cp's OEIS Frontend

A135577 Numbers that have only the digit "1" as first, central and final digit. For numbers with 5 or more digits the rest of digits are "0".

%I A135577 #25 Jun 13 2025 18:16:27
%S A135577 1,111,10101,1001001,100010001,10000100001,1000001000001,
%T A135577 100000010000001,10000000100000001,1000000001000000001,
%U A135577 100000000010000000001,10000000000100000000001,1000000000001000000000001,100000000000010000000000001,10000000000000100000000000001
%N A135577 Numbers that have only the digit "1" as first, central and final digit. For numbers with 5 or more digits the rest of digits are "0".
%C A135577 Also, equal to A135576(n), written in base 2.
%C A135577 Essentially the same as A066138. - _R. J. Mathar_ Apr 29 2008
%C A135577 a(n) has 2n-1 digits.
%H A135577 G. C. Greubel, <a href="/A135577/b135577.txt">Table of n, a(n) for n = 1..500</a>
%H A135577 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A135577 a(n) = A135576(n), written in base 2.
%F A135577 Also, a(1)=1, for n>1; a(n)=(concatenation of 1, n-2 digits 0, 1, n-2 digits 0 and 1).
%F A135577 From _Colin Barker_, Sep 16 2013: (Start)
%F A135577 a(n) = 1 + 10^(n-1) + 100^(n-1) for n>1.
%F A135577 a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>4.
%F A135577 G.f.: x*(2000*x^3 - 1110*x^2 + 1)/((1-x)*(10*x-1)*(100*x-1)). (End)
%F A135577 E.g.f.: (-111 - 200*x + 100*exp(x) + 10*exp(10*x) + exp(100*x))/100. - _Elmo R. Oliveira_, Jun 13 2025
%e A135577 ----------------------------
%e A135577 n ............ a(n)
%e A135577 ----------------------------
%e A135577 1 ............. 1
%e A135577 2 ............ 111
%e A135577 3 ........... 10101
%e A135577 4 .......... 1001001
%e A135577 5 ......... 100010001
%e A135577 6 ........ 10000100001
%e A135577 7 ....... 1000001000001
%e A135577 8 ...... 100000010000001
%e A135577 9 ..... 10000000100000001
%e A135577 10 ... 1000000001000000001
%t A135577 Join[{1}, LinearRecurrence[{111, -1110, 1000}, {111, 10101, 1001001}, 25]] (* _G. C. Greubel_, Oct 19 2016 *)
%t A135577 Join[{1},Table[FromDigits[Join[{1},PadRight[{},n,0],{1},PadRight[{},n,0],{1}]],{n,0,10}]] (* _Harvey P. Dale_, Aug 15 2022 *)
%o A135577 (PARI) Vec(-x*(2000*x^3-1110*x^2+1)/((x-1)*(10*x-1)*(100*x-1))  + O(x^100)) \\ _Colin Barker_, Sep 16 2013
%Y A135577 Cf. A001576, A066138, A135576, A138118, A138119, A138120, A138147, A138826.
%K A135577 nonn,base,less,easy
%O A135577 1,2
%A A135577 _Omar E. Pol_, Feb 24 2008