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%I M4536 N1924 #270 Mar 01 2024 05:20:12
%S 0,1,8,49,288,1681,9800,57121,332928,1940449,11309768,65918161,
%T 384199200,2239277041,13051463048,76069501249,443365544448,
%U 2584123765441,15061377048200,87784138523761,511643454094368,2982076586042449,17380816062160328,101302819786919521
%N a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1.
%C b(0)=0, c(0)=1, b(i+1)=b(i)+c(i), c(i+1)=b(i+1)+b(i); then a(i) (the number in the sequence) is 2b(i)^2 if i is even, c(i)^2 if i is odd and b(n)=A000129(n) and c(n)=A001333(n). - Darin Stephenson (stephenson(AT)cs.hope.edu) and Alan Koch
%C For n > 1 gives solutions to A007913(2x) = A007913(x+1). - _Benoit Cloitre_, Apr 07 2002
%C If (X,X+1,Z) is a Pythagorean triple, then Z-X-1 and Z+X are in the sequence.
%C For n >= 2, a(n) gives exactly the positive integers m such that 1,2,...,m has a perfect median. The sequence of associated perfect medians is A001109. Let a_1,...,a_m be an (ordered) sequence of real numbers, then a term a_k is a perfect median if Sum_{j=1..k-1} a_j = Sum_{j=k+1..m} a_j. See Puzzle 1 in MSRI Emissary, Fall 2005. - _Asher Auel_, Jan 12 2006
%C This is the r=8 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.
%C Also, 1^3 + 2^3 + 3^3 + ... + a(n)^3 = k(n)^4 where k(n) is A001109. - Anton Vrba (antonvrba(AT)yahoo.com), Nov 18 2006
%C If T_x = y^2 is a triangular number which is also a square, the least number which is both triangular and square and greater than T_x is T_(3*x + 4*y + 1) = (2*x + 3*y + 1)^2 (W. Sierpiński 1961). - _Richard Choulet_, Apr 28 2009
%C If (a,b) is a solution of the Diophantine equation 0 + 1 + 2 + ... + x = y^2, then a or (a+1) is a perfect square. If (a,b) is a solution of the Diophantine equation 0 + 1 + 2 + ... + x = y^2, then a or a/8 is a perfect square. If (a,b) and (c,d) are two consecutive solutions of the Diophantine equation 0 + 1 + 2 + ... + x = y^2 with a < c, then a+b = c-d and ((d+b)^2, d^2-b^2) is a solution, too. If (a,b), (c,d) and (e,f) are three consecutive solutions of the Diophantine equation 0 + 1 + 2 + ... + x = y^2 with a < c < e, then (8*d^2, d*(f-b)) is a solution, too. - _Mohamed Bouhamida_, Aug 29 2009
%C If (p,q) and (r,s) are two consecutive solutions of the Diophantine equation 0 + 1 + 2 + ... + x = y^2 with p < r, then r = 3p + 4q + 1 and s = 2p + 3q + 1. - _Mohamed Bouhamida_, Sep 02 2009
%C Also numbers k such that (ceiling(sqrt(k*(k+1)/2)))^2 - k*(k+1)/2 = 0. - _Ctibor O. Zizka_, Nov 10 2009
%C From _Lekraj Beedassy_, Mar 04 2011: (Start)
%C Let x=a(n) be the index of the associated triangular number T_x=1+2+3+...+x and y=A001109(n) be the base of the associated perfect square S_y=y^2. Now using the identity S_y = T_y + T_{y-1}, the defining T_x = S_y may be rewritten as T_y = T_x - T_{y-1}, or 1+2+3+...+y = y+(y+1)+...+x. This solves the Strand Magazine House Number problem mentioned in A001109 in references from Poo-Sung Park and John C. Butcher. In a variant of the problem, solving the equation 1+3+5+...+(2*x+1) = (2*x+1)+(2*x+3)+...+(2*y-1) implies S_(x+1) = S_y - S_x, i.e., with (x,x+1,y) forming a Pythagorean triple, the solutions are given by pairs of x=A001652(n), y=A001653(n). (End)
