)/5, n=1..16); # Zerinvary Lajos, Apr 20 2008

Join[{a=1,b=27},Table[c=27*b-a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[1/(1 - 27 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)

[lucas_number1(n,27,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

a(n) = S(n, 27) = U(n, 27/2) = S(2*n+1, sqrt(29))/sqrt(29) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = 27*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=27; a(-1)=0.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (27+5*sqrt(29))/2 and am = (27-5*sqrt(29))/2.
G.f.: 1/(1-27*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*26^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(29)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 5/54*(5 + sqrt(29)). - Peter Bala, Dec 23 2012

A078367 A Chebyshev T-sequence with Diophantine property.

Original entry on oeis.org

2, 17, 287, 4862, 82367, 1395377, 23639042, 400468337, 6784322687, 114933017342, 1947076972127, 32985375508817, 558804306677762, 9466687838013137, 160374888939545567, 2716906424134261502, 46027034321342899967, 779742677038695037937, 13209598475336472744962

Offset: 0

Wolfdieter Lang, Nov 29 2002

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 285*b^2 =+4 with companion sequence b(n)=A078366(n-1), n>=1.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Stefano Spezia, Table of n, a(n) for n = 0..800
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (17,-1).

Cf. A077428, A078355 (Pell +4 equations).

Cf. A049310, A053120, A078366.

a[0] = 2; a[1] = 17; a[n_] := 17a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

a(n)=if(n<0,0,subst(2*poltchebi(n),x,17/2))

[lucas_number2(n,17,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = 17*a(n-1)-a(n-2), n >= 1; a(-1)=17, a(0)=2.
a(n) = sqrt(4 + 285*A078366(n-1)^2), n>=1, (Pell equation d=285, +4).
a(n) = S(n, 17) - S(n-2, 17) = 2*T(n, 17/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 17)=A078366(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-17*x)/(1-17*x+x^2).
a(n) = ap^n + am^n, with ap := (17+sqrt(285))/2 and am := (17-sqrt(285))/2.
E.g.f.: 2*exp(17*x/2)*cosh(sqrt(285)*x/2). - Stefano Spezia, Aug 19 2023

More terms from Stefano Spezia, Aug 19 2023

A078368 A Chebyshev S-sequence with Diophantine property.

Original entry on oeis.org

1, 19, 360, 6821, 129239, 2448720, 46396441, 879083659, 16656193080, 315588584861, 5979526919279, 113295422881440, 2146633507828081, 40672741225852099, 770635449783361800, 14601400804658022101

Offset: 0

Wolfdieter Lang, Nov 29 2002

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 357*a^2 =+4 with companion sequence b(n)=A078369(n+1), n>=0.
This is the m=21 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..20 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364, A077412, A078366 and A049660. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 19's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,18}. Milan Janjic, Jan 25 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=19, q=-1.
Tanya Khovanova, Recursive Sequences
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=21.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (19,-1).

a(n) = sqrt((A078369(n+1)^2 - 4)/357), n>=0, (Pell equation d=357, +4).

Cf. A077428, A078355 (Pell +4 equations).

Join[{a=1,b=19},Table[c=19*b-a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{19,-1},{1,19},20] (* Harvey P. Dale, Feb 10 2019 *)

[lucas_number1(n,19,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

a(n) = 19*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n) = (ap^(n+1)-am^(n+1))/(ap-am) with ap = (19+sqrt(357))/2 and am = (19-sqrt(357))/2.
a(n) = S(2*n+1, sqrt(21))/sqrt(21) = S(n, 19); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
G.f.: 1/(1-19*x+x^2).
a(n) = Sum_{k=0..n} A101950(n,k)*18^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/17*(17 + sqrt(357)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/38*(17 + sqrt(357)). - Peter Bala, Dec 23 2012

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

A092499 Chebyshev polynomials S(n-1,21) with Diophantine property.

Original entry on oeis.org

0, 1, 21, 440, 9219, 193159, 4047120, 84796361, 1776676461, 37225409320, 779956919259, 16341869895119, 342399310878240, 7174043658547921, 150312517518628101, 3149388824232642200, 65986852791366858099

Offset: 0

Rainer Rosenthal, Apr 05 2004

Sequence R_21: Starts with 0,1,21 and satisfies A*C=B^2-1 for successive A,B,C.
The natural numbers a(n)=n satisfy the recurrence a(n-1)*a(n+1)=a(n)^2-1. Let R_r denote the sequence starting with 0,1,r and with this recurrence. We see that R_2 = "the natural numbers" and we find R_3 = A001906. These R_r form a "family" of sequences, which coincides with the m-family (r=m-2, n -> n+1) provided by Wolfdieter Lang (see A078368). This sequence R_21 is strongly related to A041833, which gives the denominators in the continued fraction of sqrt(437).
All positive integer solutions of Pell equation b(n)^2 - 437*a(n)^2 = +4 together with b(n)=A097777(n), n>=0.
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 21's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,20}. - Milan Janjic, Jan 25 2015

			a(3)=440 because a(1)*440 = a(2)^2-1.

Indranil Ghosh, Table of n, a(n) for n = 0..756
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (21,-1).

Cf. R_3=A001906, R_4=A001353, R_5=A004254, R_6=A001109, R_7=A004187, R_8=A001090, R_9=A018913, R_10=A004189, R_11=A004190, R_12=A004191, R_13=A078362, R_14=A007655, R_15=A078364, R_16=A077412, R_17=A078366, R_18=A049660, R_19=A078368, R_20=A075843, R_21=this, sequence, R_22=A077421. See also A041219 and A041917.

LinearRecurrence[{21,-1},{0,1},30] (* Harvey P. Dale, Apr 23 2015 *)

[lucas_number1(n,21,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

a(0)=0, a(1)=1, a(2)=21 and a(n-1)*a(n+1) = a(n)^2-1
a(n) = S(n-1, 21)=U(n-1, 21/2) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = S(2*n-1, sqrt(23))/sqrt(23), n>=1.
a(n) = 21*a(n-1)-a(n-2), n >= 1; a(0)=0, a(1)=1.
a(n) = (ap^n-am^n)/(ap-am) with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
G.f.: x/(1-21*x+x^2).
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*20^k. - Philippe Deléham, Feb 10 2012
Product {n >= 1} (1 + 1/a(n)) = 1/19*(19 + sqrt(437)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/42*(19 + sqrt(437)). - Peter Bala, Dec 23 2012

Extension, Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

Corrected by T. D. Noe, Nov 07 2006

cp's OEIS Frontend

A097831 Partial sums of Chebyshev sequence S(n,17)= U(n,17/2)=A078366(n).

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A097781 Chebyshev polynomials S(n,27) with Diophantine property.

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A078367 A Chebyshev T-sequence with Diophantine property.

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A078368 A Chebyshev S-sequence with Diophantine property.

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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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A092499 Chebyshev polynomials S(n-1,21) with Diophantine property.

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