A001085 -id:A001085 - cp's OEIS frontend

A008310 Triangle of coefficients of Chebyshev polynomials T_n(x).

1, 1, -1, 2, -3, 4, 1, -8, 8, 5, -20, 16, -1, 18, -48, 32, -7, 56, -112, 64, 1, -32, 160, -256, 128, 9, -120, 432, -576, 256, -1, 50, -400, 1120, -1280, 512, -11, 220, -1232, 2816, -2816, 1024, 1, -72, 840, -3584, 6912, -6144, 2048, 13, -364, 2912, -9984, 16640, -13312, 4096

Offset: 0

N. J. A. Sloane

The row length sequence of this irregular array is A008619(n), n >= 0. Even or odd powers appear in increasing order starting with 1 or x for even or odd row numbers n, respectively. This is the standard triangle A053120 with 0 deleted. - Wolfdieter Lang, Aug 02 2014

Let T* denote the triangle obtained by replacing each number in this triangle by its absolute value. Then T* gives the coefficients for cos(nx) as a polynomial in cos x. - Clark Kimberling, Aug 04 2024

			Rows are: (1), (1), (-1,2), (-3,4), (1,-8,8), (5,-20,16) etc., since if c = cos(x): cos(0x) = 1, cos(1x) = 1c; cos(2x) = -1+2c^2; cos(3x) = -3c+4c^3, cos(4x) = 1-8c^2+8c^4, cos(5x) = 5c-20c^3+16c^5, etc.
From _Wolfdieter Lang_, Aug 02 2014: (Start)
This irregular triangle a(n,k) begins:
  n\k   0    1     2      3      4      5      6      7 ...
  0:    1
  1:    1
  2:   -1    2
  3:   -3    4
  4:    1   -8     8
  5:    5  -20    16
  6:   -1   18   -48     32
  7:   -7   56  -112     64
  8:    1  -32   160   -256    128
  9:    9 -120   432   -576    256
 10:   -1   50  -400   1120  -1280    512
 11:  -11  220 -1232   2816  -2816   1024
 12:    1  -72   840  -3584   6912  -6144   2048
 13:   13 -364  2912  -9984  16640 -13312   4096
 14:   -1   98 -1568   9408 -26880  39424 -28672   8192
 15:  -15  560 -6048  28800 -70400  92160 -61440  16384
  ...
T(4,x) = 1 - 8*x^2 + 8*x^4, T(5,x) = 5*x - 20*x^3 +16*x^5.
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
Yaroslav Zolotaryuk, J. Chris Eilbeck, "Analytical approach to the Davydov-Scott theory with on-site potential", Physical Review B 63, p543402, Jan. 2001. The authors write, "Since the algebra of these is 'hyperbolic', contrary to the usual Chebyshev polynomials defined on the interval 0 <= x <= 1, we call the set of functions (21) the hyperbolic Chebyshev polynomials." (This refers to the triangle T* described in Comments.)

R. J. Mathar, Table of n, a(n) for n = 0..2600
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Renato Ferreira Pinto Jr. and Nathaniel Harms, Testing Support Size More Efficiently Than Learning Histograms, arXiv:2410.18915 [cs.DS], 2024. See p. 40.
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
I. Rivin, Growth in free groups (and other stories), arXiv:math/9911076 [math.CO], 1999.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

A039991 is a row reversed version, but has zeros which enable the triangle to be seen. Columns/diagonals are A011782, A001792, A001793, A001794, A006974, A006975, A006976 etc.
Reflection of A028297. Cf. A008312, A053112.
Row sums are one. Polynomial evaluations include A001075 (x=2), A001541 (x=3), A001091, A001079, A023038, A011943, A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203.
Cf. A053120.

