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A061100
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Squares with digital root 4.
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1
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4, 49, 121, 256, 400, 625, 841, 1156, 1444, 1849, 2209, 2704, 3136, 3721, 4225, 4900, 5476, 6241, 6889, 7744, 8464, 9409, 10201, 11236, 12100, 13225, 14161, 15376, 16384, 17689, 18769, 20164, 21316, 22801, 24025, 25600, 26896, 28561, 29929
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OFFSET
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1,1
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LINKS
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FORMULA
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Conjecture:
a(n) = (16-72*n+81*n^2)/4 for n even;
a(n)=(25-90*n+81*n^2)/4 for n odd;
g.f.: -x*(4*x^4+45*x^3+64*x^2+45*x+4) / ((x-1)^3*(x+1)^2). (End)
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
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EXAMPLE
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256 = 16^2, 2 + 5 + 6 = 13, 1 + 3 = 4;
1849 = 43^2, 1 + 8 + 4 + 9 = 22, 2 + 2 = 4.
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MAPLE
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seq(seq((a+9*k)^2, a=[2, 7]), k=0..20); # Robert Israel, Jun 13 2018
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MATHEMATICA
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fdsQ[n_]:=NestWhile[Total[IntegerDigits[#]]&, n, #>9&]==4; Select[Range[ 200]^2, fdsQ] (* Harvey P. Dale, Dec 15 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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