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%I #60 Sep 08 2022 08:46:13
%S 1,1001,1001001,1001001001,1001001001001,1001001001001001,
%T 1001001001001001001,1001001001001001001001,1001001001001001001001001,
%U 1001001001001001001001001001,1001001001001001001001001001001,1001001001001001001001001001001001
%N a(n) = Sum_{k=0..n} 1000^k.
%C A sequence of palindromic numbers.
%H Colin Barker, <a href="/A261544/b261544.txt">Table of n, a(n) for n = 0..333</a> (corrected by Michel Marcus, Jan 19 2019)
%H John Rafael M. Antalan, <a href="https://arxiv.org/abs/1908.06014">A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle</a>, arXiv:1908.06014 [math.HO], 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1001,-1000).
%F a(n) = (1000^(n + 1) - 1)/999.
%F a(n) = 1001*a(n-1) - 1000*a(n-2). - _Colin Barker_, Aug 24 2015
%F G.f.: 1 / ((x-1)*(1000*x-1)). - _Colin Barker_, Aug 24 2015
%F E.g.f.: (1/999)*(1000000*exp(1000*x) - exp(x)). - _G. C. Greubel_, Aug 29 2015
%e From _Bruno Berselli_, Aug 25 2015: (Start)
%e a(n) is the binary representation of A023001
%e -------------------------------------------------
%e 1 ........................................... 1
%e 1001 ........................................ 9
%e 1001001 ..................................... 73
%e 1001001001 ................................ 585
%e 1001001001001 ............................ 4681
%e 1001001001001001 ........................ 37449
%e 1001001001001001001 .................... 299593
%e 1001001001001001001001 ................ 2396745
%e 1001001001001001001001001 ............ 19173961, etc.
%e (End)
%t Table[(1000^(n + 1) - 1)/999, {n, 0, 15}]
%t LinearRecurrence[{1001, -1000}, {1, 1001}, 20] (* _Vincenzo Librandi_, Aug 24 2015 *)
%o (PARI) Vec(1 / ((x-1)*(1000*x-1)) + O(x^20)) \\ _Colin Barker_, Aug 24 2015
%o (Magma) [(1000^(n+1)-1)/999: n in [0..30]]; // _Vincenzo Librandi_, Aug 24 2015
%Y Cf. A000533, A002113, A023001.
%Y Subsequence of A033146.
%Y Sums of 100^k: A094028; sums of 10^k: A000042.
%Y Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.
%K nonn,base,easy
%O 0,2
%A _Ilya Gutkovskiy_, Aug 24 2015
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