A008310 := proc(n,m) local x ; coeftayl(simplify(ChebyshevT(n,x),'ChebyshevT'),x=0,m) ; end: i := 0 : for n from 0 to 100 do for m from n mod 2 to n by 2 do printf("%d %d ",i,A008310(n,m)) ; i := i+1 ; od ; od ; # R. J. Mathar, Apr 20 2007
# second Maple program:
b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

Flatten[{1, Table[CoefficientList[ChebyshevT[n, x], x], {n, 1, 13}]}]//DeleteCases[#, 0, Infinity]& (* or *) Flatten[{1, Table[Table[((-1)^k*2^(n-2 k-1)*n*Binomial[n-k, k])/(n-k), {k, Floor[n/2], 0, -1}], {n, 1, 13}]}] (* Eugeniy Sokol, Sep 04 2019 *)

a(n,m) = 2^(m-1) * n * (-1)^((n-m)/2) * ((n+m)/2-1)! / (((n-m)/2)! * m!) if n>0. - R. J. Mathar, Apr 20 2007
From Paul Weisenhorn, Oct 02 2019: (Start)
T_n(x) = 2*x*T_(n-1)(x) - T_(n-2)(x), T_0(x) = 1, T_1(x) = x.
T_n(x) = ((x+sqrt(x^2-1))^n + (x-sqrt(x^2-1))^n)/2. (End)
From Peter Bala, Aug 15 2022: (Start)
T(n,x) = [z^n] ( z*x + sqrt(1 + z^2*(x^2 - 1)) )^n.
Sum_{k = 0..2*n} binomial(2*n,k)*T(k,x) = (2^n)*(1 + x)^n*T(n,x).
exp( Sum_{n >= 1} T(n,x)*t^n/n ) = Sum_{n >= 0} P(n,x)*t^n, where P(n,x) denotes the n-th Legendre polynomial. (End)

Additional comments and more terms from Henry Bottomley, Dec 13 2000

Edited: Corrected Cf. A039991 statement. Cf. A053120 added. - Wolfdieter Lang, Aug 06 2014

A075843 Numbers k such that 99*k^2 + 1 is a square.

Original entry on oeis.org

0, 1, 20, 399, 7960, 158801, 3168060, 63202399, 1260879920, 25154396001, 501827040100, 10011386405999, 199725901079880, 3984506635191601, 79490406802752140, 1585823629419851199, 31636982181594271840

Offset: 0

Gregory V. Richardson, Oct 14 2002

From Wolfdieter Lang, Nov 08 2002: (Start)
Chebyshev's polynomials U(n,x) evaluated at x=10.
The a(n) give all (unsigned, integer) solutions of Pell equation b(n)^2 - 99*a(n)^2 = +1 with b(n)= A001085(n). (End)
For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 20's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imagianry 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,...,19}. - Milan Janjic, Jan 25 2015

A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (20,-1).

Cf. A001084.
Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), this sequence (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
Cf. A323182.

m:=10;; a:=[0,1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

I:=[0,1]; [n le 2 select I[n] else 20*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

seq( simplify(ChebyshevU(n-1, 10)), n=0..20); # G. C. Greubel, Dec 22 2019

Table[GegenbauerC[n-1, 1, 10], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[x/(1-20x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
ChebyshevU[Range[22] -2, 10] (* G. C. Greubel, Dec 22 2019 *)
LinearRecurrence[{20,-1},{0,1},20] (* Harvey P. Dale, Dec 03 2023 *)

vector( 22, n, polchebyshev(n-2, 2, 10) ) \\ G. C. Greubel, Dec 22 2019

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

[chebyshev_U(n-1,10) for n in (0..20)] # G. C. Greubel, Dec 22 2019

a(n) = ((10+3*sqrt(11))^n - (10-3*sqrt(11))^n) / (6*sqrt(11)).
a(n) = 20*a(n-1) - a(n-2), n>=1, a(0)=0, a(1)=1.
a(n) = S(n-1, 20), with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. S(-1, x) := 0. See A049310.
G.f.: x/(1 - 20*x + x^2).
a(n) = sqrt((A001085(n)^2 - 1)/99).
Lim_{n->inf.} a(n)/a(n-1) = 10 + 3*sqrt(11).
a(n+1) = Sum_{k=0..n} A101950(n,k)*19^k. - Philippe Deléham, Feb 10 2012
Product_{n>=1} (1 + 1/a(n)) = 1/3*(3 + sqrt(11)). - Peter Bala, Dec 23 2012
Product_{n>=2} (1 - 1/a(n)) = 3/20*(3 + sqrt(11)). - Peter Bala, Dec 23 2012

A056771 a(n) = a(-n) = 34*a(n-1) - a(n-2), and a(0)=1, a(1)=17.

