

A046177


Square numbers which are also hexagonal numbers.


1



1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, 4446390382511295358038307980025
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OFFSET

1,2


COMMENTS

Also, odd squaretriangular numbers (or bisection of A001110 = {0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ...} = Numbers that are both triangular and square: a(n) = 34a(n1)  a(n2) + 2).  Alexander Adamchuk, Nov 06 2007
Let be y^2=x*(2*x1)=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*y6)^2 = H_{17*x+12*y4}. [From Richard Choulet, May 01 2009]
As n increases, this sequence is approximately geometric with common ratio r = lim(n > Infinity, a(n)/a(n1)) = ( 1+ sqrt(2))^8 = 577 + 408 * sqrt(2).  Ant King Nov 08 2011


LINKS

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).


FORMULA

a(n) = A001110(2n1).  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(n1)  1155*a(n2) + a(n3).
a(n) = 1/32*((1 + sqrt(2))^(8*n  4) + (1  sqrt(2))^(8*n4)  2).
a(n) = floor(1/32*(1 + sqrt(2))^(8*n  4)).
a(n) = 1/32*((tan(3*pi/8))^(8*n4) + (tan(pi/8))^(8*n4)  2).
a(n) = floor(1/32*(tan(3*pi/8))^(8*n4)).
GF: x*(1 + 70*x + x^2)/((1  x)*(1  1154*x + x^2)).
(End)


MATHEMATICA

LinearRecurrence[{1155, 1155, 1}, {1, 1225, 1413721}, 11] (* Ant King, Nov 08 2011 *)


CROSSREFS

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


KEYWORD

nonn


AUTHOR

Eric W. Weisstein


STATUS

approved



