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Squares with digital root 1.
1

%I #32 Jul 15 2024 15:32:37

%S 1,64,100,289,361,676,784,1225,1369,1936,2116,2809,3025,3844,4096,

%T 5041,5329,6400,6724,7921,8281,9604,10000,11449,11881,13456,13924,

%U 15625,16129,17956,18496,20449,21025,23104,23716,25921,26569,28900,29584

%N Squares with digital root 1.

%H Harry J. Smith, <a href="/A061099/b061099.txt">Table of n, a(n) for n = 1..1001</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 From _Colin Barker_, Apr 21 2012: (Start)

%F a(n) = (9*n+2)^2/4 for n even; a(n)=(9*n+7)^2/4 for n odd.

%F G.f.: x*(1+63*x+34*x^2+63*x^3+x^4)/((1-x)^3*(1+x)^2). (End)

%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). - _Wesley Ivan Hurt_, Apr 21 2021

%e 289 = 17^2, 2+8+9 = 19, 1+9 = 1, 1369 = 37^2, 1+3+6+9 = 19, 1+9 = 1.

%o (PARI) a(n)=(n\2*9-(-1)^n)^2 \\ _Charles R Greathouse IV_, Sep 20 2012

%Y Squares of A056020.

%Y Cf. A056991.

%K nonn,base,easy

%O 1,2

%A _Amarnath Murthy_, Apr 19 2001

%E More terms from _Harry J. Smith_, Jul 17 2009