A165225 a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 5*a(n-2) for n > 1.

1, 5, 45, 425, 4025, 38125, 361125, 3420625, 32400625, 306903125, 2907028125, 27535765625, 260822515625, 2470546328125, 23401350703125, 221660775390625, 2099601000390625, 19887706126953125, 188379056267578125

Offset: 0

Author

Philippe Deléham, Sep 09 2009

Keywords

• easy
• nonn

Comments

Sum_{k=1..(m-1)/2} tan^(2n) (k*Pi/m) is an integer when m >= 3 is an odd integer (see AMM and Crux Mathematicorum links); twice this sequence is the particular case m = 5. - Bernard Schott, Apr 25 2022

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000.
• Michel Bataille and Li Zhou, A Combinatorial Sum Goes on Tangent, The American Mathematical Monthly, Vol. 112, No. 7 (Aug. - Sep., 2005), Problem 11044, pp. 657-659.
• D. J. Smeenk, Problem 3452, Crux Mathematicorum, Vol. 35, No. 5 (2009), pp. 325 and 327; Solution to Problem 3452 by Roy Barbara, ibid., Vol. 36, No. 5 (2010), pp. 341-342.
• Index entries for linear recurrences with constant coefficients, signature (10,-5).

Crossrefs

Cf. A019934, A019970.

Similar with: A000244 (m=3), 2*this sequence (m=5), A108716 (m=7), A353410 (m=9), A275546 (m=11), A353411 (m=13).

Programs

Mathematica

LinearRecurrence[{10,-5},{1,5},30] (* Harvey P. Dale, Dec 23 2019 *)

Formula

Limit_{n->oo} a(n+1)/a(n) = 5 + 2*sqrt(5) = 9.47213595...

G.f.: (1-5x)/(1-10x+5x^2).

a(n) = ((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/2. - Klaus Brockhaus, Sep 25 2009

a(n) = (tan(Pi/5)^(2*n) + tan(2*Pi/5)^(2*n))/2 (Smeenk, 2009). - Amiram Eldar, Apr 03 2022

Extensions

More terms from Klaus Brockhaus, Sep 25 2009