A322836 -id:A322836 - cp's OEIS frontend

A001091 a(n) = 8*a(n-1) - a(n-2); a(0) = 1, a(1) = 4.

1, 4, 31, 244, 1921, 15124, 119071, 937444, 7380481, 58106404, 457470751, 3601659604, 28355806081, 223244789044, 1757602506271, 13837575261124, 108942999582721, 857706421400644, 6752708371622431, 53163960551578804

Offset: 0

N. J. A. Sloane

a(15+30k)-1 and a(15+30k)+1 are consecutive odd powerful numbers. The first pair is 13837575261124 +- 1. See A076445. - T. D. Noe, May 04 2006
This sequence gives the values of x in solutions of the Diophantine equation x^2 - 15*y^2 = 1. The corresponding y values are in A001090. - Vincenzo Librandi, Nov 12 2010 [edited by Jon E. Schoenfield, May 04 2014]
The square root of 15*(n^2-1) at those numbers = 5 * A136325. - Richard R. Forberg, Nov 22 2013
For the above Diophantine equation x^2-15*y^2=1, x + y = A105426. - Richard R. Forberg, Nov 22 2013
a(n) solves for x in the Diophantine equation floor(3*x^2/5)= y^2. The corresponding y solutions are provided by A136325(n).  x + y = A070997(n). - Richard R. Forberg, Nov 22 2013
Except for the first term, values of x (or y) in the solutions to x^2 - 8xy + y^2 + 15 = 0. - Colin Barker, Feb 05 2014

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
H. Brocard, Notes élémentaires sur le problème de Peel [sic], Nouvelle Correspondance Mathématique, 4 (1878), 337-343.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-1).

Cf. A001090, A090965, A098269, A322836 (column 4).

a:=[1,4];; for n in [3..20] do a[n]:=8*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 26 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1-4*x)/(1-8*x+x^2) )); // G. C. Greubel, Aug 26 2019

LinearRecurrence[{8,-1},{1,4},20] (* Harvey P. Dale, May 01 2014 *)

PARI
```
a(n)=subst(poltchebi(n),x,4)
```

a(n)=n=abs(n); polcoeff((1-4*x)/(1-8*x+x^2)+x*O(x^n),n) /* Michael Somos, Jun 07 2005 */

def A001091_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-4*x)/(1-8*x+x^2) ).list()
A001091_list(20) # G. C. Greubel, Aug 26 2019

G.f.: A(x) = (1-4*x)/(1-8*x+x^2). - Simon Plouffe in his 1992 dissertation
For all elements x of the sequence, 15*(x^2 -1) is a square. Limit_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson, Oct 11 2002
a(n) = sqrt(15*((A001090(n))^2)+1).
a(n) = ((4+sqrt(15))^n + (4-sqrt(15))^n)/2.
a(n) = 4*S(n-1, 8) - S(n-2, 8) = (S(n, 8) - S(n-2, 8))/2, n>=1; S(n, x) := U(n, x/2) with Chebyshev's polynomials of the 2nd kind, A049310, with S(-1, x) := 0 and S(-2, x) := -1.
a(n) = T(n, 4) with Chebyshev's polynomials of the first kind; see A053120.
a(n)=a(-n). - Ralf Stephan, Jun 06 2005
a(n)*a(n+3) - a(n+1)*a(n+2) = 120. - Ralf Stephan, Jun 06 2005
From Peter Bala, Feb 19 2022: (Start)
a(n) = Sum_{k = 0..floor(n/2)} 4^(n-2*k)*15^k*binomial(n,2*k).
a(n) = [x^n] (4*x + sqrt(1 + 15*x^2))^n.
The g.f. A(x) satisfies A(2*x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 16*x + 4*x^2) is the g.f. of A098269.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p >= 3 and positive integers n and k. (End)
From Peter Bala, Aug 17 2022: (Start)
Sum_{n >= 1} 1/(a(n) - (5/2)/a(n)) = 1/3.
Sum_{n >= 1} (-1)^(n+1)/(a(n) + (3/2)/a(n)) = 1/5.
Sum_{n >= 1} 1/(a(n)^2 - 5/2) =  1/3 - 1/sqrt(15). (End)
a(n) = A001090(n+1)-4*A001090(n). - R. J. Mathar, Dec 12 2023

More terms from Larry Reeves (larryr(AT)acm.org), Aug 25 2000

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

A115066 Chebyshev polynomial of the first kind T(n,x), evaluated at x=n.

