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%I #30 Aug 15 2022 15:31:00
%S 1,7,21,73,273,1057,4161,16513,65793,262657,1049601,4196353,16781313,
%T 67117057,268451841,1073774593,4295032833,17180000257,68719738881,
%U 274878431233,1099512676353,4398048608257,17592190238721
%N Numbers whose binary expansion has only the digit "1" as first, central and final digit.
%C This sequence is essentially identical to A001576.
%C a(n) is the number whose binary representation is A135577(n), (See example). - _Omar E. Pol_, Nov 18 2008
%H G. C. Greubel, <a href="/A135576/b135576.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F a(1)=1. If n>1 then a(n) = A001576(n-1).
%F G.f.: -x*(16*x^3-14*x^2+1) / ((x-1)*(2*x-1)*(4*x-1)). - _Colin Barker_, Sep 16 2013
%e --------------------------------------
%e n ........ a(n) ..... a(n) in base 2
%e --------------------------------------
%e 1 .......... 1 ............ 1
%e 2 .......... 7 ........... 111
%e 3 ......... 21 .......... 10101
%e 4 ......... 73 ......... 1001001
%e 5 ........ 273 ........ 100010001
%e 6 ....... 1057 ....... 10000100001
%e 7 ....... 4161 ...... 1000001000001
%e 8 ...... 16513 ..... 100000010000001
%e 9 ...... 65793 .... 10000000100000001
%e 10 .... 262657 ... 1000000001000000001
%t nxt[n_]:=Module[{l=Floor[IntegerLength[n,2]/2]},FromDigits[Join[{1},Table[0,{l}],{1},Table[0,{l}],{1}],2]]
%t NestList[nxt,1,25] (* _Harvey P. Dale_, Dec 29 2010 *)
%t Join[{1},LinearRecurrence[{7,-14,8},{7,21,73},30]] (* _Harvey P. Dale_, Mar 22 2015 *)
%o (PARI) a(n)=if(n--,4^n+2^n+1,1) \\ _Charles R Greathouse IV_, Dec 28 2012
%Y Cf. A001576, A135577.
%Y Subsequence of A006995.
%K nonn,base,easy
%O 1,2
%A _Omar E. Pol_, Feb 24 2008
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