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

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

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(PARI) {a(n) = 32*n^3+48*n^2+18*n+1}
(PARI) {a(n) = polchebyshev(3, 1, 2*n+1)}
(PARI) Vec((1 + x)*(1 + 94*x + x^2) / (1 - x)^4 + O(x^40)) \\ Colin Barker, Dec 27 2018
(GAP) a:=List([0..40], n->32*n^3+48*n^2+18*n+1);; Print(a); # Muniru A Asiru, Jan 02 2019

Crossrefs

Column 3 of A322790.

Cf. A144129.

Sequence in context: A316122 A157659 A273187 * A212779 A321636 A154359

Adjacent sequences: A322827 A322828 A322829 * A322831 A322832 A322833

Keyword

nonn,easy

Author

Seiichi Manyama, Dec 27 2018

Status

approved