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A059989
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Numbers n such that 3*n+1 and 4*n+1 are both squares.
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4
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0, 56, 10920, 2118480, 410974256, 79726887240, 15466605150360, 3000441672282656, 582070217817684960, 112918621814958599640, 21905630561884150645256, 4249579410383710266580080, 824396499983877907565890320, 159928671417461930357516142056
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OFFSET
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1,2
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LINKS
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Colin Barker, Table of n, a(n) for n = 1..400
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).
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FORMULA
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a(n) = (A001570(n)^2 - 1)/3.
G.f.: 56*x^2 / (1-195*x+195*x^2-x^3).
From Colin Barker, Mar 03 2016: (Start)
a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n>3.
a(n) = (-1)*((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(7+4*sqrt(3)+(-7+4*sqrt(3))*(97+56*sqrt(3))^n))/48.
(End)
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EXAMPLE
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3*56+1=13^2 and 4*56+1=15^2.
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MAPLE
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f:= proc(n) local u;
u:= <<7, 8>|<6, 7>>^n . <1, -1>;
(u[1]^2-1)/3
end proc:
map(f, [$1..30]); # Robert Israel, Mar 03 2016
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MATHEMATICA
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CoefficientList[Series[56 x/(1 - 195 x + 195 x^2 - x^3), {x, 0, 13}], x] (* Michael De Vlieger, Mar 03 2016 *)
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PROG
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(PARI) isok(n) = issquare(3*n+1) && issquare(4*n+1) \\ Michel Marcus, Jun 08 2013
(PARI) concat(0, Vec(56*x^2/((1-x)*(1-194*x+x^2)) + O(x^20))) \\ Colin Barker, Mar 03 2016
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CROSSREFS
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Cf. A245031.
Sequence in context: A034204 A275921 A091546 * A352602 A184125 A213865
Adjacent sequences: A059986 A059987 A059988 * A059990 A059991 A059992
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KEYWORD
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nonn,easy
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AUTHOR
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David Radcliffe, Mar 07 2001
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EXTENSIONS
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Offset changed to 1 by Joerg Arndt, Mar 03 2016
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STATUS
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approved
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