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A046177
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Square numbers which are also hexagonal numbers.
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1
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1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, 4446390382511295358038307980025
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OFFSET
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1,2
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COMMENTS
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Also, odd square-triangular numbers (or bisection of A001110 = {0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ...} = Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2). - Alexander Adamchuk, Nov 06 2007
Let be y^2=x*(2*x-1)=H_x (x>=1). The least both hexagonal and square number which is greater than y^2 is given by the relation (24*x+17*y-6)^2 = H_{17*x+12*y-4}. [From Richard Choulet, May 01 2009]
As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = ( 1+ sqrt(2))^8 = 577 + 408 * sqrt(2). - Ant King Nov 08 2011
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LINKS
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Table of n, a(n) for n=1..11.
Eric Weisstein's World of Mathematics, Hexagonal Square Number.
Eric Weisstein, Link to a section of The World of Mathematics. Square Triangular Number.
Index to sequences with linear recurrences with constant coefficients, signature (1155,-1155,1).
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FORMULA
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a(n) = A001110(2n-1). - Alexander Adamchuk, Nov 06 2007
a(n+1)=577*a(n)+36+204*(8*a(n)^2+a(n))^0.5 for n>=1 (a(0)=1) [From Richard Choulet, May 01 2009]
a(n+2)=1154*a(n+1)-a(n)+72 for n>=0. [From Richard Choulet, May 01 2009]
From Ant King, Nov 07 2011: (Start)
a(n) = 1155*a(n-1) - 1155*a(n-2) + a(n-3).
a(n) = 1/32*((1 + sqrt(2))^(8*n - 4) + (1 - sqrt(2))^(8*n-4) - 2).
a(n) = floor(1/32*(1 + sqrt(2))^(8*n - 4)).
a(n) = 1/32*((tan(3*pi/8))^(8*n-4) + (tan(pi/8))^(8*n-4) - 2).
a(n) = floor(1/32*(tan(3*pi/8))^(8*n-4)).
GF: x*(1 + 70*x + x^2)/((1 - x)*(1 - 1154*x + x^2)).
(End)
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MATHEMATICA
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LinearRecurrence[{1155, -1155, 1}, {1, 1225, 1413721}, 11] (* Ant King, Nov 08 2011 *)
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CROSSREFS
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Cf. A008844, A046176.
Cf. A001110 = Numbers that are both triangular and square.
Sequence in context: A014795 A151657 A218273 * A031748 A031533 A031713
Adjacent sequences: A046174 A046175 A046176 * A046178 A046179 A046180
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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