1,1

There may be values that are not given in the recurrence shown. This sequence is suggested by Ulas, p.11, who supplied the recurrence. Abstract: In this paper we give solutions of certain Diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians - Number Theory, Krakow, 14-20 September 2008.

27*10^9 < a(19) <= 39991371446. a(20) <= 104637102152. a(21) <= 227490888350. a(22) <= 1497809326860. [From Donovan Johnson (donovan.johnson(AT)yahoo.com), Jan 24 2009]

Maciej Ulas, On certain Diophantine equations related to triangular and tetrahedral numbers, Nov 15, 2008.

a(n) = T(i)*T(j) where T(k) = A000292(k) = C(k+2,3) = k*(k+1)*(k+2)/6. If we define sequences u(n), v(n) recursively in the following way: u(0) = 1, v(0) = 1, u(n) = 17*u(n-1) + 36*v(n-1) + 20, v(n) = 8*u(n-1) + 17*v(n-1) + 10, then for each natural number n the following identity holds (v(n)*u(n)*(u(n + 1)))^2 = T(u(n))*T(2*u(n)). But this does not exclude the possibility that there are other elements of a(n) that do not come from this recurrence.

a(1) = 189070 because 189070^2 = T(73)*T(146). a(2) = 7559616818 because 7559616818^2 = T(2521)*T(5042).

2^2=4*1=T(2)*T(1). 140^2=560*35=T(14)*T(5)=19600*1=T(48)*T(1). 1092^2=3276*364=T(26)*T(12). 280^2=19600*4=T(48)*T(2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 22 2009]

Cf. A000292.

Sequence in context: A094482 A101232 A093887 this_sequence A140898 A120814 A115890

Adjacent sequences: A152002 A152003 A152004 this_sequence A152006 A152007 A152008

nonn,more

Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 19 2008

Replaced by sequence with no intermediate terms missing. R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 22 2009

a(15)-a(18) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Jan 24 2009