%C If P = 8*n +- 1 is a prime, then P divides a((P-1)/2); e.g., 7 divides a(3) and 41 divides a(20). Also, if P = 8*n +- 3 is prime, then 4*P divides (a((P-1)/2) + a((P+1)/2 +3). - _Kenneth J Ramsey_, Mar 05 2012
%C Starting at a(2), a(n) gives all the dimensions of Euclidean k-space in which the ratio of outer to inner Soddy hyperspheres' radii for k+1 identical kissing hyperspheres is rational. The formula for this ratio is (1+3k+2*sqrt(2k*(k+1)))/(k-1) where k is the dimension. So for a(3) = 49, the ratio is 6 in the 49th dimension. See comment for A010502. - _Frank M Jackson_, Feb 09 2013
%C Conjecture: For n>1 a(n) is the index of the first occurrence of -n in sequence A123737. - _Vaclav Kotesovec_, Jun 02 2015
%C For n=2*k, k>0, a(n) is divisible by 8 (deficient), so since all proper divisors of deficient numbers are deficient, then a(n) is deficient. For n=2*k+1, k>0, a(n) is odd. If a(n) is a prime number, it is deficient; otherwise a(n) has one or two distinct prime factors and is therefore deficient again. sigma(a(5)) = 1723 < 3362 = 2*a(5). In either case, a(n) is deficient. - _Muniru A Asiru_, Apr 14 2016
%C The squares of NSW numbers (A008843) interleaved with twice squares from A084703, where A008843(n) = A002315(n)^2 and A084703(n) = A001542(n)^2. Conjecture: Also numbers n such that sigma(n) = A000203(n) and sigma(n-th triangular number) = A074285(n) are both odd numbers. - _Jaroslav Krizek_, Aug 05 2016
%C For n > 0, numbers for which the number of odd divisors of both n and of n + 1 is odd. - _Gionata Neri_, Apr 30 2018
%C a(n) will be solutions to some (A000217(k) + A000217(k+1))/2. - _Art Baker_, Jul 16 2019
%C For n >= 2, a(n) is the base for which A058331(A001109(n)) is a length-3 repunit. Example: for n=2, A001109(2)=6 and A058331(6)=73 and 73 in base a(2)=8 is 111. See Grantham and Graves. - _Michel Marcus_, Sep 11 2020
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 193.
%D 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 M. S. Klamkin, "International Mathematical Olympiads 1978-1985," (Supplementary problem N.T.6)
%D W. Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, pp. 21-22 MR2002669
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Indranil Ghosh, <a href="/A001108/b001108.txt">Table of n, a(n) for n = 0..1304</a> (terms 0..200 from T. D. Noe)
%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H M. A. Asiru, <a href="http://dx.doi.org/10.1080/0020739X.2016.1164346">All square chiliagonal numbers</a>, Int J Math Edu Sci Technol, 47:7(2016), 1123-1134.
%H Elwyn Berlekamp and Joe P. Buhler, <a href="http://www.msri.org/attachments/media/news/emissary/EmissaryFall2005.pdf">Puzzle Column</a>, Emissary, MSRI Newsletter, Fall 2005. Problem 1, (6 MB).
%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">De solutione problematum diophanteorum per numeros integros</a>, Par. 19.
%H H. G. Forder, <a href="http://www.jstor.org/stable/3613403">A Simple Proof of a Result on Diophantine Approximation</a>, Math. Gaz., 47 (1963), 237-238.
%H Jon Grantham and Hester Graves, <a href="https://arxiv.org/abs/2009.04052">The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits</a>, arXiv:2009.04052 [math.NT], 2020.
%H D. B. Hayes, <a href="http://www.americanscientist.org/libraries/documents/200884115366940-2008-09Hayes.pdf">Calculemus!</a>, American Scientist, 96 (Sep-Oct 2008), 362-366.