Original entry on oeis.org

1, 17, 577, 19601, 665857, 22619537, 768398401, 26102926097, 886731088897, 30122754096401, 1023286908188737, 34761632124320657, 1180872205318713601, 40114893348711941777, 1362725501650887306817, 46292552162781456490001

Offset: 0

Henry Bottomley, Aug 16 2000

The sequence satisfies the Pell equation a(n)^2 - 18 * A202299(n+1)^2 = 1. - Vincenzo Librandi, Dec 19 2011
Also numbers n such that n - 1 and 2*n + 2 are squares. - Arkadiusz Wesolowski, Mar 15 2015
And they, n - 1 and 2*n + 2, are the squares of A005319 and A003499. - Michel Marcus, Mar 15 2015
This sequence {a(n)} gives all the nonnegative integer solutions of the Pell equation a(n)^2 - 32*(3*A091761(n))^2 = +1. - Wolfdieter Lang, Mar 09 2019

			G.f. = 1 + 17*x + 577*x^2 + 19601*x^3 + 665857*x^4 + 22619537*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..600 (terms 0..200 from Vincenzo Librandi)
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (34,-1).

Cf. A001075, A001541, A001091, A001079, A023038, A011943, A001081, A023039, A001085 and note relationship with square triangular number sequences A001110 and A001109. A091761.

Row 3 of array A188644.

I:=[1, 17]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 18 2011

LinearRecurrence[{34,-1},{1,17},30] (* Vincenzo Librandi, Dec 18 2011 *)
a[ n_] := ChebyshevT[ 2 n, 3]; (* Michael Somos, May 28 2014 *)

makelist(expand(((17+sqrt(288))^n+(17-sqrt(288))^n))/2, n, 0, 15); /* Vincenzo Librandi, Dec 18 2011 */

{a(n) = polchebyshev( n, 1, 17)}; /* Michael Somos, Apr 05 2019 */

[lucas_number2(n,34,1)/2 for n in range(0,15)] # Zerinvary Lajos, Jun 27 2008

a(n) = (r^n + 1/r^n)/2 with r = 17 + sqrt(17^2-1).
a(n) = 16*A001110(n) + 1 = A001541(2n) = (4*A001109(n))^2 + 1 = 3*A001109(2n-1) - A001109(2n-2) = A001109(2n) - 3*A001109(2n-1).
a(n) = T(n, 17) = T(2*n, 3) with T(n, x) Chebyshev's polynomials of the first kind. See A053120. T(n, 3)= A001541(n).
G.f.: (1-17*x)/(1-34*x+x^2).
G.f.: (1 - 17*x / (1 - 288*x / (17 - x))). - Michael Somos, Apr 05 2019
a(n) = cosh(2n*arcsinh(sqrt(8))). - Herbert Kociemba, Apr 24 2008
a(n) = (a^n + b^n)/2 where a = 17 + 12*sqrt(2) and b = 17 - 12*sqrt(2); sqrt(a(n)-1)/4 = A001109(n). - James R. Buddenhagen, Dec 09 2011
a(-n) = a(n). - Michael Somos, May 28 2014
a(n) = sqrt(1 + 32*9*A091761(n)^2), n >= 0. See one of the Pell comments above. - Wolfdieter Lang, Mar 09 2019

More terms from James Sellers, Sep 07 2000

Chebyshev comments from Wolfdieter Lang, Nov 29 2002

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Dec 28 2018

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 1,    2,     3,      4,      5,       6, ...
  -1, 1,    7,    17,     31,     49,      71, ...
   0, 1,   26,    99,    244,    485,     846, ...
   1, 1,   97,   577,   1921,   4801,   10081, ...
   0, 1,  362,  3363,  15124,  47525,  120126, ...
  -1, 1, 1351, 19601, 119071, 470449, 1431431, ...

