A201004 Triangular numbers, T(m), that are five-quarters of another triangular number; T(m) such that 4*T(m) = 5*T(k) for some k.

0, 45, 14535, 4680270, 1507032450, 485259768675, 156252138480945, 50312703331095660, 16200534220474321620, 5216521706289400466025, 1679703788890966475738475, 540859403501184915787322970, 174155048223592651917042257910, 56077384668593332732371819724095

Offset

0,2

Comments

For n > 1, a(n) = 322*a(n-1) - a(n-2) + 45. See A200994 for generalization.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Formula

G.f.: 45*x / ((1-x)*(x^2-322*x+1)). - R. J. Mathar, Aug 10 2014

From Colin Barker, Mar 02 2016: (Start)

a(n) = (-18 + (9-4*sqrt(5))*(161+72*sqrt(5))^(-n) + (9+4*sqrt(5))*(161+72*sqrt(5))^n)/128.

a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n > 2. (End)

Example

4*0 = 5*0.
4*45 = 5*36.
4*14535 = 5*11628.
4*4680270 = 5*3744216.

Mathematica

LinearRecurrence[{323, -323, 1}, {0, 45, 14535}, 20] (* T. D. Noe, Feb 15 2012 *)
CoefficientList[Series[-45 x/((x - 1) (x^2 - 322 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 11 2014 *)

Prog

(PARI) concat(0, Vec(45*x/((1-x)*(1-322*x+x^2)) + O(x^15))) \\ Colin Barker, Mar 02 2016
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(45*x/((1-x)*(1-322*x+x^2)))); // G. C. Greubel, Jul 15 2018

Crossrefs

Cf. A001652, A029549, A053141, A075528, A200993-A201008.

Sequence in context: A358707 A184136 A101994 * A171118 A199521 A007537

Adjacent sequences: A201001 A201002 A201003 * A201005 A201006 A201007

Keyword

nonn,easy

Author

Charlie Marion, Feb 15 2012

Extensions

a(7) corrected by R. J. Mathar, Aug 10 2014

Status

approved