A134593 a(n) = 5*n^2 + 10*n + 1. Coefficients of the rational part of (1 + sqrt(n))^5.

1, 16, 41, 76, 121, 176, 241, 316, 401, 496, 601, 716, 841, 976, 1121, 1276, 1441, 1616, 1801, 1996, 2201, 2416, 2641, 2876, 3121, 3376, 3641, 3916, 4201, 4496, 4801, 5116, 5441, 5776, 6121, 6476, 6841, 7216, 7601, 7996, 8401, 8816, 9241, 9676, 10121

Offset: 0

Author

Artur Jasinski, Nov 04 2007

Keywords

• nonn
• easy

Comments

(1+sqrt(n))^5 = (5*n^2 + 10*n + 1) + (n^2 + 10*n + 5)*sqrt(n). Coefficients of the irrational part are A134594.

Number of entries required to describe the options and constraints in Don Knuth's formulation of the n nonattacking queens on an n X n board problem (A000170) as input for his DLX (Dancing Links eXact coverage) program. Can be seen as "entries successfully read" in the video from his 2018 Annual Christmas Lecture. - Hugo Pfoertner, Jan 09 2019

Links

• D. E. Knuth, Donald Knuth's 24th Annual Christmas Lecture: Dancing Links, Stanfordonline, Video published on YouTube, Dec 12, 2018.
• Takao Komatsu, Ritika Goel, and Neha Gupta, The Frobenius number for the triple of the 2-step star numbers, arXiv:2409.14788 [math.CO], 2024. See p. 2.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000170, A134594.

Programs

Mathematica

Table[(5n^2 + 10n + 1), {n, 0, 50}]
LinearRecurrence[{3,-3,1},{1,16,41},50] (* Harvey P. Dale, Oct 20 2023 *)

PARI

a(n)=5*n^2+10*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Python

print([5*i**2-4 for i in range(1,100)])
# Ruskin Harding, Mar 27 2013

Formula

G.f.: (4*x^2 - 13*x - 1)/(x-1)^3. - R. J. Mathar, Nov 14 2007

a(n) = a(n-1) + 10*n + 5 (with a(0)=1). - Vincenzo Librandi, Nov 23 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, May 17 2021

E.g.f.: exp(x)*(1 + 15*x + 5*x^2). - Stefano Spezia, Sep 27 2024

Extensions

Edited by Charles R Greathouse IV, Aug 09 2010