%H Olcay Karaatli and Refik Keskin, <a href="https://www.researchgate.net/profile/Olcay-Karaatli/publication/256302291_ON_SOME_DIOPHANTINE_EQUATIONS_RELATED_TO_SQUARE_TRIANGULAR_AND_BALANCING_NUMBERS/links/00b7d522456f03a97a000000/ON-SOME-DIOPHANTINE-EQUATIONS-RELATED-TO-SQUARE-TRIANGULAR-AND-BALANCING-NUMBERS.pdf">On some diophantine equations related to square triangular and balancing numbers</a>, J. Alg. Number Theory 4 (2) (2010) 71-89.
%H Refik Keskin and Olcay Karaatli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Karaatli/karaatli5.html">Some New Properties of Balancing Numbers and Square Triangular Numbers</a>, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4.
%H P. Lafer, <a href="http://www.fq.math.ca/Scanned/9-1/lafer.pdf">Discovering the square-triangular numbers</a>, Fib. Quart., 9 (1971), 93-105.
%H Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.
%H MSRI newsletter, <a href="http://www.msri.org/attachments/media/news/emissary/EmissaryFall2005.pdf">Emissary</a>, Fall 2005.
%H Vladimir Pletser, <a href="https://arxiv.org/abs/2101.00998">Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers</a>, arXiv:2101.00998 [math.NT], 2020.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H B. Polster and M. Ross, <a href="http://arxiv.org/abs/1503.04658">Marching in squares</a>, arXiv preprint arXiv:1503.04658 [math.HO], 2015.
%H K. Ramsey, <a href="http://groups.yahoo.com/group/Triangular_and_Fibonacci_Numbers/message/62">Generalized Proof re Square Triangular Numbers</a>.
%H K. Ramsey, <a href="/A001108/a001108.txt">Generalized Proof re Square Triangular Numbers</a>, digest of 2 messages in Triangular_and_Fibonacci_Numbers Yahoo group, May 27, 2005 - Oct 10, 2011.
%H D. L. Vestal, <a href="http://www.maa.org/reviews/pythtriangles.html">Review of "Pythagorean Triangles" (Chapter 4) by W. Sierpiński</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareTriangularNumber.html">Square Triangular Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>.
%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F a(0) = 0, a(n+1) = 3*a(n) + 1 + 2*sqrt(2*a(n)*(a(n)+1)). - _Jim Nastos_, Jun 18 2002
%F a(n) = floor( (1/4) * (3+2*sqrt(2))^n ). - _Benoit Cloitre_, Sep 04 2002
%F a(n) = A001653(k)*A001653(k+n) - A001652(k)*A001652(k+n) - A046090(k)*A046090(k+n). - _Charlie Marion_, Jul 01 2003
%F a(n) = A001652(n-1) + A001653(n-1) = A001653(n) - A046090(n) = (A001541(n)-1)/2 = a(-n). - _Michael Somos_, Mar 03 2004
%F a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). - _Antonio Alberto Olivares_, Oct 23 2003
%F a(n) = Sum_{r=1..n} 2^(r-1)*binomial(2n, 2r). - _Lekraj Beedassy_, Aug 21 2004
%F If n > 1, then both A000203(n) and A000203(n+1) are odd numbers: n is either a square or twice a square. - _Labos Elemer_, Aug 23 2004
%F a(n) = (T(n, 3)-1)/2 with Chebyshev's polynomials of the first kind evaluated at x=3: T(n, 3) = A001541(n). - _Wolfdieter Lang_, Oct 18 2004
%F G.f.: x*(1+x)/((1-x)*(1-6*x+x^2)). Binet form: a(n) = ((3+2*sqrt(2))^n + (3-2*sqrt(2))^n - 2)/4. - Bruce Corrigan (scentman(AT)myfamily.com), Oct 26 2002
%F a(n) = floor(sqrt(2*A001110(n))) = floor(A001109(n)*sqrt(2)) = 2*(A000129(n)^2) - (n mod 2) = A001333(n)^2 - 1 + (n mod 2). - _Henry Bottomley_, Apr 19 2000, corrected by _Eric Rowland_, Jun 23 2017
%F A072221(n) = 3*a(n) + 1. - _David Scheers_, Dec 25 2006
%F A028982(a(n)) + 1 = A028982(a(n) + 1). - _Juri-Stepan Gerasimov_, Mar 28 2011
%F a(n+1)^2 + a(n)^2 + 1 = 6*a(n+1)*a(n) + 2*a(n+1) + 2*a(n). - _Charlie Marion_, Sep 28 2011
%F a(n) = 2*A001653(m)*A053141(n-m-1) + A002315(m)*A046090(n-m-1) + a(m) with m < n; otherwise, a(n) = 2*A001653(m)*A053141(m-n) - A002315(m)*A001652(m-n) + a(m). See Link to Generalized Proof re Square Triangular Numbers. - _Kenneth J Ramsey_, Oct 13 2011
%F a(n) = A048739(2n-2), n > 0. - _Richard R. Forberg_, Aug 31 2013
%F From _Peter Bala_, Jan 28 2014: (Start)
%F A divisibility sequence: that is, a(n) divides a(n*m) for all n and m. Case P1 = 8, P2 = 12, Q = 1 of the 3-parameter family of linear divisibility sequences found by Williams and Guy.
