A173121 -id:A173121 - cp's OEIS frontend

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

-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

A173171 a(n) = - sin^2((2n-1)arccos(sqrt n)) = sin^2((2n-1)arcsin(sqrt n)) - 1.

Original entry on oeis.org

-1, 0, 49, 23762, 25421763, 48225038404, 142786923879605, 608447515452613206, 3527836867501829594887, 26710782540478226038759688, 255922222218837615280903143609, 3026917140685147530327256796600410

Offset: 0

Artur Jasinski, Feb 11 2010

sign

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, A173170.

Table[Round[-N[ Sin[(2 n - 1) ArcCos[Sqrt[n]]]^2, 100]], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010; Typo fixed by Vincenzo Librandi, Jun 29 2014 *)

A211412 a(n) = 4*n^4 + 1.

Original entry on oeis.org

5, 65, 325, 1025, 2501, 5185, 9605, 16385, 26245, 40001, 58565, 82945, 114245, 153665, 202501, 262145, 334085, 419905, 521285, 640001, 777925, 937025, 1119365, 1327105, 1562501, 1827905, 2125765, 2458625, 2829125, 3240001, 3694085, 4194305, 4743685, 5345345, 6002501, 6718465, 7496645

Offset: 1

Alonso del Arte, Feb 10 2013

Except for the first term, all terms are composite. a(n) is divisible by 5 if n is not.
Long before Aurifeuille, Euler discovered that 4n^4 + 1 = (2n^2 + 2n + 1)*(2n^2 - 2n + 1). For example, 325 = 4 * 3^4 + 1 = (2 * 3^2 + 2 * 3 + 1)*(2 * 3^2 - 2 * 3 + 1) = 25 * 13. Euler shared this discovery with Goldbach in a letter dated August 28, 1742. [Euler identity corrected by Graham Holmes, Jun 02 2023]
The terms of the sequence are the arithmetic mean of eight numbers located on concentric circles (see Avilov link). - Nicolay Avilov, Jan 22 2021

Don Knuth, The Art of Computer Programming: Seminumerical Algorithms, 3rd ed., New York: Addison-Wesley Professional (1997), p. 392.
David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005), p. 15.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nicolay Avilov, Drawing with circles
P. H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, Saint-Pétersbourg, 1843, p. 145; alternative link. See in particular Lettre XLVI (Euler to Goldbach), Aug 28 1742
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A207262 (subset).
After the first term, subsequence of A121944.
Cf. A053755.

[4*n^4 + 1: n in [1..50]]; // Vincenzo Librandi, Jun 11 2017

4 Range[44]^4 + 1
Table[4 n^4 + 1, {n, 50}] (* Vincenzo Librandi, Jun 11 2017 *)
LinearRecurrence[{5,-10,10,-5,1},{5,65,325,1025,2501},40] (* Harvey P. Dale, Sep 15 2023 *)

a(n) = 4*n^4+1 \\ Felix Fröhlich, Jun 07 2017

G.f.: -x*(x^4+50*x^2+40*x+5) / (x-1)^5. - Colin Barker, Feb 11 2013
a(n) = A053755(n^2). - Michel Marcus, Sep 18 2015
a(n) = (2*n^2)^2 + 1^2 = (2*n^2-1)^2 + (2*n)^2. - Thomas Ordowski, Sep 18 2015
a(n) = A001844(n) * A001844(n+1) = A141046(n) + 1 = (A000583(n) * 4 ) + 1 = A016742(n) + A173121(n) + 1. - Bruce J. Nicholson, Jun 06 2017
From Amiram Eldar, Jul 26 2022: (Start)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/4 - 1/2.
Sum_{n>=1} (-1)^n/a(n) = 1/2 - sech(Pi/2)*Pi/4. (End)

A218486 Positive numbers differing from next 2 greater squares by squares.

Original entry on oeis.org

48, 96, 160, 240, 288, 336, 448, 480, 576, 720, 960, 1008, 1344, 1440, 1728, 2016, 2160, 2400, 2640, 2688, 3168, 3360, 3456, 3744, 4320, 4368, 4480, 5040, 5280, 5760, 6336, 6720, 7200, 7488, 8640, 8736, 8800, 9408, 10080, 10560, 10800, 11520, 12096, 12480

Offset: 1

Michel Marcus, Oct 30 2012

nonn

All terms are even. The sequence is infinite. E.g., positive terms of A173121 {48, 288, 960, 2400, 5040, 9408, 16128, 25920, 39600,...} is infinite subsequence of A218486. - Zak Seidov, Nov 26 2013

Another infinite subsequence is {96, 480, 1440, 3360, 6720, 12096, 20160, ...} = 96 *binomial(m,4) = 96*(positive terms in A000332). - Zak Seidov, Nov 26 2013

			48 = 7^2 - 1^2 = 8^2 - 4^2.

Zak Seidov, Table of n, a(n) for n = 1..1000
E. J. Barbeau, Numbers differing from consecutive squares by squares, Canad. Math. Bull. 28(1985), pp. 337-342.

Cf. A173121, A218485, A218487, A218488.

sq2(n) = {for (i=1, n, a = sqrtint(i) + 1; if (issquare(a^2-i) && issquare((a+1)^2-i), print1(i, ", ")););}

A173175 a(n) = sinh^2( 2n*arcsinh(sqrt n)).