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

Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).

Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *)

T(n,k) = polchebyshev(n,1,k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018

T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021

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

A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021

A001084 a(n) = 20*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.

Original entry on oeis.org

0, 3, 60, 1197, 23880, 476403, 9504180, 189607197, 3782639760, 75463188003, 1505481120300, 30034159217997, 599177703239640, 11953519905574803, 238471220408256420, 4757470888259553597, 94910946544782815520, 1893461460007396756803, 37774318253603152320540

Offset: 0

N. J. A. Sloane

Also 11*x^2+1 is a square. n=11 in PARI script below. - Cino Hilliard, Mar 08 2003

This sequence gives the values of y in solutions of the Diophantine equation x^2 - 11*y^2 = 1; the corresponding x values are in A001085. - Vincenzo Librandi, Nov 12 2010 [edited by Jon E. Schoenfield, May 04 2014]

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).
"Questions D'Arithmetique", Question 3686, Solution by H.L. Mennessier, Mathesis, 65(4, Supplement) 1956, pp. 1-12.

T. D. Noe, Table of n, a(n) for n = 0..200
H. Brocard, Notes élémentaires sur le problème de Peel, Nouvelle Correspondance Mathématique, 4 (1878), 161-169.
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 linear recurrences with constant coefficients, signature (20,-1).

Equals 3 * A075843.

Cf. A001085, A221762.

I:=[0,3]; [n le 2 select I[n] else 20*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 19 2017

A001084:=3*z/(1-20*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{20, -1}, {0, 3}, 20] (* T. D. Noe, Dec 19 2011 *)
CoefficientList[Series[3*x/(1 - 20*x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)
Table[3 ChebyshevU[-1 + n, 10], {n, 0, 18}] (* Herbert Kociemba, Jun 05 2022 *)

nxsqp1(m,n) = { for(x=1,m, y = n*x*x+1; if(issquare(y),print1(x" ")) ) }

x='x+O('x^30); concat([0], Vec(3*x/(1 - 20*x + x^2))) \\ G. C. Greubel, Dec 20 2017

Limit_{n->oo} a(n)/a(n-1) = 10 + 3*sqrt(11); for all n in the sequence, 11*n^2 + 1 is a perfect square. - Gregory V. Richardson, Oct 06 2002
a(n) = ((10 + 3*sqrt(11))^n - (10 - 3*sqrt(11))^n) / (2*sqrt(11)). - Gregory V. Richardson, Oct 06 2002
From Mohamed Bouhamida, Sep 20 2006: (Start)
a(n) = 19*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 21*(a(n-1) - a(n-2)) + a(n-3). (End)
G.f.: 3*x/(1 - 20*x + x^2). - G. C. Greubel, Dec 20 2017
E.g.f.: exp(10*x)*sinh(3*sqrt(11)*x)/sqrt(11). - Stefano Spezia, Aug 16 2024

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Original entry on oeis.org

0, 2, 36, 646, 11592, 208010, 3732588, 66978574, 1201881744, 21566892818, 387002188980, 6944472508822, 124613502969816, 2236098580947866, 40125160954091772, 720016798592704030

