A173121 -id:A173121 - cp's OEIS frontend

A058331 a(n) = 2*n^2 + 1.

1, 3, 9, 19, 33, 51, 73, 99, 129, 163, 201, 243, 289, 339, 393, 451, 513, 579, 649, 723, 801, 883, 969, 1059, 1153, 1251, 1353, 1459, 1569, 1683, 1801, 1923, 2049, 2179, 2313, 2451, 2593, 2739, 2889, 3043, 3201, 3363, 3529, 3699, 3873, 4051

Offset: 0

Erich Friedman, Dec 12 2000

Maximal number of regions in the plane that can be formed with n hyperbolas.
Also the number of different 2 X 2 determinants with integer entries from 0 to n.
Number of lattice points in an n-dimensional ball of radius sqrt(2). - David W. Wilson, May 03 2001
Equals A112295(unsigned) * [1, 2, 3, ...]. - Gary W. Adamson, Oct 07 2007
Binomial transform of A166926. - Gary W. Adamson, May 03 2008
a(n) = longest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (2n^2 + 1, 2n^2 + 2, 4n^2 + 1).
{a(k): 0 <= k < 3} = divisors of 9. - Reinhard Zumkeller, Jun 17 2009
Number of ways to partition a 3*n X 2 grid into 3 connected equal-area regions. - R. H. Hardin, Oct 31 2009
Let A be the Hessenberg matrix of order n defined by: A[1, j] = 1, A[i, i] := 2, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 3, a(n - 1) = coeff(charpoly(A, x), x^(n - 2)). - Milan Janjic, Jan 26 2010
Except for the first term of [A002522] and [A058331] if X = [A058331], Y = [A087113], A = [A002522], we have, for all other terms, Pell's equation: [A058331]^2 - [A002522]*[A087113]^2 = 1; (X^2 - A*Y^2 = 1); e.g., 3^2 -2*2^2 = 1; 9^2 - 5*4^2 = 1; 129^2 - 65*16^2 = 1, and so on. - Vincenzo Librandi, Aug 07 2010
Niven (1961) gives this formula as an example of a formula that does not contain all odd integers, in contrast to 2n + 1 and 2n - 1. - Alonso del Arte, Dec 05 2012
Numbers m such that 2*m-2 is a square. - Vincenzo Librandi, Apr 10 2015
Number of n-tuples from the set {1,0,-1} where at most two elements are nonzero. - Michael Somos, Oct 19 2022
a(n) gives the x-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The y-value is given by 2*n (see Tattersall). - Stefano Spezia, Jul 23 2025

			a(1) = 3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different determinants.
G.f. = 1 + 3*x + 9*x^2 + 19*x^3 + 33*x^4 + 51*x^5 + 73*x^6 + ... - _Michael Somos_, Oct 19 2022

Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Leo Tavares, Illustration: Triangular Outlines
Reinhard Zumkeller, Enumerations of Divisors.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124.
Second row of array A099597.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A112295.
Cf. A087113, A002552.
Cf. A005408, A016813, A086514, A000125, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.
Column 2 of array A188645.
Cf. A001105 and A247375. - Bruno Berselli, Sep 16 2014
Cf. A056106, A251599.
Cf. A000384, A000217, A166926.

a058331 = (+ 1) . a001105  -- Reinhard Zumkeller, Dec 13 2014

[2*n^2 + 1 : n in [0..100]]; // Wesley Ivan Hurt, Feb 02 2017

b[g_] := Length[Union[Map[Det, Flatten[ Table[{{i, j}, {k, l}}, {i, 0, g}, {j, 0, g}, {k, 0, g}, {l, 0, g}], 3]]]] Table[b[g], {g, 0, 20}]
2*Range[0, 49]^2 + 1 (* Alonso del Arte, Dec 05 2012 *)

a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011

G.f.: (1 + 3x^2)/(1 - x)^3. - Paul Barry, Apr 06 2003
a(n) = M^n * [1 1 1], leftmost term, where M = the 3 X 3 matrix [1 1 1 / 0 1 4 / 0 0 1]. a(0) = 1, a(1) = 3; a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). E.g., a(4) = 33 since M^4 *[1 1 1] = [33 17 1]. - Gary W. Adamson, Nov 11 2004
a(n) = cosh(2*arccosh(n)). - Artur Jasinski, Feb 10 2010
a(n) = 4*n + a(n-1) - 2 for n > 0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(n) = (((n-1)^2 + n^2))/2 + (n^2 + (n+1)^2)/2. - J. M. Bergot, May 31 2012
a(n) = A251599(3*n) for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = sqrt(8*(A000217(n-1)^2 + A000217(n)^2) + 1). - J. M. Bergot, Sep 03 2015
E.g.f.: (2*x^2 + 2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A002378(n) + A002061(n). - Bruce J. Nicholson, Aug 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(2))*coth(Pi/sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(2))*csch(Pi/sqrt(2)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(2))*sinh(Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(2))*csch(Pi/sqrt(2)). (End)
From Leo Tavares, May 23 2022: (Start)
a(n) = A000384(n+1) - 3*n.
a(n) = 3*A000217(n) + A000217(n-2). (End)
a(n) = a(-n) for all n in Z and A037235(n) = Sum_{k=0..n-1} a(k). - Michael Somos, Oct 19 2022

Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001

A132592 X-values of solutions to the equation X(X + 1) - 8Y^2 = 0.

