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A113449 Sum of the square root of n-th square triangular number and n-th Pell (or lambda) number (A000129). 2
2, 8, 40, 216, 1218, 7000, 40560, 235824, 1373090, 7999592, 46616920, 271683720, 1583441442, 9228858808, 53789455200, 313507253856, 1827252574658, 10650004589000, 62072766255880, 361786571934264, 2108646614622210 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 7 03:00 EST 2019. Contains 329836 sequences. (Running on oeis4.)