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

1,1

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

Table of n, a(n) for n=1..55.

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

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

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 *)

Crossrefs

Cf. A000045, A001651, A003893, A248897.

Sequence in context: A062052 A245387 A115799 * A086761 A045191 A065958

Adjacent sequences: A072700 A072701 A072702 * A072704 A072705 A072706

Keyword

nonn,base,easy

Author

Benoit Cloitre, Aug 07 2002

Extensions

Edited by M. F. Hasler, Oct 08 2014

Status

approved