A144129 -id:A144129 - cp's OEIS frontend

A243133 64n^7 - 112n^5 + 56n^3 - 7n.

0, 1, 5042, 114243, 937444, 4656965, 17057046, 50843527, 130576328, 299537289, 628855930, 1229215691, 2265463212, 3974443213, 6686381534, 10850138895, 17062657936, 26102926097, 38970776898, 56930852179, 81562047860, 114812765781, 159062294182

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(7,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A056220, A144129, A144130.

[64*n^7-112*n^5+56*n^3-7*n: n in [0..40]];

Table[ChebyshevT[7, n], {n, 0, 40}] (* or *)  Table[64 n^7 - 112 n^5 + 56 n^3 - 7 n, {n, 0, 40}]
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,5042,114243,937444,4656965,17057046,50843527},40] (* Harvey P. Dale, Mar 27 2015 *)

G.f.: (x + 5034*x^2 + 73935*x^3 + 164620*x^4 + 73935*x^5 + 5034*x^6 + x^7)/(1 - x)^8.
a(n) = n*(64*n^6 - 112*n^4 + 56*n^2 - 7).
a(0)=0, a(1)=1, a(2)=5042, a(3)=114243, a(4)=937444, a(5)=4656965, a(6)=17057046, a(7)=50843527, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)- 70*a(n-4)+ 56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8). - Harvey P. Dale, Mar 27 2015

A243135 256n^9 - 576n^7 + 432n^5 - 120n^3 + 9*n.

Original entry on oeis.org

0, 1, 70226, 3880899, 58106404, 456335045, 2421980406, 9863382151, 33165873224, 96450076809, 250283080090, 592479412811, 1300371936876, 2678768828749, 5228741809214, 9743412645135, 17438019715216, 30122754096401, 50428155189474, 82094249361619

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(9,n).

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

Cf. A056220, A144129, A144130.

[256*n^9-576*n^7+432*n^5-120*n^3+9*n: n in [0..20]];

a:= n-> simplify(ChebyshevT(9, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 31 2014

Table[ChebyshevT[9, n], {n, 0, 20}] (* or *) Table[256 n^9 - 576 n^7 + 432 n^5 - 120 n^3 + 9 n, {n, 0, 20}]

G.f.: x *(1 + 70216*x + 3178684*x^2 + 22457464*x^3 + 41484550*x^4 + 22457464*x^5 + 3178684*x^6 + 70216*x^7 + x^8)/(1 - x)^10.

a(n) = n*(4*n^2 - 3)*(64*n^6 - 96*n^4 + 36*n^2 - 3).

A322830 a(n) = 32n^3 + 48n^2 + 18*n + 1.

Original entry on oeis.org

1, 99, 485, 1351, 2889, 5291, 8749, 13455, 19601, 27379, 36981, 48599, 62425, 78651, 97469, 119071, 143649, 171395, 202501, 237159, 275561, 317899, 364365, 415151, 470449, 530451, 595349, 665335, 740601, 821339, 907741, 999999, 1098305, 1202851, 1313829, 1431431, 1555849

Offset: 0

Seiichi Manyama, Dec 27 2018

			(sqrt(2) + sqrt(1))^6 = 99 + 70*sqrt(2) = 99 + sqrt(99^2 - 1). So a(1) = 99.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Column 3 of A322790.

Cf. A144129.

a:=List([0..40],n->32*n^3+48*n^2+18*n+1);; Print(a); # Muniru A Asiru, Jan 02 2019

[32*n^3+48*n^2+18*n+1$n=0..40]; # Muniru A Asiru, Jan 02 2019

CoefficientList[Series[(1 + x) (1 + 94 x + x^2)/(1 - x)^4, {x, 0, 36}], x] (* or *)
Array[ChebyshevT[3, 2 # + 1] &, 37, 0] (* Michael De Vlieger, Jan 01 2019 *)
Table[32n^3+48n^2+18n+1,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,99,485,1351},40] (* Harvey P. Dale, Mar 11 2019 *)

PARI
```
{a(n) = 32*n^3+48*n^2+18*n+1}
```
PARI
```
{a(n) = polchebyshev(3, 1, 2*n+1)}
```

Vec((1 + x)*(1 + 94*x + x^2) / (1 - x)^4 + O(x^40)) \\ Colin Barker, Dec 27 2018

a(n) + sqrt(a(n)^2 - 1) = (sqrt(n+1) + sqrt(n))^6.
a(n) - sqrt(a(n)^2 - 1) = (sqrt(n+1) - sqrt(n))^6.
a(n) = A144129(2*n+1) = T_3(2*n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind.
From Colin Barker, Dec 27 2018: (Start)
G.f.: (1 + x)*(1 + 94*x + x^2) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
a(n) = (2*n + 1)*(16*n^2 + 16*n + 1). - Bruno Berselli, Jan 02 2019

A369922 a(n) = 8n^3 - 6n - 1.

Original entry on oeis.org

-1, 1, 51, 197, 487, 969, 1691, 2701, 4047, 5777, 7939, 10581, 13751, 17497, 21867, 26909, 32671, 39201, 46547, 54757, 63879, 73961, 85051, 97197, 110447, 124849, 140451, 157301, 175447, 194937, 215819, 238141, 261951, 287297, 314227, 342789, 373031, 405001

Offset: 0

Darío Clavijo, Feb 05 2024

This polynomial (evaluated over the rationals) arises in demonstrating the impossibility of trisecting an arbitrary angle by compass and straightedge.

Proofwiki, Trisecting the Angle by Compass and Straightedge Construction is Impossible.
Wikipedia, Angle trisection.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A144129.

Table[8n^3-6n-1,{n,0,37}] (* James C. McMahon, Feb 05 2024 *)

a(n) = 2*A144129(n) - 1.
From Elmo R. Oliveira, Sep 04 2025: (Start)
G.f.: (-1 + 5*x + 41*x^2 + 3*x^3)/(-1+x)^4.
E.g.f.: (-1 + 2*x + 24*x^2 + 8*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

cp's OEIS Frontend

A243133 64*n^7 - 112*n^5 + 56*n^3 - 7*n.

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A243135 256*n^9 - 576*n^7 + 432*n^5 - 120*n^3 + 9*n.

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A322830 a(n) = 32*n^3 + 48*n^2 + 18*n + 1.

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PARI

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A369922 a(n) = 8*n^3 - 6*n - 1.

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Mathematica

Formula

A243133 64n^7 - 112n^5 + 56n^3 - 7n.

A243135 256n^9 - 576n^7 + 432n^5 - 120n^3 + 9*n.

A322830 a(n) = 32n^3 + 48n^2 + 18*n + 1.

A369922 a(n) = 8n^3 - 6n - 1.