Original entry on oeis.org

0, 8, 288, 9800, 332928, 11309768, 384199200, 13051463048, 443365544448, 15061377048200, 511643454094368, 17380816062160328, 590436102659356800, 20057446674355970888, 681362750825443653408, 23146276081390728245000, 786292024016459316676608, 26710782540478226038759688

Offset: 0

Mohamed Bouhamida, Nov 14 2007

Equivalently, numbers k such that both k/2 and k+1 are squares. - Karl-Heinz Hofmann, Sep 20 2022

Equivalently, numbers k such that the k-dimensional volume and total (k-1)-dimensional volume are equal, with side length being a positive integer, for all regular polyhedra constructible in k dimensions. - Matt Moir, Jul 09 2024

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Intersection between A132411 and A001105.
Cf. A007654.
Cf. A001541, A058331, A001079, A037270, A055792, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.

I:=[0,8,288]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2)+ Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 24 2018

Table[Round[N[Sinh[2 n ArcCosh[Sqrt[2]]]^2, 100]], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *)
LinearRecurrence[{35, -35, 1}, {0, 8, 288}, 30] (* Vincenzo Librandi, Dec 24 2018 *)

A132592 = [0, 8]
for n in range(2, 18): A132592.append(34 * A132592[-1] - A132592[-2] + 16)
print(A132592) # Karl-Heinz Hofmann, Sep 20 2022

a(0)=0, a(1)=8 and a(n) = 34*a(n-1) - a(n-2) + 16.
a(n) = (A056771(n) - 1)/2. - Max Alekseyev, Nov 13 2009
a(n) = sinh(2*n*arccosh(sqrt(2))^2) (n=0,1,2,3,...). - Artur Jasinski, Feb 10 2010
G.f.: -8*x*(x+1)/((x-1)*(x^2-34*x+1)). - Colin Barker, Oct 24 2012
a(n) = A055792(n+1)-1 = A001541(n)^2 - 1. - Antti Karttunen, Oct 03 2016

More terms from Max Alekseyev, Nov 13 2009

A173116 a(n) = sinh(2arcsinh(n))^2 = 4n^2*(n^2 + 1).

Original entry on oeis.org

0, 8, 80, 360, 1088, 2600, 5328, 9800, 16640, 26568, 40400, 59048, 83520, 114920, 154448, 203400, 263168, 335240, 421200, 522728, 641600, 779688, 938960, 1121480, 1329408, 1565000, 1830608, 2128680, 2461760, 2832488, 3243600

Offset: 0

Artur Jasinski, Feb 10 2010

			G.f. = 8*x + 80*x^2 + 360*x^3 + 1088*x^4 + 2600*x^5 + 5328*x^6 + 9800*x^7 + ... - _Michael Somos_, Jul 05 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173121.

[4*n^2*(n^2+1): n in [0..40]]; // Vincenzo Librandi, Jun 15 2011

Table[4*n^2*(n^2 + 1), {n, 0, 30}] (* OR *)
Table[Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}]
a[ n_] := TrigExpand @ Sinh[ 2 ArcSinh @ n]^2; (* Michael Somos, Jul 05 2018 *)

a(n)=4*n^2*(n^2+1) \\ Charles R Greathouse IV, Apr 17 2012

a(n)=8*binomial(n^2+1,2) \\ Charles R Greathouse IV, Apr 17 2012

a(n) = 4*A071253(n) = 8*A037270(n).
G.f.: 8*x*(1 + 5*x + 5*x^2 + x^3)/(1 - x)^5. - Colin Barker, Jan 08 2012
E.g.f.: 4*x*(2 + 8*x + 6*x^2 + x^3)*exp(x). - Michael Somos, Jul 05 2018
a(n) = a(-n) = (2*n)^2 + (2*n^2)^2 = (2*n^2 + 1)^2 - 1. - Michael Somos, Jul 05 2018
From Amiram Eldar, Oct 25 2024: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/24 + (1-Pi*coth(Pi))/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 + (Pi*cosech(Pi)-1)/8. (End)

Name corrected by Jianing Song, Nov 23 2018

A173129 a(n) = cosh(2 * n * arccosh(n)).

