0,2

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

Triangular numbers that are also pronics. 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 (guptass(AT)rediffmail.com), 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 (genewardsmith(AT)gmail.com), Sep 30 2006

Shyam Sunder Gupta Fascinating Triangular Numbers

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

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 (genewardsmith(AT)gmail.com), Sep 30 2006

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

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

Table[ Floor[ N[ (Sqrt[ 2 ]+1)^(4n+2)/32 ]], {n, 0, 20} ],

Cf. A123478, A123479, A123480, A123482.

Cf. A075528 = triangular numbers that are half other triangular numbers.

Equals A029546/6.

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

Sequence in context: A084694 A065945 A076715 this_sequence A087639 A028350 A099788

Adjacent sequences: A029546 A029547 A029548 this_sequence A029550 A029551 A029552

nonn

Don N. Page (don(AT)phys.ualberta.ca)

Additional comments from Christian G. Bower (bowerc(AT)usa.net), Sep 19 2002; Shyam Sunder Gupta (guptass(AT)rediffmail.com), Oct 26 2002; T. D. Noe (noe(AT)sspectra.com), Nov 07 2006; and others.

Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 18 2007, following suggestion from Andrew Plewe and Tanya Khovanova.