A200994 Triangular numbers, T(m), that are three-halves of another triangular number; T(m) such that 2*T(m) = 3*T(k) for some k.
0, 15, 1485, 145530, 14260470, 1397380545, 136929032955, 13417647849060, 1314792560174940, 128836253249295075, 12624638025870742425, 1237085690282083462590, 121221773009618308591410, 11878496669252312158495605, 1163971451813716973223977895
Offset: 0
Examples
2*0 = 3*0. 2*15 = 3*10. 2*1485 = 3*990. 2*145530 = 3*97020.
Links
- Colin Barker, Table of n, a(n) for n = 0..500
- Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
- Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
- Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
- Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
- Index entries for linear recurrences with constant coefficients, signature (99,-99,1).
Programs
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Magma
m:=25; R
:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(15*x/((1-x)*(1-98*x+x^2)))); // G. C. Greubel, Jul 15 2018 -
Mathematica
LinearRecurrence[{99, -99, 1}, {0, 15, 1485}, 20] (* T. D. Noe, Feb 15 2012 *) -
PARI
concat(0, Vec(15*x/((1-x)*(1-98*x+x^2)) + O(x^20))) \\ Colin Barker, Mar 02 2016
Formula
From Colin Barker, Mar 02 2016: (Start)
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>2.
G.f.: 15*x / ((1-x)*(1-98*x+x^2)). (End)
a(n) = (-10+(5-2*sqrt(6))*(49+20*sqrt(6))^(-n)+(5+2*sqrt(6))*(49+20*sqrt(6))^n)/64. - Colin Barker, Mar 03 2016
Comments