Original entry on oeis.org

1, 1, 97, 19601, 7380481, 4517251249, 4097989415521, 5170128475599457, 8661355881006882817, 18605234632923999244961, 49862414878754347585980001, 163104845048002042971670685041, 639582975902942936737758325440001

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..193
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A001079, A037270, A053120 (Chebyshev polynomial), A058331, A115066, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173148.

Cf. A349070, A349071, A349073.

seq(orthopoly[T](2*n,n), n=0..50); # Robert Israel, Dec 27 2018

Table[Round[Cosh[2 n ArcCosh[n]]], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *)
Round[Table[1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x), {x, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)
Table[ChebyshevT[2*n, n], {n, 0, 15}] (* Vaclav Kotesovec, Nov 07 2021 *)

{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2-1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018

{a(n) = polchebyshev(2*n, 1, n)} \\ Seiichi Manyama, Dec 28 2018

{a(n) = polchebyshev(n, 1, 2*n^2-1)} \\ Seiichi Manyama, Dec 29 2018

a(n) = (1/2)*((n+sqrt(n^2-1))^(2*n) + (n-sqrt(n^2-1))^(2*n)). - Artur Jasinski, Feb 14 2010, corrected by Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2-1)^(n-k)*n^(2*k). - Seiichi Manyama, Dec 27 2018
a(n) = T_{2n}(n) where T_{2n} is a Chebyshev polynomial of the first kind. - Robert Israel, Dec 27 2018
a(n) = T_{n}(2*n^2-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

A173127 a(n) = sinh((2n-1)*arcsinh(3)).

Original entry on oeis.org

-3, 3, 117, 4443, 168717, 6406803, 243289797, 9238605483, 350823718557, 13322062699683, 505887558869397, 19210405174337403, 729489509065951917, 27701390939331835443, 1051923366185543794917, 39945386524111332371403

Offset: 0

Artur Jasinski, Feb 10 2010

Numbers n such that ((n^2 + 1)/10) is a square. - Vincenzo Librandi, Jan 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (38,-1).

Cf. A058331 A001079, A037270, A071253, A108741, A132592, A146311-A146313, A173115, A173116, A173121.

[-3] cat [n: n in [0..10^7]|IsSquare((n^2+1)/10)]; // Vincenzo Librandi, Jan 02 2012

LinearRecurrence[{38,-1},{-3,3},30] (* Harvey P. Dale, Jan 14 2015 *)

from itertools import islice
def A173127_gen(): # generator of terms
    x, y = -30, 10
    while True:
        yield x//10
        x, y = x*19+y*60, x*6+y*19
A173127_list = list(islice(A173127_gen(),20)) # Chai Wah Wu, Apr 24 2025

a(n) = (1/2)*((-3+sqrt(10))*(19+6*sqrt(10))^n + (-3-sqrt(10))*(19-6*sqrt(10))^n).
a(n) = -a(-n+1).
G.f.: -3*(1-39*x)/(1-38*x+x^2). - Bruno Berselli, Jan 03 2011
E.g.f.: exp(19*x)*(-3*cosh(6*sqrt(10)*x) + sqrt(10)*sinh(6*sqrt(10)*x)). - Stefano Spezia, Apr 24 2025

A173128 a(n) = cosh(2narcsinh(n)).

Original entry on oeis.org

1, 3, 161, 27379, 9478657, 5517751251, 4841332221601, 5964153172084899, 9814664424981012481, 20791777842234580902499, 55106605639755476546020001, 178627672869645203363556318483, 695165908550906808156689590141441

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Robert Israel, Table of n, a(n) for n = 0..192
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173129, A173174.

seq(expand( (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n))), n=0..30); # Robert Israel, Apr 05 2016

Round[Table[Cosh[2 n ArcSinh[n]], {n, 0, 20}]] (* Artur Jasinski *)
Round[Table[1/2 (x - Sqrt[1 + x^2])^(2 x) + 1/2 (x + Sqrt[1 + x^2])^(2 x), {x, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *)

{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2+1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018

{a(n) = polchebyshev(n, 1, 2*n^2+1)} \\ Seiichi Manyama, Dec 29 2018

a(n) = (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n)). - Artur Jasinski, Feb 14 2010, corrected by Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2+1)^(n-k)*n^(2*k). - Seiichi Manyama, Dec 27 2018
a(n) = T_{n}(2*n^2+1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

A173130 a(n) = Cosh[(2 n - 1) ArcCosh[n]].

Original entry on oeis.org

0, 1, 26, 3363, 937444, 456335045, 343904160606, 371198523608647, 543466014742175624, 1036834190110356583689, 2499384905955651114739810, 7429238104512325157021090411, 26695718139185294187938997247212

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128.

