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Squares of A002277.
20

%I #30 Feb 12 2023 16:00:24

%S 0,9,1089,110889,11108889,1111088889,111110888889,11111108888889,

%T 1111111088888889,111111110888888889,11111111108888888889,

%U 1111111111088888888889,111111111110888888888889

%N Squares of A002277.

%C A transformation of the Wonderful Demlo numbers (A002477).

%H Vincenzo Librandi, <a href="/A075412/b075412.txt">Table of n, a(n) for n = 0..200</a>

%H G. Villemin, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Addition/P100a500/Carrerep.htm">Variations sur les carrés</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11, -10).

%F a(n) = A002277(n)^2 = (3 * A002275(n) )^2 = 9 * (A002275(n) )^2

%F a(n) = {111111... (2n times)} - 2*{ 111... (n times)} a(n) = A000042(2n) - 2*A000042(n). - _Amarnath Murthy_, Jul 21 2003

%F a(n) = {333... (n times)}^2 ={111...(n times)}{000... (n times)} - {111... (n times)}. For example, 333^2 = 111000 - 111 = 110889. - _Kyle D. Balliet_, Mar 07 2009

%F a(n) = A002283(n)*A002275(n). [_Reinhard Zumkeller_, May 31 2010]

%F For n>0, a(n) = (A002275(n-1)*10^n + A002282(n-1))*10 + 9. - _Reinhard Zumkeller_, May 31 2010

%F a(n) = (10^(n+1)-10)^2/900. - _José de Jesús Camacho Medina_, Apr 01 2016

%e a(2) = 33^2 = 1089.

%e Contribution from _Reinhard Zumkeller_, May 31 2010: (Start)

%e n=1: ...................... 9 = 9 * 1;

%e n=2: ................... 1089 = 99 * 11;

%e n=3: ................. 110889 = 999 * 111;

%e n=4: ............... 11108889 = 9999 * 1111;

%e n=5: ............. 1111088889 = 99999 * 11111;

%e n=6: ........... 111110888889 = 999999 * 111111;

%e n=7: ......... 11111108888889 = 9999999 * 1111111;

%e n=8: ....... 1111111088888889 = 99999999 * 11111111;

%e n=9: ..... 111111110888888889 = 999999999 * 111111111. (End)

%t LinearRecurrence[{11, -10}, {0, 3}, 20]^2 (* _Vincenzo Librandi_, Mar 20 2014 *)

%t Table[FromDigits[PadRight[{},n,9]]FromDigits[PadRight[{},n,1]],{n,0,15}] (* _Harvey P. Dale_, Feb 12 2023 *)

%Y Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417, A002283, A178630, A178631, A178632, A178633, A178634, A178635, A059988.

%K nonn,easy

%O 0,2

%A Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002