%F a(2*n+1) = A002315(n)^2 = Sum_{k = 0..4*n + 1} Pell(n), where Pell(n) = A000129(n).
%F a(2*n) = (1/2)*A005319(n)^2 = 8*A001109(n)^2.
%F (2,1) entry of the 2 X 2 matrix T(n,M), where M = [0, -3; 1, 4] and T(n,x) is the Chebyshev polynomial of the first kind. (End)
%e a(1) = ((3 + 2*sqrt(2)) + (3 - 2*sqrt(2)) - 2) / 4 = (3 + 3 - 2) / 4 = 4 / 4 = 1;
%e a(2) = ((3 + 2*sqrt(2))^2 + (3 - 2*sqrt(2))^2 - 2) / 4 = (9 + 4*sqrt(2) + 8 + 9 - 4*sqrt(2) + 8 - 2) / 4 = (18 + 16 - 2) / 4 = (34 - 2) / 4 = 32 / 4 = 8, etc.
%p A001108:=-(1+z)/(z-1)/(z**2-6*z+1); # _Simon Plouffe_ in his 1992 dissertation, without the leading 0
%t Table[(1/2)(-1 + Sqrt[1 + Expand[8(((3 + 2Sqrt[2])^n - (3 - 2Sqrt[2])^n)/(4Sqrt[2]))^2]]), {n, 0, 100}] (* _Artur Jasinski_, Dec 10 2006 *)
%t Transpose[NestList[{#[[2]],#[[3]],6#[[3]]-#[[2]]+2}&,{0,1,8},20]][[1]] (* _Harvey P. Dale_, Sep 04 2011 *)
%t LinearRecurrence[{7, -7, 1}, {0, 1, 8}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2012 *)
%o (PARI) a(n)=(real((3+quadgen(32))^n)-1)/2
%o (PARI) a(n)=(subst(poltchebi(abs(n)),x,3)-1)/2
%o (PARI) a(n)=if(n<0,a(-n),(polsym(1-6*x+x^2,n)[n+1]-2)/4)
%o (PARI) x='x+O('x^99); concat(0, Vec(x*(1+x)/((1-x)*(1-6*x+x^2)))) \\ _Altug Alkan_, May 01 2018
%o (Haskell)
%o a001108 n = a001108_list !! n
%o a001108_list = 0 : 1 : map (+ 2)
%o (zipWith (-) (map (* 6) (tail a001108_list)) a001108_list)
%o -- _Reinhard Zumkeller_, Jan 10 2012
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)/((1-x)*(1-6*x+x^2)))); // _G. C. Greubel_, Jul 15 2018
%Y Cf. A001109, A001110, A007913, A000203, A084301, A001652, A072221.
%Y Partial sums of A002315. A000129, A005319.
%Y a(n) = A115598(n), n > 0. - _Hermann Stamm-Wilbrandt_, Jul 27 2014
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 19 2000
%E More terms from _Lekraj Beedassy_, Aug 21 2004
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