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URL: https://oeis.org/A200998

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A200998
Triangular numbers, T(m), that are three-quarters of another triangular number: T(m) such that 4*T(m)=3*T(k) for some k.
2
0, 21, 4095, 794430, 154115346, 29897582715, 5799976931385, 1125165627105996, 218276331681631860, 42344483180609474865, 8214611460706556491971, 1593592278893891349967530, 309148687493954215337208870, 59973251781548223884068553271, 11634501696932861479293962125725
OFFSET
0,2
COMMENTS
For n>1, a(n) = 194*a(n-1) - a (n-2) + 21. See A200993 for generalization.
FORMULA
G.f.: (21*x)/(1 - 195*x + 195*x^2 - x^3).
From Colin Barker, Mar 02 2016: (Start)
a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n > 2.
a(n) = ((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(-7+4*sqrt(3)+(7+4*sqrt(3))*(97+56*sqrt(3))^n))/128. (End)
Sum_{n>=1} 1/a(n) = (2/3) * (7 - 4*sqrt(3)). - Amiram Eldar, Jan 27 2026
EXAMPLE
4*0 = 3*0.
4*21 = 3*28.
4*4095 = 3*5640.
4*794430 = 3*1059240.
MATHEMATICA
LinearRecurrence[{195, -195, 1}, {0, 21, 4095}, 30] (* Vincenzo Librandi, Mar 03 2016 *)
PROG
(PARI) concat(0, Vec(21/(1 - 195*x + 195*x^2 - x^3) + O(x^99))) \\ Charles R Greathouse IV, Dec 20 2011
(Magma) I:=[0, 21]; [n le 2 select I[n] else 194*Self(n-1) - Self(n-2) + 21: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016
KEYWORD
nonn,easy
AUTHOR
Charlie Marion, Dec 20 2011
STATUS
approved