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 A029549 a(n + 3) = 35*a(n + 2) - 35*a(n + 1) + a(n), with a(0) = 0, a(1) = 6, a(2) = 210. 37
 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 This sequence satisfy the same recurrence as 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. Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021. Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021. Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021. Vladimir Pletser, Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2103.03019 [math.GM], 2021. Vladimir Pletser, Searching for multiple of triangular numbers being triangular numbers, 2021. Index entries for linear recurrences with constant coefficients, signature (35,-35,1). FORMULA G.f.: 6*x/(1 - 35*x + 35*x^2 - x^3) = 6*x /( (1-x)*(1 - 34*x + x^2) ). a(n) = 6*A029546(n-1) = 2*A075528(n). a(n) = -3/16 + ((3+2*sqrt(2))/32) *(17 + 12*sqrt(2))^n + ((3-2*sqrt(2))/32) *(17 - 12*sqrt(2))^n. - Gene Ward Smith, Sep 30 2006 From Bill Gosper, Feb 07 2010: (Start) a(n) = (cosh((4*n + 2)*log(1 + sqrt(2))) - 3)/16. a(n) = binomial(A001652(n) + 1, 2) = 2*binomial(A053141(n) + 1, 2). (End) a(n) = binomial(A001652(n), 2) = A000217(A001652(n)). - Mitch Harris, Apr 19 2007, R. J. Mathar, Jun 26 2009 a(n) = ceiling((3 + 2*sqrt(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 a(n) = (Pell(2*n + 1)^2 - 1)/4 = (Q(4*n + 2) - 6)/32, where Q(n) are the Pell-Lucas numbers (A002203). - G. C. Greubel, Jan 13 2020 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 + 1)^(4n + 2)/32], {n, 0, 20} ] (* Original program from author, corrected by Ray Chandler, Jul 09 2015 *) CoefficientList[Series[6/(1 - 35x + 35x^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 *) (LucasL[4Range - 2, 2] -6)/32 (* G. C. Greubel, Jan 13 2020 *) 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^25))) \\ Charles R Greathouse IV, Jun 13 2013 (MAGMA) R:=PowerSeriesRing(Integers(), 25);  cat  Coefficients(R!(6/(1-35*x+35*x^2-x^3))); // G. C. Greubel, Jul 15 2018 (Scala) val triNums = (0 to 39999).map(n => (n * n + n)/2) triNums.filter(_ % 2 == 0).filter(n => (triNums.contains(n/2))) // Alonso del Arte, Jan 12 2020 (Sage) [(lucas_number2(4*n+2, 2, -1) -6)/32 for n in (0..20)] # G. C. Greubel, Jan 13 2020 (GAP) List([0..20], n-> (Lucas(2, -1, 4*n+2) -6)/32 ); # G. C. Greubel, Jan 13 2020 CROSSREFS Cf. A123478, A123479, A123480, A123482, A075528, A082405 (first differences). Cf. A000129, A001108, A002203, A009111, A245031. Sequence in context: A285149 A065945 A076715 * A183252 A183287 A087639 Adjacent sequences:  A029546 A029547 A029548 * A029550 A029551 A029552 KEYWORD nonn,easy AUTHOR EXTENSIONS Additional comments from Christian G. Bower, Sep 19 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 October 19 21:02 EDT 2021. Contains 348091 sequences. (Running on oeis4.)