Original entry on oeis.org

1, 1, 7, 99, 1921, 47525, 1431431, 50843527, 2081028097, 96450076809, 4993116004999, 285573847759211, 17882714781360001, 1216895030905226413, 89415396036432386311, 7055673735003659189775, 595077930963909484707841, 53421565080956452077519377

Offset: 0

Roger L. Bagula, Mar 01 2006

			a(3) = 99 because T[3, x] = 4x^3 - 3x and T[3, 3] = 4*3^3 - 3*3 = 99.

G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.
M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 18.
G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.

Seiichi Manyama, Table of n, a(n) for n = 0..351

Main diagonal of A322836.

Cf. A053120, A107995, A173129, A323118, A349070, A349071, A349072.

Maple
```
with(orthopoly): seq(T(n,n),n=0..17);
```

Table[ChebyshevT[n, n], {n, 0, 17}] (* Arkadiusz Wesolowski, Nov 17 2012 *)

A115066(n)=cos(n*acos(n))  \\ M. F. Hasler, Apr 06 2012

a(n) = polchebyshev(n, 1, n); \\ Seiichi Manyama, Dec 28 2018

a(n) = if(n==0, 1, n*sum(k=0, n, (2*n-2)^k*binomial(n+k, 2*k)/(n+k))); \\ Seiichi Manyama, Mar 05 2021

a(n) = cos(n*arccos(n)).
a(n) ~ 2^(n-1) * n^n. - Vaclav Kotesovec, Jan 19 2019
a(n) = (A323118(n) - A107995(n-2))/2 for n > 1. - Seiichi Manyama, Mar 05 2021
a(n) = n * Sum_{k=0..n} (2*n-2)^k * binomial(n+k,2*k)/(n+k) for n > 0. - Seiichi Manyama, Mar 05 2021
It appears that a(2*n+1) == 0 (mod (2*n+1)^2) and 2*a(4*n+2) == -2 (mod (4*n+2)^4), while for k > 1, 2*a(2^k*(2*n+1)) == 2 (mod (2^k*(2*n+1))^4). - Peter Bala, Feb 01 2022

Edited by N. J. A. Sloane, Apr 05 2006

A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 2, -1, 1, 4, 3, 0, 1, 6, 15, 4, 1, 1, 8, 35, 56, 5, 0, 1, 10, 63, 204, 209, 6, -1, 1, 12, 99, 496, 1189, 780, 7, 0, 1, 14, 143, 980, 3905, 6930, 2911, 8, 1, 1, 16, 195, 1704, 9701, 30744, 40391, 10864, 9, 0, 1, 18, 255, 2716, 20305, 96030, 242047, 235416, 40545, 10, -1

Offset: 0

Seiichi Manyama, Jan 06 2019

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 2,    4,     6,      8,     10,      12, ...
  -1, 3,   15,    35,     63,     99,     143, ...
   0, 4,   56,   204,    496,    980,    1704, ...
   1, 5,  209,  1189,   3905,   9701,   20305, ...
   0, 6,  780,  6930,  30744,  96030,  241956, ...
  -1, 7, 2911, 40391, 242047, 950599, 2883167, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Mirror of A228161.
Columns 0-19 give A056594, A000027(n+1), A001353(n+1), A001109(n+1), A001090(n+1), A004189(n+1), A004191, A007655(n+2), A077412, A049660(n+1), A075843(n+1), A077421, A077423, A097309, A097311, A097313, A029548, A029547, A144128(n+1), A078987.
Rows 0-10 give A000012, A005843, A000466, A144138, A144139, A242850, A242851, A242852, A242853, A242854, A243130.
Main diagonal gives A323118.
Cf. A179943, A322836 (Chebyshev polynomial of the first kind).

