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A001108 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.
(Formerly M4536 N1924)
43
0, 1, 8, 49, 288, 1681, 9800, 57121, 332928, 1940449, 11309768, 65918161, 384199200, 2239277041, 13051463048, 76069501249, 443365544448, 2584123765441, 15061377048200, 87784138523761, 511643454094368, 2982076586042449, 17380816062160328, 101302819786919521 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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) - from stephenson(AT)cs.hope.edu (Darin Stephenson and Alan Koch)

For n>1 gives solutions to A007913(2x)=A007913(x+1) - Benoit Cloitre, Apr 07 2002

If (X,X+1,Z) is a Pythagorean triple, then Z-X-1 and Z+X is in the sequence.

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_{1<=j<k} a_j = sum_{k<j<=m} a_j. See Puzzle 1 in MSRI Emissary, Fall 2005. - Asher Auel (auela(AT)math.upenn.edu), Jan 12 2006

This is the r=8 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.

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

The sequence lists the numbers n for which Sum_{i=0..n}{i} is a perfect square. - Paolo P. Lava, Nov 28 2007

If T_x=y^2 is a triangular number which is also a square, the least both triangular and square number which is greater as T_x is T_(3*x+4*y+1)=(2*x+3*y+1)^2 (W. Sierpinski 1961). [Richard Choulet, Apr 28 2009]

The remainder of the division of a(n) by 5 is: 0, 1, 3 or 4. The remainder of the division of a(n) by 7 is: 0 or 1. [Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Aug 26 2009]

Number of units of a(n) belongs to a periodic sequence: 0, 1, 8, 9, 8, 1. The remainder of the division of a(n) by 5 belongs to a periodic sequence: 0, 1, 3, 4, 3, 1. [Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 01 2009]

If (a,b) is a solution of the Diophantine equation: 0+1+2+...+x=y^2 then a or (a+1) are perfect squares. If (a,b) is a solution of the Diophantine equation: 0+1+2+...+x=y^2 then a or a/8 are perfect squares. 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 (bhmd95(AT)yahoo.fr), Aug 29 2009]

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 (bhmd95(AT)yahoo.fr), Sep 02 2009]

Also numbers n such that (ceiling(sqrt(n*(n+1)/2)))^2 - n*(n+1)/2 = 0. [Ctibor O. Zizka, Nov 10 2009]

Contribution from Lekraj Beedassy, Mar 04 2011: (Start)

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)

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 Ramsey, Mar 05 2012]

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

REFERENCES

I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.

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

Elwyn Berlekamp and Joe P. Buhler, Puzzle Column, Emissary, MSRI Newsletter, Fall 2005. Problem 1.

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.

Refik Keskin and Olcay Karaatli, Some New Properties of Balancing Numbers and Square Triangular Numbers, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4

M. S. Klamkin, "International Mathematical Olympiads 1978-1985," (Supplementary problem N.T.6)

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

W. Sierpinski, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, pp. 21-22 MR2002669

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

L. Euler, De solutione problematum diophanteorum per numeros integros, Par. 19

MSRI newsletter, Emissary, Fall 2005

D. B. Hayes, Calculemus!, American Scientist, 96 (Sep-Oct 2008), 362-366.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

K. Ramsey, Generalized Proof re Square Triangular Numbers

D. L. Vestal, Review of "Pythagorean Triangles" (Chapter 4) by W. Sierpinski

Eric Weisstein's World of Mathematics, Square Triangular Number.

Eric Weisstein's World of Mathematics, Triangular Number.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for sequences related to Chebyshev polynomials.

Index entries for sequences related to linear recurrences with constant coefficients, signature (7,-7,1).

Index entries for two-way infinite sequences

FORMULA

a(0) = 0, a(n+1) = 3*a(n) + 1 + 2*sqrt(2*a(n)*(a(n)+1)). - Jim Nastos, Jun 18 2002

a(n) = floor( (1/4) * (3+2*sqrt(2))^n ). - Benoit Cloitre, Sep 04 2002

a(n) = A001653(k)*A001653(k+n) - A001652(k)*A001652(k+n) -A046090(k)*A046090(k+n). - Charlie Marion, Jul 01 2003

a(n) = A001652(n-1)+A001653(n-1) =A001653(n)-A046090(n) =(A001541(n)-1)/2 =a(-n). - Michael Somos Mar 03 2004

a(n) = 7*a(n-1) -7*a(n-2) +a(n-3). - Antonio Olivares, Oct 23 2003

a(n) = sum_(r=1, ..., n) 2^(r-1)*binomial(2n, 2r). - Lekraj Beedassy, Aug 21 2004

If n>1, then both A000203[n] and A000203[n+1] are odd numbers: n is either square or twice square. - Labos Elemer, Aug 23 2004

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

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

a(n) = floor(sqrt(2*A001110(n))) = floor(A001109(n)*sqrt(2)) = 2*(A00012 9(n)^2)+[n mod 2]=A001333^2+1-[n mod 2] - Henry Bottomley, Apr 19 2000

A072221(n) = 3*a(n) + 1. - David Scheers, Dec 25 2006

A028982(a(n))+1=A028982(a(n)+1). [From Juri-Stepan Gerasimov, Mar 28 2011]

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

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 Ramsey_, Oct 13 2011

a(n) = A048739(2n-2), n > 0. - Richard R. Forberg, Aug 31 2013

From Peter Bala, Jan 28 2014: (Start)

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.

a(2*n+1) = A002315(n)^2 = sum {k = 0..4*n + 1} Pell(n), where Pell(n) = A000129(n).

a(2*n) = 1/2*A005319(n)^2 = 8*A001109(n)^2.

(2,1) entry of the 2X2 matrix T(n,M), where M = [0, -3; 1, 4] and T(n,x) is the Chebyshev polynomial of the first kind. (End)

EXAMPLE

a(1)=((3+2*sqrt(2))+(3-2*sqrt(2))-2)/4=(3+3-2)/4=4/4=1 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.

MAPLE

A001108:=-(1+z)/(z-1)/(z**2-6*z+1); [Simon Plouffe in his 1992 dissertation without the leading 0.]

MATHEMATICA

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

Transpose[NestList[{#[[2]], #[[3]], 6#[[3]]-#[[2]]+2}&, {0, 1, 8}, 20]][[1]] (* Harvey P. Dale, Sep 04 2011 *)

LinearRecurrence[{7, -7, 1}, {0, 1, 8}, 50] (* From Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)

PROG

(PARI) a(n)=(real((3+quadgen(32))^n)-1)/2

(PARI) a(n)=(subst(poltchebi(abs(n)), x, 3)-1)/2

(PARI) a(n)=if(n<0, a(-n), (polsym(1-6*x+x^2, n)[n+1]-2)/4)

(Haskell)

a001108 n = a001108_list !! n

a001108_list = 0 : 1 : map (+ 2)

   (zipWith (-) (map (* 6) (tail a001108_list)) a001108_list)

-- Reinhard Zumkeller, Jan 10 2012

CROSSREFS

Cf. A001109, A001110, A007913, A000203, A084301, A001652, A072221.

Partial sums of A002315. A000129, A005319.

Sequence in context: A200660 A028443 * A115598 A097204 A037539 A037483

Adjacent sequences:  A001105 A001106 A001107 * A001109 A001110 A001111

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

More terms from Lekraj Beedassy, Aug 21 2004

STATUS

approved

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Last modified April 17 20:01 EDT 2014. Contains 240655 sequences.