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Numbers with 5n binary digits where every run length is 5, written in binary.
3

%I #27 Apr 20 2024 10:23:19

%S 11111,1111100000,111110000011111,11111000001111100000,

%T 1111100000111110000011111,111110000011111000001111100000,

%U 11111000001111100000111110000011111,1111100000111110000011111000001111100000,111110000011111000001111100000111110000011111

%N Numbers with 5n binary digits where every run length is 5, written in binary.

%C A154808 written in base 2.

%H Vincenzo Librandi, <a href="/A154807/b154807.txt">Table of n, a(n) for n = 1..100</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (100000,1,-100000).

%F From _Colin Barker_, Apr 20 2014: (Start)

%F a(n) = (-100001-99999*(-1)^n+2^(6+5*n)*3125^(1+n))/1800018.

%F a(n) = 100000*a(n-1)+a(n-2)-100000*a(n-3).

%F G.f.: 11111*x / ((x-1)*(x+1)*(100000*x-1)). (End)

%e n ... a(n) ........................ A154808(n)

%e 1 ... 11111 ....................... 31

%e 2 ... 1111100000 .................. 992

%e 3 ... 111110000011111 ............. 31775

%e 4 ... 11111000001111100000 ........ 1016800

%e 5 ... 1111100000111110000011111 ... 32537631

%t CoefficientList[Series[11111/((x - 1) (x + 1) (100000 x - 1)), {x, 0, 10}], x] (* _Vincenzo Librandi_, Apr 22 2014 *)

%t LinearRecurrence[{100000,1,-100000},{11111,1111100000,111110000011111},20] (* _Harvey P. Dale_, Aug 08 2023 *)

%o (PARI) Vec(11111*x/((x-1)*(x+1)*(100000*x-1)) + O(x^100)) \\ _Colin Barker_, Apr 20 2014

%Y Cf. A152775, A153435, A154805, A154808.

%K easy,nonn,base

%O 1,1

%A _Omar E. Pol_, Jan 25 2009

%E More terms from _Colin Barker_, Apr 20 2014