Original entry on oeis.org

0, 8, 2400, 1825200, 2687489280, 6503780163000, 23436548406180000, 117725514040791821024, 786292024016459316676608, 6739465778247681589030301160, 72110357818535214970387726284000, 942092946853627620313318842336862608, 14758709413836719039368938494112056160000

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..175
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A322699.

A173175 := proc(n) sinh(2*n*arcsinh(sqrt(n))) ; %^2 ; expand(%); simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011

Table[Round[N[Sinh[(2 n) ArcSinh[Sqrt[n]]]^2, 100]], {n, 0, 20}]

{a(n) = (polchebyshev(2*n, 1, 2*n+1)-1)/2} \\ Seiichi Manyama, Jan 02 2019

{a(n) = 1/2*(-1+sum(k=0, 2*n, binomial(4*n, 2*k)*(n+1)^(2*n-k)*n^k))} \\ Seiichi Manyama, Jan 02 2019

From Seiichi Manyama, Jan 02 2019: (Start)
a(n) = A322699(n,2*n).
a(n) = (T_{2*n}(2*n+1) - 1)/2 where T_{n}(x) is a Chebyshev polynomial of the first kind.
a(n) = 1/2 * (-1 + Sum_{k=0..2*n} binomial(4*n,2*k)*(n+1)^(2*n-k)*n^k). (End)
a(n) ~ exp(1) * 2^(4*n - 2) * n^(2*n). - Vaclav Kotesovec, Jan 02 2019

a(11)-a(12) from Seiichi Manyama, Jan 02 2019

A173194 a(n) = -sin^2 (2narccos n) = - sin^2 (2narcsin n).

Original entry on oeis.org

0, 0, 9408, 384199200, 54471499791360, 20405558846592060000, 16793517249722147195701440, 26730228454204365035835498694848, 75019085697452515216001640927169855488, 346154755746154620929434271983392498083891520

Offset: 0

Artur Jasinski, Feb 12 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..107
Wikipedia, Chebyshev polynomials.

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A173174, A173175, A173176.

A173194 := proc(n) ((n+sqrt(n^2-1))^(2*n)-(n-sqrt(n^2-1))^(2*n))^2 ; expand(%/4) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011

Round[Table[ -N[Sin[2 n ArcSin[n]], 100]^2, {n, 0, 15}]] (* Artur Jasinski *)
Table[FullSimplify[(-1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x))^2], {x, 0, 7}] (* Artur Jasinski, Feb 17 2010 *)
Table[(n^2-1)*ChebyshevU[2*n-1, n]^2, {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)

{a(n) = (n^2-1)*n^2*(sum(k=0, n-1, binomial(2*n, 2*k+1)*(n^2-1)^(n-1-k)*n^(2*k)))^2} \\ Seiichi Manyama, Jan 05 2019

{a(n) = (n^2-1)*polchebyshev(2*n-1, 2, n)^2} \\ Seiichi Manyama, Jan 05 2019

4*a(n) = ( (n+sqrt(n^2-1))^(2*n) - (n-sqrt(n^2-1))^(2*n) )^2. - Artur Jasinski, Feb 17 2010
From Seiichi Manyama, Jan 05 2019: (Start)
a(n) = (n^2-1) * n^2 * (Sum_{k=0..n-1} binomial(2*n,2*k+1)*(n^2-1)^(n-1-k)*n^(2*k))^2.
For n > 0, a(n) = (n^2-1) * U_{2*n-1}(n)^2 where U_{n}(x) is a Chebyshev polynomial of the second kind. (End)
a(n) ~ 2^(4*n - 2) * n^(4*n). - Vaclav Kotesovec, Jan 05 2019

a(9) from Seiichi Manyama, Jan 05 2019

A173150 a(n) = sinh^2 (2n*arccosh(sqrt n)).

Original entry on oeis.org

0, 0, 288, 235224, 354079488, 865363202000, 3134808545188320, 15796198853361763368, 105717380511014096025600, 907380314352243226001152800, 9718304978537581699085289156000

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

Also a(n) = -sin(2n*arccos(sqrt(n)))^2 = -sin(2n*arcsin(sqrt(n)))^2.

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.

A173150 := proc(n) sinh(2*n*arccosh(sqrt(n))) ; %^2 ; expand(%) ; simplify(%) ;end proc: # R. J. Mathar, Feb 26 2011

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

Minor edits by Vaclav Kotesovec, Apr 05 2016

cp's OEIS Frontend

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

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A173174 a(n) = cosh(2*n*arcsinh(sqrt(n))).

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A173171 a(n) = - sin^2((2n-1)*arccos(sqrt n)) = sin^2((2n-1)*arcsin(sqrt n)) - 1.

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A211412 a(n) = 4*n^4 + 1.

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A218486 Positive numbers differing from next 2 greater squares by squares.

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A173175 a(n) = sinh^2( 2n*arcsinh(sqrt n)).

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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))).

A173171 a(n) = - sin^2((2n-1)arccos(sqrt n)) = sin^2((2n-1)arcsin(sqrt n)) - 1.

A173194 a(n) = -sin^2 (2narccos n) = - sin^2 (2narcsin n).