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A029549 a(0) = 0, a(1) = 6, a(2) = 210; for n >= 0, a(n+3) = 35*a(n+2) - 35*a(n+1) + a(n). 30
0, 6, 210, 7140, 242556, 8239770, 279909630, 9508687656, 323015470680, 10973017315470, 372759573255306, 12662852473364940, 430164224521152660, 14612920781245825506, 496409142337836914550, 16863297918705209269200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Triangular numbers that are twice other triangular numbers. - Don N. Page

Triangular numbers that are also pronic numbers. These will be shown to have a Pythagorean connection in a paper in preparation. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Mar 09 2002

In other words, triangular numbers which are products of two consecutive numbers. E.g., a(2)=210: 210 is a triangular number which is the product of two consecutive numbers: 14*15. - Shyam Sunder Gupta, Oct 26 2002

Coefficients of the series giving the best rational approximations to sqrt(8). The partial sums of the series 3 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(8) = 2 sqrt(2), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [2;1,4,1], [2;1,4,1,4,1], [2;1,4,1,4,1,4,1], [2;1,4,1,4,1,4,1,4,1] and so forth. - Gene Ward Smith, Sep 30 2006

The a(n) satisfy the same recurrence relation that defines the terms of A165518. - Ant King, Dec 13 2010

Intersection of A000217 and A002378.

This is the sequence of areas, x(n)*y(n)/2, of the ordered Pythagorean triples (x(n),y(n)=x(n)+1,z(n)) with x(0)=0,y(0)=1,z(0)=1,a(0)=0 and x(1)=3,y(1)=4,z(1)=5,a(1)=6. - George F. Johnson, Aug 20 2012

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

H. J. Hindin, Stars, hexes, triangular numbers and Pythagorean triples, J. Rec. Math., 16 (1983/1984), 191-193. (Annotated scanned copy)

Shyam Sunder Gupta Fascinating Triangular Numbers

Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

FORMULA

G.f.: 6x/(1-35*x+35*x^2-x^3) =  -6*x / ( (x-1)*(x^2-34*x+1) ).

a(n) = 6*A029546(n-1) = 2*A075528(n).

a(n) = -3/16 + (3/32 + (1/16)*2^(1/2)) *(17 + 12*2^(1/2))^n + (3/32 - (1/16)*2^(1/2)) *(17 - 12*2^(1/2))^n. - Gene Ward Smith, Sep 30 2006

a(n) = (cosh((4*n+2)*log(1+sqrt(2)))-3)/16. a(n) = binomial(A001652+1,2) = 2*binomial(A053141+1,2). - Bill Gosper, Feb 07 2010

a(n) = binomial(A001652(n), 2) = A000217(A001652(n)). - Mitch Harris, Apr 19 2007, R. J. Mathar, Jun 26 2009

a(n) = ceiling((3 + 2sqrt(2))^(2n+1) - 6)/32 = floor((1/32) (1+sqrt(2))^(4n+2)). - Ant King Dec 13 2010

Sum_{n>=1} 1/a(n) = 3 - 2*sqrt(2) = A157259 - 4. - Ant King, Dec 13 2010

a(n) = a(n-1) + A001109(2n). - Charlie Marion, Feb 10 2011

a(n+2) = 34*a(n+1) - a(n) + 6. - Charlie Marion, Feb 11 2011

From George F. Johnson, Aug 20 2012: (Start)

a(n) = ((3 + 2*sqrt(2))^(2*n+1) + (3 - 2*sqrt(2))^(2*n+1)-6)/32.

8*a(n)+1 = (A002315(n))^2, 4*a(n) + 1 = (A000129(2*n+1))^2, 32*a(n)^2 + 12*a(n) + 1 are perfect squares.

a(n+1) = 17*a(n) + 3 + 3*sqrt((8*a(n)+1)*(4*a(n)+1)).

a(n-1) = 17*a(n) + 3 - 3*sqrt((8*a(n)+1)*(4*a(n)+1)).

a(n-1)*a(n+1) = a(n)*(a(n)-6), a(n) = A096979(2*n).

a(n) = (1/2)*A084159(n)*A046729(n) = (1/2)*A001652(n)*A046090(n).

Lim_{n -> inf} a(n)/a(n-1) = 17 + 12*sqrt(2),

Lim_{n -> inf} a(n)/a(n-2) = (17 + 12*sqrt(2))^2 = 577 + 408*sqrt(2),

Lim_{n -> inf} a(n)/a(n-r) = (17 + 12*sqrt(2))^r,

Lim_{n -> inf} a(n-r)/a(n) = (17 + 12*sqrt(2))^(-r) = (17 - 12*sqrt(2))^r. (End)

a(n) = 3 * T( b(n) ) + (2*b(n) + 1)*sqrt( T( b(n) ) ) where b(n) = A001108(n) (indices of the square triangular numbers), T(n) = A000217(n) (the n-th triangular number). - Dimitri Papadopoulos, Jul 07 2017

MAPLE

A029549 := proc(n)

    option remember;

    if n <= 1 then

        op(n+1, [0, 6]) ;

    else

        34*procname(n-1)-procname(n-2)+6 ;

    end if;

end proc: # R. J. Mathar, Feb 05 2016

MATHEMATICA

Table[ Floor[ (Sqrt[ 2 ]+1)^(4n+2)/32 ], {n, 0, 20} ] (* Original program from author, corrected by Ray Chandler, Jul 09 2015 *)

CoefficientList[Series[6/(1 - 35*x + 35*x^2 - x^3), {x, 0, 14}], x]

Intersection[ #, 2*# ]&@Table[ Binomial[ n, 2 ], {n, 999999} ] (* Bill Gosper, Feb 07 2010 *)

LinearRecurrence[{35, -35, 1}, {0, 6, 210}, 20] (* Harvey P. Dale, Jun 06 2011 *)

PROG

(Macsyma) (makelist(binom(n, 2), n, 1, 999999), intersection(%%, 2*%%)) /* Bill Gosper, Feb 07 2010 */

(Haskell)

a029549 n = a029549_list !! n

a029549_list = [0, 6, 210] ++

   zipWith (+) a029549_list

               (map (* 35) $ tail delta)

   where delta = zipWith (-) (tail a029549_list) a029549_list

-- Reinhard Zumkeller, Sep 19 2011

(PARI) concat(0, Vec(6/(1-35*x+35*x^2-x^3)+O(x^98))) \\ Charles R Greathouse IV, Jun 13 2013

(MAGMA) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat  Coefficients(R!(6/(1-35*x+35*x^2-x^3))); // G. C. Greubel, Jul 15 2018

CROSSREFS

Cf. A123478, A123479, A123480, A123482, A075528, A082405 (first differences).

Cf. A245031, A009111.

Cf. A001108.

Sequence in context: A285149 A065945 A076715 * A183252 A183287 A087639

Adjacent sequences:  A029546 A029547 A029548 * A029550 A029551 A029552

KEYWORD

nonn,easy

AUTHOR

Don N. Page

EXTENSIONS

Additional comments from Christian G. Bower, Sep 19 2002; Shyam Sunder Gupta, Oct 26 2002; T. D. Noe, Nov 07 2006; and others

Edited by N. J. A. Sloane, Apr 18 2007, following suggestions from Andrew S. Plewe and Tanya Khovanova

STATUS

approved

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Last modified December 10 05:49 EST 2018. Contains 318044 sequences. (Running on oeis4.)