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

1,2

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) = (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

Example

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

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 *)

Prog

(PARI) concat(0, Vec(9*x^2/(1-6*x+x^2) + O(x^30)))
(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
(Sage) [(9/2)*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), A276598 (k=3), A276599 (k=5), A276600 (k=6), A276601 (k=9), where k is the value added to n^2.

Sequence in context: A001392 A188428 A243415 * A079764 A079761 A115784

Adjacent sequences: A276599 A276600 A276601 * A276603 A276604 A276605

Keyword

nonn,easy

Author

Colin Barker, Sep 07 2016

Status

approved