A239610 -id:A239610 - cp's OEIS frontend

A239607 a(n) = (1-2*n^2)^2.

1, 1, 49, 289, 961, 2401, 5041, 9409, 16129, 25921, 39601, 58081, 82369, 113569, 152881, 201601, 261121, 332929, 418609, 519841, 638401, 776161, 935089, 1117249, 1324801, 1560001, 1825201, 2122849, 2455489, 2825761, 3236401, 3690241, 4190209, 4739329

Offset: 0

José María Grau Ribas, Mar 22 2014

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

Cf. A056220, A239608, A239609, A239610.

Mathematica
```
Table[(1-2*n^2)^2 , {n, 0, 43}]
```

vector(100, n, round(sin(asin(n-1) - acos(n-1))^2)) \\ Colin Barker, May 24 2014

a(n)=(1-2*n^2)^2 \\ Charles R Greathouse IV, Jun 04 2014

From Colin Barker, May 24 2014: (Start)
a(n) = sin(arcsin(n) - arccos(n))^2.
G.f.: -(x^4+44*x^3+54*x^2-4*x+1) / (x-1)^5. (End)
a(n) = A056220(n)^2. - Michel Marcus, May 27 2014
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi^2*cosec(Pi/sqrt(2))^2/8 + (Pi/(4*sqrt(2))*cot(Pi/sqrt(2))) + 1/2.
Sum_{n>=0} (-1)^n/a(n) = Pi^2*cosec(Pi/sqrt(2))*cot(Pi/sqrt(2))/8 + (Pi/(4*sqrt(2)))*cosec(Pi/sqrt(2)) + 1/2. (End)
E.g.f.: exp(x)*(1 + 24*x^2 + 24*x^3 + 4*x^4). - Stefano Spezia, Feb 22 2025

A239608 Sin( arcsin(n)- 2*arccos(n) )^2.

Original entry on oeis.org

0, 1, 676, 9801, 59536, 235225, 715716, 1825201, 4096576, 8346321, 15760900, 27994681, 47279376, 76545001, 119552356, 181037025, 266864896, 384199201, 541679076, 749609641, 1020163600, 1367594361, 1808460676, 2361862801, 3049690176, 3896880625, 4931691076

Offset: 0

José María Grau Ribas, Mar 22 2014

nonn

The terms are integers.

This is assuming the "standard branch" of arcsin and arccos, where sin(arccos(n)) = cos(arcsin(n)) = sqrt(1-n^2). - Robert Israel, May 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A239607, A239609, A239610.

[n^2*(3-4*n^2)^2 : n in [0..50]]; // Vincenzo Librandi, May 30 2014

G[n_, a_, b_] := G[n, a, b] = Sin[a ArcSin[ n] + b ArcCos[n]]^2 // ComplexExpand // FullSimplify; Table[G[n, 1, -2], {n, 0, 43}]
CoefficientList[Series[- x (x + 1) (x^4 + 668 x^3 + 4422 x^2 + 668 x + 1)/(x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2014 *)
Table[n^2*(3-4*n^2)^2,{n,0,30}] (* Harvey P. Dale, Aug 05 2016 *)

vector(100, n, round(sin(asin(n-1) - 2*acos(n-1))^2)) \\ Colin Barker, May 24 2014

a(n) = n^2*(3-4*n^2)^2. G.f.: -x*(x+1)*(x^4+668*x^3+4422*x^2+668*x+1) / (x-1)^7. - Colin Barker, May 24 2014

a(n) = A144129(n)^2. - Robert Israel, May 25 2014

A239609 Sin(arcsin(n)- 3 arccos(n))^2.

Original entry on oeis.org

1, 1, 9409, 332929, 3690241, 23049601, 101626561, 354079489, 1040514049, 2687489281, 6272798401, 13493377921, 27138279169, 51591216769, 93489789121, 162571046401, 272735662081, 443365544449, 700932305089, 1080936581761, 1630220793601, 2409700487041

Offset: 0

José María Grau Ribas, Mar 22 2014

The terms are integers.

This is assuming the "standard branch" of arcsin and arccos, so that sin(arccos(n)) = cos(arcsin(n)) = sqrt(1-n^2). - Robert Israel, May 25 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A239607, A239608, A239610.

G[n_, a_, b_] := G[n, a, b] = Sin[a ArcSin[ n] + b ArcCos[n]]^2 // ComplexExpand // FullSimplify; Table[G[n, 1, -3], {n, 0, 43}]

vector(100, n, round(sin(asin(n-1) - 3*acos(n-1))^2)) \\ Colin Barker, May 24 2014

G.f.: -(x^8 +9400*x^7 +248284*x^6 +1032520*x^5 +1032646*x^4 +248200*x^3 +9436*x^2 -8*x +1) / (x -1)^9. - Colin Barker, May 24 2014

a(n) = A144130(n)^2. - Robert Israel, May 25 2014

cp's OEIS Frontend

A239607 a(n) = (1-2*n^2)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A239608 Sin( arcsin(n)- 2*arccos(n) )^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A239609 Sin(arcsin(n)- 3 arccos(n))^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula