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A072703 Indices of Fibonacci numbers whose last digit is 5.

%I A072703 #40 Nov 27 2024 06:28:51
%S A072703 5,10,20,25,35,40,50,55,65,70,80,85,95,100,110,115,125,130,140,145,
%T A072703 155,160,170,175,185,190,200,205,215,220,230,235,245,250,260,265,275,
%U A072703 280,290,295,305,310,320,325,335,340,350,355,365,370,380,385,395,400,410
%N A072703 Indices of Fibonacci numbers whose last digit is 5.
%C A072703 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.
%C A072703 Numbers that are congruent to {5, 10} mod 15. - _Amiram Eldar_, Jan 01 2022, Nov 25 2024
%H A072703 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A072703 a(n) = 15*(n-1)-a(n-1), with a(1) = 5. - _Vincenzo Librandi_, Aug 08 2010
%F A072703 From _Harvey P. Dale_, May 15 2011: (Start)
%F A072703 a(1) = 5, a(2) = 10, a(3) = 20, a(n) = a(n-1)+a(n-2)-a(n-3).
%F A072703 a(n) = -(5/4)*(3+(-1)^n-6*n). (End)
%F A072703 G.f.: 5*x*(x^2+x+1) / ((x-1)^2*(x+1)). - _Colin Barker_, Jun 16 2013
%F A072703 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(15*sqrt(3)) = A248897 / 10. - _Amiram Eldar_, Jan 01 2022
%F A072703 From _Amiram Eldar_, Nov 25 2024: (Start)
%F A072703 Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/10)*sec(Pi/6) = sqrt((5+sqrt(5))/6).
%F A072703 Product_{n>=1} (1 + (-1)^n/a(n)) = (2/sqrt(3))*cos(7*Pi/30). (End)
%F A072703 a(n) = 5 * A001651(n). - _Alois P. Heinz_, Nov 27 2024
%t A072703 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 *)
%Y A072703 Cf. A000045, A001651, A003893, A248897.
%K A072703 nonn,base,easy
%O A072703 1,1
%A A072703 _Benoit Cloitre_, Aug 07 2002
%E A072703 Edited by _M. F. Hasler_, Oct 08 2014