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A252763
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Numbers n such that the hexagonal number H(n) is equal to the sum of the pentagonal numbers P(m), P(m+1), P(m+2) and P(m+3) for some m.
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2
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8, 1480, 287064, 55688888, 10803357160, 2095795600104, 406573543062968, 78873171558615640, 15300988708828371144, 2968312936341145386248, 575837408661473376560920, 111709488967389493907432184, 21671065022264900344665282728, 4204074904830423277371157417000
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OFFSET
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1,1
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COMMENTS
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Also positive integers y in the solutions to 12*x^2-4*y^2+32*x+2*y+36 = 0, the corresponding values of x being A252762.
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LINKS
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Colin Barker, Table of n, a(n) for n = 1..437
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).
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FORMULA
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a(n) = 195*a(n-1)-195*a(n-2)+a(n-3).
G.f.: -8*x*(3*x^2-10*x+1) / ((x-1)*(x^2-194*x+1)).
a(n) = (6+(285-164*sqrt(3))*(97+56*sqrt(3))^n+(97+56*sqrt(3))^(-n)*(285+164*sqrt(3)))/24. - Colin Barker, Mar 02 2016
a(n) = 194*a(n-1)-a(n-2)-48. - Vincenzo Librandi, Mar 03 2016
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EXAMPLE
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8 is in the sequence because H(8) = 120 = 12+22+35+51 = P(3)+P(4)+P(5)+P(6).
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MATHEMATICA
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LinearRecurrence[{195, -195, 1}, {8, 1480, 287064}, 30] (* Vincenzo Librandi, Mar 03 2016 *)
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PROG
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(PARI) Vec(-8*x*(3*x^2-10*x+1)/((x-1)*(x^2-194*x+1)) + O(x^100))
(Magma) I:=[8, 1480]; [n le 2 select I[n] else 194*Self(n-1)- Self(n-2)-48: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016
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CROSSREFS
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Cf. A000326, A000384, A252762.
Sequence in context: A201492 A172938 A252176 * A096970 A248386 A114617
Adjacent sequences: A252760 A252761 A252762 * A252764 A252765 A252766
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KEYWORD
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nonn,easy
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AUTHOR
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Colin Barker, Dec 21 2014
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STATUS
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approved
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