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A061100 Squares with digital root 4.

%I A061100 #36 Dec 06 2024 21:44:16
%S A061100 4,49,121,256,400,625,841,1156,1444,1849,2209,2704,3136,3721,4225,
%T A061100 4900,5476,6241,6889,7744,8464,9409,10201,11236,12100,13225,14161,
%U A061100 15376,16384,17689,18769,20164,21316,22801,24025,25600,26896,28561,29929
%N A061100 Squares with digital root 4.
%H A061100 Harry J. Smith, <a href="/A061100/b061100.txt">Table of n, a(n) for n = 1..1000</a>
%H A061100 Amarnath Murthy & Charles Ashbacher, <a href="http://fs.gallup.unm.edu/murthybook.pdf">Fabricating a perfect square with a given valid digit sum</a>, in Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences, pp. 154-156.
%F A061100 From _Colin Barker_, Feb 18 2013: (Start)
%F A061100 Conjecture:
%F A061100 a(n) = (16-72*n+81*n^2)/4 for n even;
%F A061100 a(n)=(25-90*n+81*n^2)/4 for n odd;
%F A061100 g.f.: -x*(4*x^4+45*x^3+64*x^2+45*x+4) / ((x-1)^3*(x+1)^2). (End)
%F A061100 Conjecture is true since x^2 == 4 (mod 9) if and only if x == 2 or 7 (mod 9). The odd-numbered terms are (2+9*k)^2 and the even-numbered terms are (7+9*k)^2. - _Robert Israel_, Jun 13 2018
%e A061100 256 = 16^2, 2 + 5 + 6 = 13, 1 + 3 = 4;
%e A061100 1849 = 43^2, 1 + 8 + 4 + 9 = 22, 2 + 2 = 4.
%p A061100 seq(seq((a+9*k)^2,a=[2,7]),k=0..20); # _Robert Israel_, Jun 13 2018
%t A061100 fdsQ[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]==4; Select[Range[ 200]^2,fdsQ] (* _Harvey P. Dale_, Dec 15 2011 *)
%o A061100 (PARI) a(n)=(n\2*9-2*(-1)^n)^2 \\ _Charles R Greathouse IV_, Sep 21 2012
%Y A061100 Cf. A056991.
%K A061100 nonn,base
%O A061100 1,1
%A A061100 _Amarnath Murthy_, Apr 19 2001
%E A061100 More terms from _Harry J. Smith_, Jul 18 2009