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 A077290 Triangular numbers that are 6 times other triangular numbers. 11
 0, 6, 36, 630, 3570, 61776, 349866, 6053460, 34283340, 593177346, 3359417496, 58125326490, 329188631310, 5695688818716, 32257126450926, 558119378907720, 3160869203559480, 54690003444137886, 309732924822378156, 5359062218146605150, 30350665763389499850 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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. Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1). 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 EXAMPLE 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 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 *) PROG (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 (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 CROSSREFS Cf. A077288, A077289, A077291. Subsequence of A000217. Sequence in context: A195229 A339300 A296389 * A222780 A206636 A244841 Adjacent sequences:  A077287 A077288 A077289 * A077291 A077292 A077293 KEYWORD easy,nonn AUTHOR Bruce Corrigan (scentman(AT)myfamily.com), Nov 03 2002 EXTENSIONS More terms from Joerg Arndt, Jul 03 2013 STATUS approved

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Last modified July 24 11:17 EDT 2021. Contains 346273 sequences. (Running on oeis4.)