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A252762
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Numbers n such that the sum of the pentagonal numbers P(n), P(n+1), P(n+2) and P(n+3) is equal to the hexagonal number H(m) for some m.
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2
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3, 853, 165735, 32151993, 6237321163, 1210008153885, 234735344532783, 45537446831206273, 8834029949909484435, 1713756272835608774373, 332459882900158192744183, 64495503526357853783597385, 12511795224230523475825148763, 2427223777997195196456295262893
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OFFSET
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1,1
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COMMENTS
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Also positive integers x in the solutions to 12*x^2-4*y^2+32*x+2*y+36 = 0, the corresponding values of y being A252763.
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LINKS
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FORMULA
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a(n) = 195*a(n-1)-195*a(n-2)+a(n-3).
G.f.: x*(15*x^2-268*x-3) / ((x-1)*(x^2-194*x+1)).
a(n) = -4/3+1/24*(97+56*sqrt(3))^(-n)*(-164-95*sqrt(3)+(97+56*sqrt(3))^(2*n)*(-164+95*sqrt(3))). - Colin Barker, Mar 02 2016
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EXAMPLE
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3 is in the sequence because P(3)+P(4)+P(5)+P(6) = 12+22+35+51 = 120 = H(8).
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MATHEMATICA
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LinearRecurrence[{195, -195, 1}, {3, 853, 165735}, 30] (* Vincenzo Librandi, Mar 03 2016 *)
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PROG
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(PARI) Vec(x*(15*x^2-268*x-3)/((x-1)*(x^2-194*x+1)) + O(x^100))
(Magma) I:=[3, 853]; [n le 2 select I[n] else 194*Self(n-1) - Self(n-2)+256: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016
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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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