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Squares with digital root 4.
1

%I #33 Jul 02 2018 01:42:23

%S 4,49,121,256,400,625,841,1156,1444,1849,2209,2704,3136,3721,4225,

%T 4900,5476,6241,6889,7744,8464,9409,10201,11236,12100,13225,14161,

%U 15376,16384,17689,18769,20164,21316,22801,24025,25600,26896,28561,29929

%N Squares with digital root 4.

%H Harry J. Smith, <a href="/A061100/b061100.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 From _Colin Barker_, Feb 18 2013: (Start)

%F Conjecture:

%F a(n) = (16-72*n+81*n^2)/4 for n even;

%F a(n)=(25-90*n+81*n^2)/4 for n odd;

%F g.f.: -x*(4*x^4+45*x^3+64*x^2+45*x+4) / ((x-1)^3*(x+1)^2). (End)

%F 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 256 = 16^2, 2 + 5 + 6 = 13, 1 + 3 = 4;

%e 1849 = 43^2, 1 + 8 + 4 + 9 = 22, 2 + 2 = 4.

%p seq(seq((a+9*k)^2,a=[2,7]),k=0..20); # _Robert Israel_, Jun 13 2018

%t fdsQ[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]==4; Select[Range[ 200]^2,fdsQ] (* _Harvey P. Dale_, Dec 15 2011 *)

%o (PARI) { b=0; for (n=0, 1000, until (s==4, b++; s=b^2; s-=9*(s\9)); write("b061100.txt", n, " ", b^2) ) } \\ _Harry J. Smith_, Jul 18 2009

%o (PARI) a(n)=(n\2*9-2*(-1)^n)^2 \\ _Charles R Greathouse IV_, Sep 21 2012

%Y Cf. A056991.

%K nonn,base

%O 1,1

%A _Amarnath Murthy_, Apr 19 2001

%E More terms from _Harry J. Smith_, Jul 18 2009