A157081 a(n) = 32805000*n^2 + 16200*n + 1.

32821201, 131252401, 295293601, 524944801, 820206001, 1181077201, 1607558401, 2099649601, 2657350801, 3280662001, 3969583201, 4724114401, 5544255601, 6430006801, 7381368001, 8398339201, 9480920401, 10629111601, 11842912801

Offset

1,1

Comments

The identity (32805000*n^2 + 16200*n + 1)^2 - (2025*n^2 + n)*(729000*n + 180)^2 = 1 can be written as a(n)^2 - A156856(n)*A156868(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: x*(32821201 + 32788798*x + x^2)/(1-x)^3.

E.g.f.: -1 + (1 + 32821200*x + 32805000*x^2)*exp(x). - G. C. Greubel, Jan 27 2022

Mathematica

LinearRecurrence[{3, -3, 1}, {32821201, 131252401, 295293601}, 40]

Prog

(Magma) I:=[32821201, 131252401, 295293601]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
(PARI) a(n)=32805000*n^2+16200*n+1 \\ Charles R Greathouse IV, Dec 23 2011
(Sage) [16200*n*(2025*n + 1) + 1 for n in (1..30)] # G. C. Greubel, Jan 27 2022

Crossrefs

Cf. A156856, A156868, A157078, A157079, A157080.

Sequence in context: A290264 A206043 A157080 * A015354 A256356 A338607

Adjacent sequences: A157078 A157079 A157080 * A157082 A157083 A157084

Keyword

nonn,easy

Author

Vincenzo Librandi, Feb 22 2009

Status

approved