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%I #25 Mar 07 2017 16:40:31
%S 16,25,169,196,484,529,961,1024,1600,1681,2401,2500,3364,3481,4489,
%T 4624,5776,5929,7225,7396,8836,9025,10609,10816,12544,12769,14641,
%U 14884,16900,17161,19321,19600,21904,22201,24649,24964,27556,27889,30625
%N Squares with digital root 7.
%H Harry J. Smith, <a href="/A061101/b061101.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.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F Conjecture: a(n)=(9*n-8)^2/4 for n even. a(n)=(9*n-1)^2/4 for n odd. G.f.: x*(16+9*x+112*x^2+9*x^3+16*x^4)/((1-x)^3*(1+x)^2). - _Colin Barker_, Apr 21 2012
%F Conjecture is true, because x^2 == 7 (mod 9) if and only if x == 4 or 5 (mod 9). - _Robert Israel_, Jan 31 2017
%e 1681=41^2, 1+6+8+1 = 16, 1+6 =7, 4624=68^2, 4+6+2+4 = 16, 1+6 =7.
%p seq(seq((9*i+j)^2, j=4..5), i=0..100); # _Robert Israel_, Jan 31 2017
%o (PARI) b=0; for (n=1, 1000, until (s==7, b++; s=b^2; s-=9*(s\9)); write("b061101.txt", n, " ", b^2)) \\ _Harry J. Smith_, Jul 18 2009
%o (PARI) a(n)=(n\2*9-4*(-1)^n)^2 \\ _Charles R Greathouse IV_, Sep 21 2012
%Y Cf. A056991.
%K nonn,base,easy
%O 1,1
%A _Amarnath Murthy_, Apr 19 2001
%E More terms from _Harry J. Smith_, Jul 18 2009
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