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%I #36 Sep 08 2022 08:46:17
%S 0,5,30,175,1020,5945,34650,201955,1177080,6860525,39986070,233055895,
%T 1358349300,7917039905,46143890130,268946300875,1567533915120,
%U 9136257189845,53250009223950,310363798153855,1808932779699180,10543232880041225,61450464500548170
%N Values of m such that m^2 + 3 is a triangular number (A000217).
%H Colin Barker, <a href="/A276598/b276598.txt">Table of n, a(n) for n = 1..1000</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, <a href="https://hal.archives-ouvertes.fr/hal-02918958/document#page=18">Integer sequences and ellipse chains inside a hyperbola</a>, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(n) = 5*A001109(n-1).
%F a(n) = 5*( (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n - (3 + 2*sqrt(2))*(3 - 2*sqrt(2))^n )/(4*sqrt(2)).
%F a(n) = 6*a(n-1) - a(n-2) for n>2.
%F G.f.: 5*x^2 / (1-6*x+x^2).
%F a(n) = (5/2)*A000129(2*n-2). - _G. C. Greubel_, Sep 15 2021
%e 5 is in the sequence because 5^2 + 3 = 28, which is a triangular number.
%t CoefficientList[Series[5*x/(1 - 6*x + x^2), {x, 0, 20}], x] (* _Wesley Ivan Hurt_, Sep 07 2016 *)
%t LinearRecurrence[{6,-1},{0,5},30] (* _Harvey P. Dale_, Apr 26 2019 *)
%t (5/2)*Fibonacci[2*Range[30] -2, 2] (* _G. C. Greubel_, Sep 15 2021 *)
%o (PARI) concat(0, Vec(5*x^2/(1-6*x+x^2) + O(x^30)))
%o (PARI) a(n)=([0,1;-1,6]^n*[-5;0])[1,1] \\ _Charles R Greathouse IV_, Sep 07 2016
%o (Magma) [n le 2 select 5*(n-1) else 6*Self(n-1) - Self(n-2): n in [1..31]]; // _G. C. Greubel_, Sep 15 2021
%o (Sage) [(5/2)*lucas_number1(2*n-2, 2, -1) for n in (1..30)] # _G. C. Greubel_, Sep 15 2021
%Y Cf. A000129, A000217, A230044.
%Y Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276599 (k=5), A276600 (k=6), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
%Y Cf. A328791 (the resulting triangular numbers).
%K nonn,easy
%O 1,2
%A _Colin Barker_, Sep 07 2016
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