Table[Round[Cosh[(2 n - 1) ArcCosh[n]]], {n, 0, 20}] (* Artur Jasinski *)

a(n) ~ 2^(2*n-2) * n^(2*n-1). - Vaclav Kotesovec, Apr 05 2016

A173131 a(n) = (Cosh[(2n-1)ArcSinh[n]])^2.

Original entry on oeis.org

1, 2, 1445, 19740250, 1361599599377, 298514762397852026, 160545187370375075046277, 179656719395983409634002348450, 373368546362937441101158606899394625

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130.

Table[Round[Cosh[(2 n - 1) ArcSinh[n]]^2], {n, 0, 10}] (* Artur Jasinski *)

a(n) ~ 2^(4*n-4) * n^(4*n-2). - Vaclav Kotesovec, Apr 05 2016

A173148 a(n) = cos(2narccos(sqrt(n))).

Original entry on oeis.org

1, 1, 17, 485, 18817, 930249, 55989361, 3974443213, 325142092801, 30122754096401, 3117419602578001, 356452534779818421, 44627167107085622401, 6071840759403431812825, 892064955046043465408177, 140751338790698080509966749, 23737154316161495960243527681

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

The Chebyshev polynomial T_n is defined by cos(nx) = T_n(cos(x)). So T_2n(cos(x)) = cos(2nx) = cos^2(nx) - 1 = (T_n(x))^2 - 1 consists of only even powers of x. As a result, a(n) = T_2n(sqrt(n)) is an integer. - Michael B. Porter, Jan 01 2019

Seiichi Manyama, Table of n, a(n) for n = 0..321
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A053120 (Chebyshev polynomial), A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A322790.

a:=List([0..20],n->Sum([0..n],k->Binomial(2*n,2*k)*(n-1)^(n-k)*n^k));; Print(a); # Muniru A Asiru, Jan 03 2019

[&+[Binomial(2*n,2*k)*(n-1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jan 03 2019

Table[Round[Cos[2 n ArcCos[Sqrt[n]]]], {n, 0, 30}] (* Artur Jasinski, Feb 11 2010 *)

{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n-1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 27 2018

{a(n) = round(cosh(2*n*acosh(sqrt(n))))} \\ Seiichi Manyama, Dec 27 2018

{a(n) = polchebyshev(n, 1, 2*n-1)} \\ Seiichi Manyama, Dec 29 2018

a(n) ~ exp(-1/2) * 2^(2*n-1) * n^n. - Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n-1)^(n-k)*n^k. - Seiichi Manyama, Dec 27 2018
a(n) = cosh(2*n*arccosh(sqrt(n))). - Seiichi Manyama, Dec 27 2018
a(n) = T_{2*n}(sqrt(n)) = T_{n}(2*n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018
a(n) = A322790(n-1, n) for n > 0. - Seiichi Manyama, Dec 29 2018

Minor edits by Vaclav Kotesovec, Apr 05 2016

A173133 a(n) = Sinh[(2n-1) ArcSinh[n]].

Original entry on oeis.org

0, 1, 38, 4443, 1166876, 546365045, 400680904674, 423859315570607, 611038907405197432, 1151555487914640463209, 2748476184146759127540190, 8102732939160371170806346243, 28915133156938367486730067779348

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131.

Table[Round[Sinh[(2 n - 1) ArcSinh[n]]], {n, 0, 20}] (* Artur Jasinski *)
Round[Table[1/2 (n - Sqrt[1 + n^2])^(2 n - 1) + 1/2 (n + Sqrt[1 + n^2])^(2 n - 1), {n, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)

a(n) = 1/2 (n - sqrt(1 + n^2))^(2 n - 1) + 1/2 (n + sqrt(1 + n^2))^(2 n - 1). - Artur Jasinski, Feb 14 2010

Minor edits by Vaclav Kotesovec, Apr 05 2016

cp's OEIS Frontend

A058331 a(n) = 2*n^2 + 1.

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A132592 X-values of solutions to the equation X*(X + 1) - 8*Y^2 = 0.

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A173116 a(n) = sinh(2*arcsinh(n))^2 = 4*n^2*(n^2 + 1).

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A173129 a(n) = cosh(2 * n * arccosh(n)).

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A173127 a(n) = sinh((2n-1)*arcsinh(3)).

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A173128 a(n) = cosh(2*n*arcsinh(n)).

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Mathematica

PARI

PARI

A132592 X-values of solutions to the equation X(X + 1) - 8Y^2 = 0.

A173116 a(n) = sinh(2arcsinh(n))^2 = 4n^2*(n^2 + 1).

A173128 a(n) = cosh(2narcsinh(n)).

A173148 a(n) = cos(2narccos(sqrt(n))).