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A046178
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Indices of pentagonal numbers which are also hexagonal.
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3
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1, 165, 31977, 6203341, 1203416145, 233456528757, 45289363162681, 8785902997031325, 1704419892060914337, 330648673156820350021, 64144138172531086989705
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The reason is that we obtain the same Diophantine equation with various parameters is the following: the number which is written 361 in base 4*A046179(n)-2 is the square of 6*A046178(n)-1. That is, 361 in base 110770 is 3*110770^2+6*110770+1=36810643321 i.e. the square of 191861 if we consider the third terms of A046179 and A046178 which are 27693 and 31977 respectively. - Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 03 2007
As n increases, this sequence is approximately geometric with common ratio r=lim(n->Infinity,a(n)/a(n-1))=(2+sqrt(3))^4=97+56*sqrt(3). - Ant King, Dec 14 2011
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LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| a(n) = 194*a(n-1) - a(n-2) - 32; g.f.: (1-30*x-3*x^2)/((1-x)*(1-194*x+x^2)). - Warut Roonguthai (warut822(AT)yahoo.com) Jan 08 2001
a(n+1)=97*a(n)-16+28*(12*a(n)^2-4*a(n)+1)^0.5. - R. Choulet (richardchoulet(AT)yahoo.fr), Oct 09 2007
a(n)=(1/6)+(5/12)*[97-56*sqrt(3)]^n+(5/12)*[97+56*sqrt(3)]^n-(1/4)*[97-56*sqrt(3)]^n*sqrt(3) +(1/4)*sqrt(3)*[97+56*sqrt(3)]^n, with n>=0. [From Paolo P. Lava (paoloplava(AT)gmail.com), Sep 26 2008]
From Ant King, Dec 14 2011: (Start)
a(n) = 195*a(n-1) - 195*a(n-2) + a(n-3).
a(n) = 1/12*((sqrt(3)-1)*(2+sqrt(3))^(4n-2)-(sqrt(3)+1)* (2-sqrt(3))^(4n-2)+2).
a(n) = ceiling(1/12*(sqrt(3)-1)*(2+sqrt(3))^(4n-2)).
(End)
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MATHEMATICA
| LinearRecurrence[{195, -195, 1}, {1, 165, 31977}, 11] (* Ant King, Dec 14 2011 *)
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CROSSREFS
| Cf. A046179, A046180.
Sequence in context: A071576 A140912 A132055 * A176018 A184287 A203183
Adjacent sequences: A046175 A046176 A046177 * A046179 A046180 A046181
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KEYWORD
| nonn,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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