T(n,k)  = polchebyshev(n, 2, k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Jan 07 2019

T(n, k) = sum(j=0, n, (2*k-2)^j*binomial(n+1+j, 2*j+1)); \\ Seiichi Manyama, Mar 03 2021

T(0,k) = 1, T(1,k) = 2 * k and T(n,k) = 2 * k * T(n-1,k) - T(n-2,k) for n > 1.

T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - Seiichi Manyama, Mar 03 2021

A188644 Array of (k^n + k^(-n))/2 where k = (sqrt(x^2-1) + x)^2 for integers x >= 1.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 97, 17, 1, 1, 1351, 577, 31, 1, 1, 18817, 19601, 1921, 49, 1, 1, 262087, 665857, 119071, 4801, 71, 1, 1, 3650401, 22619537, 7380481, 470449, 10081, 97, 1, 1, 50843527, 768398401, 457470751, 46099201, 1431431, 18817, 127, 1

Offset: 0

Charles L. Hohn, Apr 06 2011

Conjecture: Given the function f(x,y) = (sqrt(x^2+y) + x)^2 and constant k=f(x,y), then for all integers x >= 1 and y=[+-]1, k may be irrational, but (k^n + k^(-n))/2 always produces integer sequences; y=-1 results shown here; y=1 results are A188645.

Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{2*k}(x), evaluated at x=n. - Seiichi Manyama, Dec 30 2018

			Row 2 gives {( (2+sqrt(3))^(2*n) + (2-sqrt(3))^(2*n) )/2}.
Square array begins:
     | 0    1       2          3             4
-----+---------------------------------------------
   1 | 1,   1,      1,         1,            1, ...
   2 | 1,   7,     97,      1351,        18817, ...
   3 | 1,  17,    577,     19601,       665857, ...
   4 | 1,  31,   1921,    119071,      7380481, ...
   5 | 1,  49,   4801,    470449,     46099201, ...
   6 | 1,  71,  10081,   1431431,    203253121, ...
   7 | 1,  97,  18817,   3650401,    708158977, ...
   8 | 1, 127,  32257,   8193151,   2081028097, ...
   9 | 1, 161,  51841,  16692641,   5374978561, ...
  10 | 1, 199,  79201,  31521799,  12545596801, ...
  11 | 1, 241, 116161,  55989361,  26986755841, ...
  12 | 1, 287, 164737,  94558751,  54276558337, ...
  13 | 1, 337, 227137, 153090001, 103182433537, ...
  14 | 1, 391, 305761, 239104711, 186979578241, ...
  15 | 1, 449, 403201, 362074049, 325142092801, ...
  ...

Row 2 is A011943, row 3 is A056771, row 8 is A175633, (row 2)*2 is A067902, (row 9)*2 is A089775.
Column 0-5 give A000012, A056220, A144130, A243132, A243134, A243136.
(column 1)*2 is A060626.
Cf. A188645 (f(x, y) as above with y=1).
Diagonals give A173129, A322899.
Cf. A188646, A322836.

max = 9; y = -1; t = Table[k = ((x^2 + y)^(1/2) + x)^2; ((k^n) + (k^(-n)))/2 // FullSimplify, {n, 0, max - 1}, {x, 1, max}]; Table[ t[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)

A(n,k) = (A188646(n,k-1) + A188646(n,k))/2.

A(n,k) = Sum_{j=0..k} binomial(2*k,2*j)*(n^2-1)^(k-j)*n^(2*j). - Seiichi Manyama, Jan 01 2019

Edited by Seiichi Manyama, Dec 30 2018

More terms from Seiichi Manyama, Jan 01 2019

A101124 Number triangle associated to Chebyshev polynomials of first kind.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, 0, 1, 2, 1, 1, 1, 7, 3, 1, 0, 1, 26, 17, 4, 1, -1, 1, 97, 99, 31, 5, 1, 0, 1, 362, 577, 244, 49, 6, 1, 1, 1, 1351, 3363, 1921, 485, 71, 7, 1, 0, 1, 5042, 19601, 15124, 4801, 846, 97, 8, 1, -1, 1, 18817, 114243, 119071, 47525, 10081, 1351, 127, 9, 1, 0, 1, 70226, 665857, 937444, 470449, 120126, 18817, 2024, 161

