%I A001110 M5259 N2291
%S A001110 0,1,36,1225,41616,1413721,48024900,1631432881,55420693056,
%T A001110 1882672131025,63955431761796,2172602007770041,73804512832419600,
%U A001110 2507180834294496361,85170343853180456676,2893284510173841030625
%N A001110 a(0) = 0, a(1) = 1; for n >= 2, a(n) = 34a(n-1) - a(n-2) + 2.
%C A001110 These are the numbers that are both triangular and square.
%C A001110 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
%C A001110 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
%D A001110 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001110 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001110 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 193.
%D A001110 Tom Beldon and Tony Gardiner, ``Triangular numbers and perfect squares'', 
               The Mathematical Gazette, 2002, pp. 423--431, esp pp. 424--426.
%D A001110 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.
%D A001110 H. G. Forder, A Simple Proof of a Result on Diophantine Approximation, 
               Math. Gaz., 47 (1963), 237-238.
%D A001110 Martin Gardner, Time Travel and other Mathematical Bewilderments, pp. 
               16-17, Freeman 1988
%D A001110 P. Lafer, Discovering the square-triangular numbers, Fib. Quart., 9 (1971), 
               93-105.
%D A001110 D. A. Q., Triangular square numbers - a postscript, Math. Gaz., 56 (1972), 
               311-314.
%D A001110 J. H. Silverman, A Friendly Introduction to Number Theory, p 196, Prentice 
               Hall 2001
%H A001110 T. D. Noe, <a href="b001110.txt">Table of n, a(n) for n=0..100</a>
%H A001110 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001110 K. S. Brown, <a href="http://www.mathpages.com/home/kmath159.htm">Square 
               Triangular Numbers</a>
%H A001110 Yong-Gao Chen and Jin-Hui Fang, <a href="http://www.integers-ejcnt.org/
               vol7.html">Triangular numbers in geometric progression,</a> INTEGERS 
               7 (2007), #A19.
%H A001110 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001110 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001110 R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof 
               of a nonlinearity</a>
%H A001110 Chris Thatcher, <a href="http://webpages.shepherd.edu/CTHATC01/numbertheory1_11.html">
               Square Triangular Numbers</a> [Broken link?]
%H A001110 Eric Weisstein, CRC Online Dictionary, <a href="http://br.crashed.net/
               ~akrowne/crc/math/s/s649.htm">Square Triangular Number</a>
%H A001110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareTriangularNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A001110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TriangularNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A001110 Wikipedia, <a href="http://en.wikipedia.org/wiki/Triangular_square_number">
               Square triangular number</a>
%F A001110 G.f.: x*(1 + x) / (( 1 - x )*( 1 - 34 x + x^2 )).
%F A001110 a(n-1) * a(n+1) = (a(n)-1)^2. - Colin Dickson, posting to alt.math.recreational, 
               circa Mar 13 2004
%F A001110 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
%F A001110 a(n)-a(n-1)=A001109(2n-1). - Sophie Kuo (ejiqj_6(AT)yahoo.com.tw), May 
               27 2006
%F A001110 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.
%F A001110 a(n)=(((17+12*sqrt(2))^n)+((17-12*sqrt(2))^n)-2)/32 - Bruce Corrigan 
               (scentman(AT)myfamily.com), Oct 26 2002
%F A001110 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
%F A001110 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
%F A001110 a(n+1)=[17*A029547(n)-A091761(n)-1]/16. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 16 2007
%F A001110 a(n) = A001333^2 * A000129^2 = A000129[2n]^2/4 = binom(A001108,2). - 
               R. W. Gosper (rwg(AT)sdf.lonestar.org), Jul 28 2008
%F A001110 Comment from R. W. Gosper (rwg(AT)sdf.lonestar.org), Jul 25 2008: Closed 
               form (as square = triangular): ( (sqrt(2) + 1)^(2n)/(4 sqrt(2)) - 
               (1 - sqrt(2))^(2n)/(4 sqrt(2)) )^2 = (1/2) * ( ( (sqrt(2) + 1)^n 
               / 2 - (sqrt(2) - 1)^n / 2 )^2 + 1 ) ( (sqrt(2) + 1)^n / 2 - (sqrt(2) 
               - 1)^n / 2 )^2.
%e A001110 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
%p A001110 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)
%p A001110 A001110:=-(1+z)/((z-1)*(z**2-34*z+1)); [S. Plouffe in his 1992 dissertation.]
%Y A001110 Cf. A001108, A001109.
%Y A001110 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.
%Y A001110 A001014; intersection of A000217 and A000290; A010052(a(n))*A010054(a(n))=1. 
               [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 27 
               2008]
%Y A001110 Sequence in context: A004294 A075760 A113938 this_sequence A064196 A060786 
               A063819
%Y A001110 Adjacent sequences: A001107 A001108 A001109 this_sequence A001111 A001112 
               A001113
%K A001110 nonn,easy,nice
%O A001110 0,3
%A A001110 N. J. A. Sloane (njas(AT)research.att.com).
%E A001110 More terms from Larry Reeves (larryr(AT)acm.org), Apr 19 2000