Offset: 0

Gary Detlefs, Feb 20 2012

Denote as {a,b,c,d} the second-order linear recurrence a(n) = c*a(n-1) + d*a(n-2) with initial terms a, b. The following sequences and recurrence formulas are related to integer solutions of k*x^2 + 1 = y^2.
.
   k             x                        y
   -  -----------------------  -----------------------
A001542 {0,2,6,-1}       A001541 {1,3,6,-1}
A001353 {0,1,4,-1}       A001075 {1,2,4,-1}
A060645 {0,4,18,-1}      A023039 {1,9,18,-1}
A001078 {0,2,10,-1}      A001079 {1,5,10,-1}
A001080 {0,3,16,-1}      A001081 {1,8,16,-1}
A001109 {0,1,6,-1}       A001541 {1,3,6,-1}
A084070 {0,1,38,-1}      A078986 {1,19,38,-1}
A001084 {0,3,20,-1}      A001085 {1,10,20,-1}
A011944 {0,2,14,-1}      A011943 {1,7,14,-1}
A075871 {0,180,1298,-1}  A114047 {1,649,1298,-1}
A068204 {0,4,30,-1}      A069203 {1,15,30,-1}
A001090 {0,1,8,-1}       A001091 {1,4,8,-1}
A121740 {0,8,66,-1}      A099370 {1,33,66,-1}
A202299 {0,4,34,-1}      A056771 {1,17,34,-1}
A174765 {0,39,340,-1}    A114048 {1,179,340,-1}
a(n)    {0,2,18,-1}      A023039 {1,9,18,-1}
A174745 {0,12,110,-1}    A114049 {1,55,110,-1}
A174766 {0,42,394,-1}    A114050 {1,197,394,-1}
A174767 {0,5,48,-1}      A114051 {1,24,48,-1}
A004189 {0,1,10,-1}      A001079 {1,5,10,-1}
A174768 {0,10,102,-1}    A099397 {1,51,102,-1}
The sequence of the c parameter is listed in A180495.

Bruno Berselli, Table of n, a(n) for n = 0..500
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (18,-1).

Cf. A023039, A049660, A079962.

m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // Bruno Berselli, Jun 19 2019

readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od;

LinearRecurrence[{18, -1}, {0, 2}, 16] (* Bruno Berselli, Feb 21 2012 *)
Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}]  (* Herbert Kociemba, Jun 05 2022 *)

makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* Bruno Berselli, Jun 19 2019 */

a(n) = 18*a(n-1) - a(n-2).
From Bruno Berselli, Feb 21 2012: (Start)
G.f.: 2*x/(1-18*x+x^2).
a(n) = -a(-n) = 2*A049660(n) = ((2 + sqrt(5))^(2*n)-(2 - sqrt(5))^(2*n))/(4*sqrt(5)). (End)
a(n) = Fibonacci(6*n)/4. - Bruno Berselli, Jun 19 2019
For n>=1, a(n) = A079962(6n-3). - Christopher Hohl, Aug 22 2021

A238240 Positive integers n such that x^2 - 20xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

18, 35, 50, 63, 72, 74, 83, 90, 95, 98, 99, 107, 140, 162, 171, 200, 215, 227, 252, 266, 275, 288, 296, 315, 332, 347, 359, 360, 362, 371, 380, 387, 392, 395, 396, 407, 428, 450, 491, 495, 530, 539, 560, 567, 602, 623, 626, 635, 648, 666, 684, 695, 711, 722, 743, 747, 755, 770, 791, 794, 800, 810

Offset: 1

Colin Barker, Feb 20 2014

nonn

Positive integers n such that x^2 - 99 y^2 + n = 0 has integer solutions. - Robert Israel, Oct 22 2024

			63 is in the sequence because x^2 - 20xy + y^2 + 63 = 0 has integer solutions, for example (x, y) = (1, 16).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A075839 (n = 18), A221763 (n = 63), A198947 (n = 90), A001085 (n = 99).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330, A236331.

filter:= t -> [isolve(99*y^2 - z^2 = t)] <> []:
select(filter, [$1..1000]); # Robert Israel, Oct 22 2024

Corrected by Robert Israel, Oct 22 2024

A090728 a(n) = 20*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 20.

