A134593 - OEIS

%I #52 Sep 27 2024 05:40:32

%S 1,16,41,76,121,176,241,316,401,496,601,716,841,976,1121,1276,1441,

%T 1616,1801,1996,2201,2416,2641,2876,3121,3376,3641,3916,4201,4496,

%U 4801,5116,5441,5776,6121,6476,6841,7216,7601,7996,8401,8816,9241,9676,10121

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

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

%C 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

%H D. E. Knuth, <a href="https://youtu.be/_cR9zDlvP88?t=3177">Donald Knuth's 24th Annual Christmas Lecture: Dancing Links</a>, Stanfordonline, Video published on YouTube, Dec 12, 2018.

%H Takao Komatsu, Ritika Goel, and Neha Gupta, <a href="https://arxiv.org/abs/2409.14788">The Frobenius number for the triple of the 2-step star numbers</a>, arXiv:2409.14788 [math.CO], 2024. See p. 2.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (4*x^2 - 13*x - 1)/(x-1)^3. - _R. J. Mathar_, Nov 14 2007

%F a(n) = a(n-1) + 10*n + 5 (with a(0)=1). - _Vincenzo Librandi_, Nov 23 2010

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Wesley Ivan Hurt_, May 17 2021

%F E.g.f.: exp(x)*(1 + 15*x + 5*x^2). - _Stefano Spezia_, Sep 27 2024

%t Table[(5n^2 + 10n + 1), {n, 0, 50}]

%t LinearRecurrence[{3,-3,1},{1,16,41},50] (* _Harvey P. Dale_, Oct 20 2023 *)

%o (Python)

%o print([5*i**2-4 for i in range(1,100)])

%o # _Ruskin Harding_, Mar 27 2013

%o (PARI) a(n)=5*n^2+10*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000170, A134594.

%K nonn,easy

%O 0,2

%A _Artur Jasinski_, Nov 04 2007

%E Edited by _Charles R Greathouse IV_, Aug 09 2010