A276602 Values of k such that k^2 + 10 is a triangular number (A000217).

0, 9, 54, 315, 1836, 10701, 62370, 363519, 2118744, 12348945, 71974926, 419500611, 2445028740, 14250671829, 83059002234, 484103341575, 2821561047216, 16445262941721, 95850016603110, 558654836676939, 3256079003458524, 18977819184074205, 110610836100986706

Offset: 1

Author

Colin Barker, Sep 07 2016

Keywords

• nonn
• easy

Examples

			9 is in the sequence because 9^2+10 = 91, which is a triangular number.
		

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).

Crossrefs

Cf. A000129, A000217, A230044.

Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276599 (k=5), A276600 (k=6), A276601 (k=9), where k is the value added to n^2.

Programs

Magma

[n le 2 select 9*(n-1) else 6*Self(n-1) - Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 15 2021

Mathematica

CoefficientList[Series[9*x/(1 - 6*x + x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 07 2016 *)
(9/2)*Fibonacci[2*(Range[30] -1), 2] (* G. C. Greubel, Sep 15 2021 *)

PARI

concat(0, Vec(9*x^2/(1-6*x+x^2) + O(x^30)))

Sage

[(9/2)*lucas_number1(2*n-2, 2, -1) for n in (1..30)] # G. C. Greubel, Sep 15 2021

Formula

a(n) = (9/(4*sqrt(2))*( (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n - (3 + 2*sqrt(2))*(3 - 2*sqrt(2))^n) ).

a(n) = 9*A001109(n-1).

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

G.f.: 9*x^2 / (1-6*x+x^2).

a(n) = (9/2)*A000129(2*n-2). - G. C. Greubel, Sep 15 2021