A113450 Difference between the square root of n-th square triangular number and n-th lambda number given by the recurrence f(n) = 2f(n-1) + f(n-2), f(1) = 1, f(2)= 2.

0, 4, 30, 192, 1160, 6860, 40222, 235008, 1371120, 7994836, 46605438, 271656000, 1583374520, 9228697244, 53789065150, 313506312192, 1827250301280, 10649999100580, 62072753005662, 361786539945408, 2108646537394280

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (8,-12,-4,1).

Formula

a(n) = A001109(n) - A000129(n). - R. J. Mathar, Feb 08 2008

From G. C. Greubel, Mar 11 2017: (Start)

a(n) = 8*a(n-1) - 12*a(n-2) - 4*a(n-3) + a(n-4).

G.f.: (2*x^2)*(2-x) / ((1-2*x-x^2)*(1-6*x+x^2)). (End)

Maple

A001110 := proc(n) coeftayl( x*(1+x)/(1-x)/(1-34*x+x^2), x=0, n) ; end: A001109 := proc(n) sqrt(A001110(n)) ; end: A000129 := proc(n) coeftayl( x/(1-2*x-x^2), x=0, n) ; end: A113450 := proc(n) A001109(n) - A000129(n) ; end: seq(A113450(n), n=1..40) ; # R. J. Mathar, Feb 08 2008

Mathematica

LinearRecurrence[{8, -12, -4, 1}, {0, 4, 30, 192}, 50] (* or *) CoefficientList[Series[(2*x^2)*(2-x)/((1-2*x-x^2)*(1-6*x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Mar 11 2017 *)

Prog

(PARI) x='x+O('x^50); concat([0], Vec((2*x^2)*(2-x)/((1-2*x-x^2)*(1-6*x+x^2)))) \\ G. C. Greubel, Mar 11 2017

Crossrefs

Cf. A113449.

Sequence in context: A115867 A057416 A089154 * A344399 A268218 A272493

Adjacent sequences: A113447 A113448 A113449 * A113451 A113452 A113453

Keyword

easy,nonn

Author

K. B. Subramaniam (subramaniam_kb05(AT)yahoo.co.in), Nov 02 2005

Extensions

Corrected and extended by R. J. Mathar, Feb 08 2008

Status

approved