A077261 Triangular numbers that are 5 times another triangular number.
0, 15, 105, 4950, 33930, 1594005, 10925475, 513264780, 3517969140, 165269665275, 1132775137725, 53216318953890, 364750076378430, 17135489433487425, 117448391818716855, 5517574381263997080, 37818017415550449000, 1776641815277573572455, 12177284159415425861265
Offset: 0
Examples
a(3)=5*990=4950.
Links
- Colin Barker, Table of n, a(n) for n = 0..797
- 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 (1,322,-322,-1,1).
Programs
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Mathematica
CoefficientList[Series[(-15 x (x^2 + 6 x + 1))/((x - 1) (x^2 - 18 x + 1) (x^2 + 18 x + 1)), {x, 0, 18}], x] (* Michael De Vlieger, Apr 21 2021 *)
Formula
a(n) = 5*A077260(n).
G.f.: (-15*x*(x^2+6*x+1))/((x-1)*(x^2-18*x+1)*(x^2+18*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = 322*a(n-2) - a(n-4) + 120. - Vladimir Pletser, Feb 09 2021
E.g.f.: (-6*cosh(x) - (-3 + sqrt(5))*cosh((9 - 4*sqrt(5))*x) + (3 + sqrt(5))*cosh((9 + 4*sqrt(5))*x) - 6*sinh(x) + (7 - 3*sqrt(5))*sinh((9 - 4*sqrt(5))*x) + (7 + 3*sqrt(5))*sinh((9 + 4*sqrt(5))*x))/16. - Stefano Spezia, Aug 15 2024