0,3

These are the numbers that are both triangular and square.

Satisfies a recurrence of S_r type for r=36: 0, 1, 36 and a(n-1)*a(n+1)=(a(n)-1)^2. First observed by Colin Dickson in alt.math.recreational March 7th 2004. - Rainer Rosenthal (r.rosenthal(AT)web.de), Mar 14 2004

For every n, a(n) is the first of three triangular numbers in geometric progression. The third number in the progression is a(n+1). The middle triangular number is sqrt(a(n)*a(n+1)). Chen and Fang prove that four distinct triangular numbers are never in geometric progression. - T. D. Noe (noe(AT)sspectra.com), Apr 30 2007

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 193.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 10.

H. G. Forder, A Simple Proof of a Result on Diophantine Approximation, Math. Gaz., 47 (1963), 237-238.

Martin Gardner, Time Travel and other Mathematical Bewilderments, pp. 16-17, Freeman 1988

P. Lafer, Discovering the square-triangular numbers, Fib. Quart., 9 (1971), 93-105.

D. A. Q., Triangular square numbers - a postscript, Math. Gaz., 56 (1972), 311-314.

J. H. Silverman, A Friendly Introduction to Number Theory, p 196, Prentice Hall 2001

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

Joerg Arndt, Fxtbook

Kevin Browne, Square Triangular Numbers

Yong-Gao Chen and Jin-Hui Fang, Triangular numbers in geometric progression, INTEGERS 7 (2007), #A19.

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.

R. Stephan, Boring proof of a nonlinearity

Chris Thatcher, Square Triangular Numbers [Broken link?]

Eric Weisstein, CRC Online Dictionary, Square Triangular Number

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Square triangular number

G.f.: x*(1 + x) / (( 1 - x )*( 1 - 34 x + x^2 )).

a(n-1) * a(n+1) = (a(n)-1)^2. - Colin Dickson, posting to alt.math.recreational, circa Mar 13 2004

If L is a square-triangular number, then the next one is 1 + 17*L + 6*sqrt(L + 8*L^2) - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 27 2001

a(n)-a(n-1)=A001109(2n-1). - Sophie Kuo (ejiqj_6(AT)yahoo.com.tw), May 27 2006

a(n) = A001109(n)^2 = A001108(n)*(A001108(n)+1)/2 = (A000129(n)*A001333(n))^2 = (A000129(n)*(A000129(n) + A000129(n-1)))^2 - Henry Bottomley, Apr 19, 2000.

a(n)=(((17+12*sqrt(2))^n)+((17-12*sqrt(2))^n)-2)/32 - Bruce Corrigan (scentman(AT)myfamily.com), Oct 26 2002

As n goes to infinity the ratio a(n+1)/a(n) goes to 17 + 12*sqrt(2). See Problem A of Nieuw Archief voor Wiskunde http://www.math.leidenuniv.nl/~naw/serie5/deel05/dec2004/pdf/uwc.pdf After Feb 01 2005 (submission deadline) a solution can be found at http://www.jaapspies.nl/mathfiles/problem2004-4A.pdf - Jaap Spies (j.spies(AT)hccnet.nl), Dec 12 2004

a(n) = 35(a(n-1)-a(n-2)) + a(n-3); a(n) = -1/16 +((-24+17*2^(1/2))/2^(11/2))*(17-12*2^(1/2))^(n-1) +((24+17*2^(1/2))/2^(11/2))*(17+12*2^(1/2))^(n-1) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Nov 07 2003

a(n+1)=[17*A029547(n)-A091761(n)-1]/16. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

a(2) = ((17+12*sqrt(2))^2+(17-12*sqrt(2))^2-2)/32 = (289+24*sqrt(2)+288+289-24*sqrt(2)+288-2)/32 = (578+576-2)/32 = 1152/32 = 36 and 6^2 = 36 = 8*9/2 = >a(2) is both the sixth square and the 8th triangular number

a:=17+12*sqrt(2); b:=17-12*sqrt(2); A001110:=n -> expand((a^n + b^n - 2)/32); seq(A001110(n), n=0..20); (Spies)

A001110:=-(1+z)/((z-1)*(z**2-34*z+1)); [S. Plouffe in his 1992 dissertation.]

Cf. A001108, A001109.

Other S_r type sequences are S_4=A000290, S_5=A004146, S_7=A054493, S_8=A001108, S_9=A049684, S_20=A049683, S_36=this sequence, S_49=A049682, S_144=A004191^2.

Sequence in context: A004294 A075760 A113938 this_sequence A064196 A060786 A063819

Adjacent sequences: A001107 A001108 A001109 this_sequence A001111 A001112 A001113

nonn,easy,nice

njas

More terms from Larry Reeves (larryr(AT)acm.org), Apr 19 2000