A156640 a(n) = 169*n^2 + 140*n + 29.

29, 338, 985, 1970, 3293, 4954, 6953, 9290, 11965, 14978, 18329, 22018, 26045, 30410, 35113, 40154, 45533, 51250, 57305, 63698, 70429, 77498, 84905, 92650, 100733, 109154, 117913, 127010, 136445, 146218, 156329, 166778

Offset

0,1

Comments

The identity (57122*n^2 +47320*n +9801)^2 - (169*n^2 +140*n +29)*(4394*n +1820)^2 = 1 can be written as A156735(n)^2 - a(n)*A156636(n)^2 = 1.

The continued fraction expansion of sqrt(a(n)) is [13n+5; {2, 1, 1, 2, 26n+10}]. - Magus K. Chu, Sep 15 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..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) for n>2.

G.f.: (29 + 251*x + 58*x^2)/(1-x)^3. - Vincenzo Librandi, May 03 2014

E.g.f.: (29 +309*x +169*x^2)*exp(x). - G. C. Greubel, Feb 28 2021

Maple

A156640:= n-> 169*n^2 + 140*n + 29; seq(A156640(n), n=0..50); # G. C. Greubel, Feb 28 2021

Mathematica

LinearRecurrence[{3, -3, 1}, {29, 338, 985}, 50]
CoefficientList[Series[(29 +251x +58x^2)/(1-x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, May 03 2014 *)

Prog

(Magma) I:=[29, 338, 985]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];
(PARI) a(n)=169*n^2+140*n+29 \\ Charles R Greathouse IV, Dec 23 2011
(Sage) [169*n^2 + 140*n + 29 for n in (0..50)] # G. C. Greubel, Feb 28 2021

Crossrefs

Cf. A156636, A156718, A156735.

Cf. A154609 (13n+5).

Sequence in context: A057131 A160442 A125417 * A239743 A022689 A234810

Adjacent sequences: A156637 A156638 A156639 * A156641 A156642 A156643

Keyword

nonn,easy

Author

Vincenzo Librandi, Feb 15 2009

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010

Status

approved