Original entry on oeis.org

2, 20, 398, 7940, 158402, 3160100, 63043598, 1257711860, 25091193602, 500566160180, 9986232009998, 199224074039780, 3974495248785602, 79290680901672260, 1581839122784659598, 31557491774791519700, 629567996373045734402, 12559802435686123168340

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

Except for the first term, positive values of x (or y) satisfying x^2 - 20xy + y^2 + 396 = 0. - Colin Barker, Feb 28 2014

Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (20,-1).

Cf. A080959, A037035, A054877.

Cf. A001085.

a[0] = 2; a[1] = 20; a[n_] := 20a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

Vec((2-20*x)/(1-20*x+x^2) + O(x^100)) \\ Colin Barker, Feb 28 2014

[lucas_number2(n,20,1) for n in range(0,20)] # Zerinvary Lajos, Jun 27 2008

a(n) = p^n + q^n, where p = 10 + 3*sqrt(11) and q = 10 - 3*sqrt(11). - Tanya Khovanova, Feb 06 2007

G.f.: (2-20*x)/(1-20*x+x^2). - Philippe Deléham, Nov 02 2008

More terms from Robert G. Wilson v, Jan 30 2004

More terms from Colin Barker, Feb 28 2014

A118894 Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution with x even.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51, 59, 63, 67, 71, 75, 79, 83, 87, 91, 99, 103, 107, 115, 119, 123, 127, 131, 135, 139, 143, 151, 159, 163, 167, 171, 175, 179, 187, 191, 195, 199, 211, 215, 219, 223, 227, 231, 235, 239, 243, 247, 251, 255, 263, 267

Offset: 1

T. D. Noe, May 04 2006

nonn

Numbers m such that A002350(m) is even. These m can be used to generate consecutive odd powerful numbers, as in A076445. As shown by Lang, the solution of Pell's equation is greatly simplified by Chebyshev polynomials of the first kind T(n,x), which is illustrated in A001075 for the case m=3. In that case, the solutions are x=T(n,2), for integer n>0. For any m in this sequence, let E(k)=T(m+2mk,A002350(m)). Then E(k)-1 and E(k)+1 are consecutive odd powerful numbers for k=0,1,2,...

Wolfdieter Lang, Chebyshev Polynomials and Certain Quadratic Diophantine Equations
H. W. Lenstra Jr., Solving the Pell equation, Notices AMS, 49 (2002), 182-192.

Cf. A001075, A001091, A023038, A001081, A001085, A077424, A097310 (x solutions for m=3, 15, 35, 63, 99, 143, 195).

A099276 Unsigned member r=-18 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 18, 361, 7200, 143641, 2865618, 57168721, 1140508800, 22753007281, 453919636818, 9055639729081, 180658874944800, 3604121859166921, 71901778308393618, 1434431444308705441, 28616727107865715200

Offset: 0

Wolfdieter Lang, Oct 18 2004

((-1)^(n+1))*a(n) = S_{-18}(n), n>=0, defined in A092184.

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (19,19,-1)>

LinearRecurrence[{19,19,-1},{0,1,18},30] (* Harvey P. Dale, Sep 08 2024 *)

a(n)= 20*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n)= 19*a(n-1) + 19*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=18.
G.f.: x*(1-x)/((1+x)*(1-20*x+x^2)) = x*(1-x)/(1-19*x-19*x^2+x^3) (from the Stephan link, see A092184).
a(n)= (T(n, 10)-(-1)^n)/11, with Chebyshev's polynomials of the first kind evaluated at x=10: T(n, 10)=A001085(n)=((10+3*sqrt(11))^n + (10-3*sqrt(11))^n)/2.

cp's OEIS Frontend

A008310 Triangle of coefficients of Chebyshev polynomials T_n(x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A075843 Numbers k such that 99*k^2 + 1 is a square.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Sage

Formula

A056771 a(n) = a(-n) = 34*a(n-1) - a(n-2), and a(0)=1, a(1)=17.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Sage

Formula

Extensions

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A001084 a(n) = 20*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Views

Author

Keywords