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A039300
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Number of distinct quadratic residues mod 3^n.
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8
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1, 2, 4, 11, 31, 92, 274, 821, 2461, 7382, 22144, 66431, 199291, 597872, 1793614, 5380841, 16142521, 48427562, 145282684, 435848051, 1307544151, 3922632452, 11767897354, 35303692061, 105911076181, 317733228542, 953199685624
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OFFSET
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0,2
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COMMENTS
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Number of distinct n-digit suffixes of base 3 squares.
In general, for any odd prime p, the number s of quadratic residues mod p^n is given by s = (p^(n+1) + p + 2)/(2p + 2) for even n and s = (p^(n+1) + 2*p + 1)/(2p + 2) for odd n, see A000224. - Lekraj Beedassy, Jan 07 2005
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REFERENCES
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J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 324.
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LINKS
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FORMULA
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a(n) = floor(3*(3^n + 3)/8).
G.f.: (1 - x - 3*x^2)/((1 - x)*(1 + x)*(1 - 3*x)). - Michael Somos, Mar 27 2005
a(n) = 2*a(n-1) + 3*a(n-2) - 3 with n > 1, a(0) = 1, a(1) = 1. - Zerinvary Lajos, Dec 14 2008
E.g.f.: (1/8)*exp(-x)*(-1+6*exp(2*x)+3*exp(4*x)). - Stefano Spezia, Sep 04 2018
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MAPLE
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floor((3^n+3)*3/8) ;
end proc:
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MATHEMATICA
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CoefficientList[Series[(1-x-3x^2)/((1-x)(1+x)(1-3x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 21 2012 *)
Table[Floor((3^n+3)*3/8), {n, 0, 30}] (* Bruno Berselli, Apr 21 2012 *)
CoefficientList[Series[1/8 E^-x (-1 + 6 E^(2 x) + 3 E^(4 x)), {x, 0, 30}], x]*Table[k!, {k, 0, 30}] (* Stefano Spezia, Sep 04 2018 *)
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PROG
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(PARI) {a(n) = if(n<0, 0, 3^n*3\8 + 1)}; /* Michael Somos, Mar 27 2005 */
(PARI) {a(n) = if(n<1, n==0, 3*a(n-1) - 2 + n%2)}; /* Michael Somos, Mar 27 2005 */
(Magma) [(3^(n+1) + 6 + (-1)^(n+1))/8: n in [0..30]]; // Vincenzo Librandi, Apr 21 2012
(Sage) [(3^(n+1) +6 -(-1)^n)/8 for n in (0..30)] # G. C. Greubel, Jul 14 2019
(GAP) List([0..30], n-> (3^(n+1) +6 -(-1)^n)/8) # G. C. Greubel, Jul 14 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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