A113449 Sum of the square root of n-th square triangular number and n-th Pell (or lambda) number (A000129).

2, 8, 40, 216, 1218, 7000, 40560, 235824, 1373090, 7999592, 46616920, 271683720, 1583441442, 9228858808, 53789455200, 313507253856, 1827252574658, 10650004589000, 62072766255880, 361786571934264, 2108646614622210

Offset

1,1

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) = sqrt(((17 + 12*sqrt(2))^n + (17 - 12*sqrt(2))^n - 2)/32) + ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/(2*sqrt(2)). - Stefan Steinerberger, Jun 17 2007

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

a(n) = sqrt((Q_{4*n} - 2)/32) + P_{n}, where P_{n} and the Pell numbers and Q_{n} are the Pell-Lucas numbers.

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

G.f.: (2*x)*(1-4*x) / ((1-2*x-x^2)*(1-6*x+x^2)). (End)

Mathematica

Simplify[Table[Sqrt[((17 + 12*Sqrt[2])^n + (17 - 12*Sqrt[2])^n - 2)/32] + ((1 + Sqrt[2])^n - (1 - Sqrt[2])^n)/(2*Sqrt[2]), {n, 1, 25}]] (* Stefan Steinerberger, Jun 17 2007 *)
Table[Sqrt[(LucasL[4*n, 2] - 2)/32] + Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Mar 11 2017 *)

Prog

(PARI) x='x+O('x^50); Vec((2*x)*(1-4*x) / ((1-2*x-x^2)*(1-6*x+x^2))) \\ G. C. Greubel, Mar 11 2017

Crossrefs

Cf. A000129, A001110, A002203.

Sequence in context: A119817 A025570 A227081 * A234938 A143388 A027282

Adjacent sequences: A113446 A113447 A113448 * A113450 A113451 A113452

Keyword

easy,nonn

Author

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

Extensions

More terms from Stefan Steinerberger, Jun 17 2007

Status

approved