Offset: 0

Paul Barry, Dec 02 2004

			As a number triangle, rows begin:
  {1},
  {0,1},
  {-1,1,1},
  {0,1,2,1},
  ...
As a square array, rows begin
   1, 1,  1,   1,    1, ...
   0, 1,  2,   3,    4, ...
  -1, 1,  7,  17,   31, ...
   0, 1, 26,  99,  244, ...
   1, 1, 97, 577, 1921, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns include A001075, A001541, A001091, A001079, A023038, A011943.
Row sums are A101125.
Diagonal sums are A101126.
Main diagonal gives A115066.
Mirror of A322836.
Cf. A053120.

T[n_, k_] := SeriesCoefficient[x^k (1 - k x)/(1 - 2 k x + x^2), {x, 0, n}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 12 2017 *)

Number triangle S(n, k)=T(n-k, k), k

Columns have g.f. x^k(1-kx)/(1-2kx+x^2).

Also, square array if(n=0, 1, T(n, k)) read by antidiagonals.

A323117 a(n) = T_{n}(n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 0, 1, 26, 577, 15124, 470449, 17057046, 708158977, 33165873224, 1730726404001, 99612037019890, 6269617090376641, 428438743526336412, 31592397706723526737, 2500433598371461203374, 211434761022028192051201, 19023879409608991280267536

Offset: 0

Seiichi Manyama, Jan 05 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..351
Wikipedia, Chebyshev polynomials.

Cf. A115066, A322836, A323118.

Table[ChebyshevT[n, n - 1], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)

PARI
```
a(n) = polchebyshev(n, 1, n-1);
```

a(n) = round(cos(n*acos(n-1))); \\ Seiichi Manyama, Mar 05 2021

a(n) = if(n==0, 1, n*sum(k=0, n, (2*n-4)^k*binomial(n+k, 2*k)/(n+k))); \\ Seiichi Manyama, Mar 05 2021

a(n)^2 - ((n - 1)^2 - 1) * A323118(n-1)^2 = 1 for n > 0.
a(n) = A322836(n,n-1) for n > 0.
a(n) ~ exp(-1) * 2^(n-1) * n^n. - Vaclav Kotesovec, Jan 05 2019
a(n) = cos(n*arccos(n-1)). - Seiichi Manyama, Mar 05 2021
a(n) = n * Sum_{k=0..n} (2*n-4)^k * binomial(n+k,2*k)/(n+k) for n > 0. - Seiichi Manyama, Mar 05 2021

A322888 Chebyshev T-polynomials T_n(16).

Original entry on oeis.org

1, 16, 511, 16336, 522241, 16695376, 533729791, 17062657936, 545471324161, 17438019715216, 557471159562751, 17821639086292816, 569734979601807361, 18213697708171542736, 582268591681887560191, 18614381236112230383376, 595077930963909484707841

Offset: 0

Seiichi Manyama, Dec 29 2018

Colin Barker, Table of n, a(n) for n = 0..600
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (32, -1).

Column 16 of A322836.

a:=[1,16];; for n in [3..20] do a[n]:=32*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 31 2018

I:=[1, 16]; [n le 2 select I[n] else 32*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 02 2019

seq(coeff(series((1-16*x)/(1-32*x+x^2),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 31 2018

Array[ChebyshevT[#, 16] &, 17, 0] (* or *)
With[{k = 16}, CoefficientList[Series[(1 - k x)/(1 - 2 k x + x^2), {x, 0, 16}], x]] (* Michael De Vlieger, Jan 01 2019 *)

PARI
```
{a(n) = polchebyshev(n, 1, 16)}
```

Vec((1 - 16*x) / (1 - 32*x + x^2) + O(x^20)) \\ Colin Barker, Dec 30 2018

a(0) = 1, a(1) = 16 and a(n) = 32*a(n-1) - a(n-2) for n > 1.
From Colin Barker, Dec 30 2018: (Start)
G.f.: (1 - 16*x) / (1 - 32*x + x^2).
a(n) = ((16+sqrt(255))^(-n) * (1+(16+sqrt(255))^(2*n))) / 2.
(End)

A322889 Chebyshev T-polynomials T_n(18).

