

A046177


Squares (A000290) which are also hexagonal numbers (A000384).


4



1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, 4446390382511295358038307980025, 5131130648390546663702275158894481
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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}.  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
Also centered octagonal numbers (A016754) which are also triangular numbers (A000217).  Colin Barker, Jan 16 2015
Also hexagonal numbers (A000384) which are also centered octagonal numbers (A016754).  Colin Barker, Jan 25 2015


LINKS

Colin Barker, Table of n, a(n) for n = 1..327
Eric Weisstein's World of Mathematics, Hexagonal Square Number.
Eric Weisstein's World of Mathematics, Square Triangular Number.
Index entries for 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).  Richard Choulet, May 01 2009
a(n+2) = 1154*a(n+1)a(n)+72 for n>=0.  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)).
G.f.: 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 *)


PROG

(PARI) Vec(x*(1+70*x+x^2)/((1x)*(11154*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 16 2015


CROSSREFS

Cf. A008844, A046176, A253826.
Cf. A001110 (Numbers that are both triangular and square).
Cf. A000290, A000384, A016754, A253826.
Sequence in context: A014795 A151657 A218273 * A031748 A031533 A031713
Adjacent sequences: A046174 A046175 A046176 * A046178 A046179 A046180


KEYWORD

nonn,easy


AUTHOR

Eric W. Weisstein


STATUS

approved



