A146311 -id:A146311 - cp's OEIS frontend

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

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

A146535 Numerator of (2*n-1)/3.

Original entry on oeis.org

1, 1, 5, 7, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9, 29, 31, 11, 35, 37, 13, 41, 43, 15, 47, 49, 17, 53, 55, 19, 59, 61, 21, 65, 67, 23, 71, 73, 25, 77, 79, 27, 83, 85, 29, 89, 91, 31, 95, 97, 33, 101, 103, 35, 107, 109, 37, 113, 115, 39, 119, 121, 41, 125, 127, 43, 131, 133, 45

Offset: 1

Artur Jasinski, Oct 31 2008

From Jaroslav Krizek, May 28 2010: (Start)
a(n+1) = numerators of antiharmonic mean of the first n positive integers for n >= 1.
See A169609(n-1) - denominators of antiharmonic mean of the first n positive integers for n >= 1. (End)

			Fractions begin with 1/6, 1/2, 5/6, 7/6, 3/2, 11/6, 13/6, 5/2, 17/6, 19/6, 7/2, 23/6, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A146306, A146307, A146308, A146309, A146310, A146311, A146312, A146313.
Cf. A061347, A102283.
Trisections: A016921, A005408, A016969. [R. J. Mathar, Nov 21 2008]

Table[Numerator[(2 n - 1)/6], {n, 1, 100}]
LinearRecurrence[{0,0,2,0,0,-1},{1,1,5,7,3,11},100] (* Harvey P. Dale, Feb 24 2015 *)

a(n) = numerator((2*n-1)/3); \\ Altug Alkan, Apr 13 2018

From R. J. Mathar, Nov 21 2008: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x(1+x)(1+5x^2+x^4)/((1-x)^2*(1+x+x^2)^2). (End)
Sum_{k=1..n} a(k) ~ (7/9) * n^2. - Amiram Eldar, Apr 04 2024
a(n) = (2*n - 1)*(7 - A061347(n) +3*A102283(n))/9. - Stefano Spezia, Feb 14 2025

Name edited by Altug Alkan, Apr 13 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

A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

Original entry on oeis.org

-1, 0, 675, 11309768, 878801253135, 208241673295152024, 118270071682117442287235, 137788343929239264227213170608, 295355309179742652677310128859789375

Offset: 0

Artur Jasinski, Feb 10 2010

sign

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133.

Table[Round[Sinh[(2 n - 1) ArcCosh[n]]^2], {n, 0, 20}]

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

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

Original entry on oeis.org

1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623, 1279323224145592925115025, 130478425272533383961791927, 13307520054574259571177661529

Offset: 0

Wolfdieter Lang, Aug 31 2004

a(-1) = -1. - Artur Jasinski, Feb 10 2010

5*a(n) gives the x-values in the solution to the Pell equation x^2 - 26*y^2 = -1. - Colin Barker, Aug 24 2013

			(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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 (102,-1).

Cf. A097725 for S(n, 102).
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121. - Artur Jasinski, Feb 10 2010
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* Artur Jasinski, Feb 10 2010 *)
CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)
LinearRecurrence[{102,-1},{1,103},20] (* Harvey P. Dale, Aug 20 2017 *)

x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 102*x + x^2).
a(n) = S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 5*i)/(5*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=103. - Philippe Deléham, Nov 18 2008
a(n) = (1/5)*sinh((2*n-1)*arcsinh(5)), n >= 1. - Artur Jasinski, Feb 10 2010

More terms from Harvey P. Dale, Aug 20 2017

A173170 a(n) = sin^2((2n-1)arcsin(sqrt n)) = 1 - sin^2( (2n-1)arccos(sqrt n)).

Original entry on oeis.org

0, 1, 50, 23763, 25421764, 48225038405, 142786923879606, 608447515452613207, 3527836867501829594888, 26710782540478226038759689, 255922222218837615280903143610, 3026917140685147530327256796600411

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..170

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151.

Table[Round[Sin[(2 n - 1) ArcSin[Sqrt[n]]]^2], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010 *)

a(n) ~ exp(-1) * 2^(4*n-4) * n^(2*n-1). - Vaclav Kotesovec, Apr 05 2016

Minor edits by Vaclav Kotesovec, Apr 05 2016

A173174 a(n) = cosh(2narcsinh(sqrt(n))).

Original entry on oeis.org

1, 3, 49, 1351, 51841, 2550251, 153090001, 10850138895, 886731088897, 82094249361619, 8491781781142001, 970614726270742103, 121485428812828080001, 16525390478051500325307, 2427469037137019032095121, 382956978214541873571486751, 64576903826545426454350012417, 11591229031806966336496244914595

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

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

Cf. A132592, A146311 - A146313, A173115, A173116 A173121, A173127 - A173131, A173133, A173134, A173148, A173151, A173170, A173171.
Cf. A322746.
Main diagonal of A322790.

[&+[Binomial(2*n, 2*k)*(n+1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 29 2018

A173174 := proc(n) cosh(2*n*arcsinh(sqrt(n))) ; expand(%) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011

Table[Round[N[Cosh[(2 n) ArcSinh[Sqrt[n]]], 100]], {n, 0, 30}] (* Artur Jasinski *)
Join[{1}, a[n_]:=Sum[Binomial[2 n, 2 k] (n + 1)^(n - k) n^k, {k, 0, n}]; Array[a, 25]] (* Vincenzo Librandi, Dec 29 2018 *)

{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n+1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 26 2018

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

a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k. - Seiichi Manyama, Dec 26 2018

a(n) = T_{n}(2*n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

More terms from Seiichi Manyama, Dec 26 2018

cp's OEIS Frontend

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

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A146535 Numerator of (2*n-1)/3.

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

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

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A173148 a(n) = cos(2*n*arccos(sqrt(n))).

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

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A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

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A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

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

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

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

A173170 a(n) = sin^2((2n-1)arcsin(sqrt n)) = 1 - sin^2( (2n-1)arccos(sqrt n)).

A173174 a(n) = cosh(2narcsinh(sqrt(n))).