0,2

Also 6*x^2+1 is a square. - Cino Hilliard (hillcino368(AT)gmail.com), Mar 08 2003

This sequence has the following property. For each n, if A = a(n), B = 2*a(n+1), C = 3*a(n+1) then A*B+1, A*C+1, B*C+1 are perfect squares. - Deshpande M.N. (dpratap_ngp(AT)sancharnet.in), Sep 22 2004

n such that 6*n^2=floor(sqrt(6)*n*ceil(sqrt(6)*n)). - Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003

Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 19 2005

(sqrt(2)+sqrt(3))^(2*n)=A001079(n)+a(n)*sqrt(6); a(n)=A054320(n)+A138288(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

O. Bottema: Verscheidenheden XXVI. Het vraagstuk van Malfatti, Euclides 25 (1949-50), pp. 144-149 [in Dutch].

O. Bottema, The Malfatti problem (translation of Het vraagstuk van Malfatti), Forum Geom. 1 (2001) 43-50.

V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 281.

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 283, 302, P_{16}).

T. D. Noe, Table of n, a(n) for n=0..100

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Tanya Khovanova, Recursive Sequences

Bottema article in Forum Geometricorum

Bottema article, from Euclides

L. Euler, De solutione problematum diophanteorum per numeros integros, par. 18

a(n)=[(5+2sqrt(6))^n-(5-2sqrt(6))^n]/[2sqrt(6)]. G.f.=2z/(1-10z+z^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 19 2005

a(-n)=-a(n).

a(n) = 9*(a(n-1)+a(n-2))-a(n-3). a(n) = 11*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 20 2006

a(n+1) = A054320(n) + A138288(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008

a(n) = Sinh[2n*ArcSinh[Sqrt[2]]]/Sqrt[6] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008

A001078 := proc(n) option remember; if n=0 then 0 elif n=1 then 2 else 10*A001078(n-1)-A001078(n-2); fi; end;

A001078:=2*z/(1-10*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

(PARI) nxsqp1(m, n) = { for(x=1, m, y = n*x*x+1; if(issquare(y), print1(x" ")) ) }

(PARI) a(n)=imag((5+2*quadgen(24))^n) /* Michael Somos Jul 05 2005 */

(PARI) a(n)=subst(poltchebi(n+1)-5*poltchebi(n), x, 5)/12 /* Michael Somos Jul 05 2005 */

Cf. A053410.

Cf. A138281.

Sequence in context: A067641 A037566 A125857 this_sequence A001253 A085586 A136902

Adjacent sequences: A001075 A001076 A001077 this_sequence A001079 A001080 A001081

nonn,easy,nice

njas

Thanks to Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr) and Floor van Lamoen (fvlamoen(AT)wxs.nl) for the Bottema references.