A276598 Values of m such that m^2 + 3 is a triangular number (A000217).

0, 5, 30, 175, 1020, 5945, 34650, 201955, 1177080, 6860525, 39986070, 233055895, 1358349300, 7917039905, 46143890130, 268946300875, 1567533915120, 9136257189845, 53250009223950, 310363798153855, 1808932779699180, 10543232880041225, 61450464500548170

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.

Index entries for linear recurrences with constant coefficients, signature (6,-1).

Formula

a(n) = 5*A001109(n-1).

a(n) = 5*( (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n - (3 + 2*sqrt(2))*(3 - 2*sqrt(2))^n )/(4*sqrt(2)).

a(n) = 6*a(n-1) - a(n-2) for n>2.

G.f.: 5*x^2 / (1-6*x+x^2).

a(n) = (5/2)*A000129(2*n-2). - G. C. Greubel, Sep 15 2021

Example

5 is in the sequence because 5^2 + 3 = 28, which is a triangular number.

Mathematica

CoefficientList[Series[5*x/(1 - 6*x + x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 07 2016 *)
LinearRecurrence[{6, -1}, {0, 5}, 30] (* Harvey P. Dale, Apr 26 2019 *)
(5/2)*Fibonacci[2*Range[30] -2, 2] (* G. C. Greubel, Sep 15 2021 *)

Prog

(PARI) concat(0, Vec(5*x^2/(1-6*x+x^2) + O(x^30)))
(PARI) a(n)=([0, 1; -1, 6]^n*[-5; 0])[1, 1] \\ Charles R Greathouse IV, Sep 07 2016
(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
(Sage) [(5/2)*lucas_number1(2*n-2, 2, -1) for n in (1..30)] # G. C. Greubel, Sep 15 2021

Crossrefs

Cf. A000129, A000217, A230044.

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.

Cf. A328791 (the resulting triangular numbers).

Sequence in context: A111469 A241588 A229246 * A057088 A156195 A105481

Adjacent sequences: A276595 A276596 A276597 * A276599 A276600 A276601

Keyword

nonn,easy

Author

Colin Barker, Sep 07 2016

Status

approved