A072703 Indices of Fibonacci numbers whose last digit is 5.

5, 10, 20, 25, 35, 40, 50, 55, 65, 70, 80, 85, 95, 100, 110, 115, 125, 130, 140, 145, 155, 160, 170, 175, 185, 190, 200, 205, 215, 220, 230, 235, 245, 250, 260, 265, 275, 280, 290, 295, 305, 310, 320, 325, 335, 340, 350, 355, 365, 370, 380, 385, 395, 400, 410

Offset: 1

Author

Benoit Cloitre, Aug 07 2002

Keywords

• nonn
• base
• easy

Comments

Sequence contains numbers of the forms 5 + 60*k, 10 + 60*k, 20 + 60*k, 25 + 60*k, 35 + 60*k, 40 + 60*k, 50 + 60*k, 55 + 60*k, where k>=0.

Numbers that are congruent to {5, 10} mod 15. - Amiram Eldar, Jan 01 2022, Nov 25 2024

Links

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

Crossrefs

Cf. A000045, A001651, A003893, A248897.

Programs

Mathematica

Flatten[Position[Fibonacci[Range[400]],?(Last[IntegerDigits[#]]==5&)]] (* or *) LinearRecurrence[{1,1,-1},{5,10,20},60] (* or *) Table[-(5/4) (3+(-1)^n-6 n),{n,60}] (* _Harvey P. Dale, May 15 2011 *)

Formula

a(n) = 15*(n-1)-a(n-1), with a(1) = 5. - Vincenzo Librandi, Aug 08 2010

From Harvey P. Dale, May 15 2011: (Start)

a(1) = 5, a(2) = 10, a(3) = 20, a(n) = a(n-1)+a(n-2)-a(n-3).

a(n) = -(5/4)*(3+(-1)^n-6*n). (End)

G.f.: 5*x*(x^2+x+1) / ((x-1)^2*(x+1)). - Colin Barker, Jun 16 2013

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(15*sqrt(3)) = A248897 / 10. - Amiram Eldar, Jan 01 2022

From Amiram Eldar, Nov 25 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/10)*sec(Pi/6) = sqrt((5+sqrt(5))/6).

Product_{n>=1} (1 + (-1)^n/a(n)) = (2/sqrt(3))*cos(7*Pi/30). (End)

a(n) = 5 * A001651(n). - Alois P. Heinz, Nov 27 2024

Extensions

Edited by M. F. Hasler, Oct 08 2014