A077290 Triangular numbers that are 6 times other triangular numbers.
0, 6, 36, 630, 3570, 61776, 349866, 6053460, 34283340, 593177346, 3359417496, 58125326490, 329188631310, 5695688818716, 32257126450926, 558119378907720, 3160869203559480, 54690003444137886, 309732924822378156, 5359062218146605150, 30350665763389499850
Offset: 0
Examples
The k-th triangular number is T(k) = k*(k+1)/2, so T(35)/T(14) = (35*36/2)/(14*15/2) = 630/105 = 6, so T(35)=630 is a term. - _Jon E. Schoenfield_, Feb 20 2021
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
- Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
- Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2022.
- Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).
Programs
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Maple
f := gfun:-rectoproc({a(-2) = 6, a(-1) = 0, a(0) = 0, a(1) = 6, a(n) = 98*a(n-2)-a(n-4)+42}, a(n), remember); map(f, [`$`(0 .. 1000)])[]; # Vladimir Pletser, Feb 20 2021 -
Mathematica
CoefficientList[Series[-6 x (x^2 + 5 x + 1)/((x - 1) (x^2 - 10 x + 1) (x^2 + 10 x + 1)), {x, 0, 20}], x] (* Michael De Vlieger, Apr 21 2021 *) -
PARI
T(n)=n*(n+1)\2; istriang(n)=issquare(8*n+1); for(n=0,10^10, t=T(n); if ( t%6==0 && istriang(t\6), print1(t,", ") ) ); \\ Joerg Arndt, Jul 03 2013
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PARI
concat(0, Vec(-6*x*(x^2+5*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))) \\ Colin Barker, May 15 2015
Formula
a(n) = 6*A077289(n).
G.f.: -6*x*(x^2+5*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)). - Colin Barker, Jul 02 2013
a(n) = 98*a(n-2) - a(n-1) + 42. - Vladimir Pletser, Feb 20 2021
Extensions
More terms from Joerg Arndt, Jul 03 2013