Original entry on oeis.org

1, 18, 647, 23274, 837217, 30116538, 1083358151, 38970776898, 1401864610177, 50428155189474, 1814011722210887, 65253993844402458, 2347329766676277601, 84438617606501591178, 3037442904067381004807, 109263505928819214581874

Offset: 0

Seiichi Manyama, Dec 29 2018

Colin Barker, Table of n, a(n) for n = 0..600
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (36,-1).

Column 18 of A322836.

a:=[1,18];; for n in [3..20] do a[n]:=36*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 31 2018

I:=[1, 18]; [n le 2 select I[n] else 36*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 02 2019

seq(coeff(series((1-18*x)/(1-36*x+x^2),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 31 2018

Array[ChebyshevT[#, 18] &, 16, 0] (* or *)
With[{k = 18}, CoefficientList[Series[(1 - k x)/(1 - 2 k x + x^2), {x, 0, 15}], x]] (* Michael De Vlieger, Jan 01 2019 *)

PARI
```
{a(n) = polchebyshev(n, 1, 18)}
```

Vec((1 - 18*x) / (1 - 36*x + x^2) + O(x^20)) \\ Colin Barker, Dec 30 2018

a(0) = 1, a(1) = 18 and a(n) = 36*a(n-1) - a(n-2) for n > 1.
From Colin Barker, Dec 30 2018: (Start)
G.f.: (1 - 18*x) / (1 - 36*x + x^2).
a(n) = ((18+sqrt(323))^(-n) * (1+(18+sqrt(323))^(2*n))) / 2. (End)
E.g.f.: exp(18*x)*cosh(sqrt(323)*x). - Stefano Spezia, Aug 02 2025

A322890 a(n) = value of Chebyshev T-polynomial T_n(20).

Original entry on oeis.org

1, 20, 799, 31940, 1276801, 51040100, 2040327199, 81562047860, 3260441587201, 130336101440180, 5210183616019999, 208277008539359780, 8325870157958371201, 332826529309795488260, 13304735302233861159199, 531856585560044650879700

Offset: 0

Seiichi Manyama, Dec 29 2018

Seiichi Manyama, Table of n, a(n) for n = 0..624
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (40, -1).

Column 20 of A322836.

Cf. A041758.

a:=[1,20];; for n in [3..20] do a[n]:=40*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 31 2018

seq(coeff(series((1-20*x)/(1-40*x+x^2),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 31 2018

CoefficientList[Series[(1 - 20 x)/(1 - 40 x + x^2), {x, 0, 15}], x] (* or *)
Array[ChebyshevT[#, 20] &, 16, 0] (* Michael De Vlieger, Jan 01 2019 *)

PARI
```
{a(n) = polchebyshev(n, 1, 20)}
```

Vec((1 - 20*x) / (1 - 40*x + x^2) + O(x^20)) \\ Colin Barker, Dec 30 2018

a(0) = 1, a(1) = 20 and a(n) = 40*a(n-1) - a(n-2) for n > 1.
From Colin Barker, Dec 30 2018: (Start)
G.f.: (1 - 20*x) / (1 - 40*x + x^2).
a(n) = ((20+sqrt(399))^(-n) * (1+(20+sqrt(399))^(2*n))) / 2.
(End)

Showing 1-9 of 9 results.

cp's OEIS Frontend

A001091 a(n) = 8*a(n-1) - a(n-2); a(0) = 1, a(1) = 4.

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A115066 Chebyshev polynomial of the first kind T(n,x), evaluated at x=n.

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A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

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A188644 Array of (k^n + k^(-n))/2 where k = (sqrt(x^2-1) + x)^2 for integers x >= 1.

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A101124 Number triangle associated to Chebyshev polynomials of first kind.

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A323117 a(n) = T_{n}(n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind.

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PARI

PARI

PARI

Formula

A322888 Chebyshev T-polynomials T_n(16).