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a(n) = n! + n + 1.
2

%I #21 Jun 06 2024 14:36:27

%S 2,3,5,10,29,126,727,5048,40329,362890,3628811,39916812,479001613,

%T 6227020814,87178291215,1307674368016,20922789888017,355687428096018,

%U 6402373705728019,121645100408832020,2432902008176640021

%N a(n) = n! + n + 1.

%C Can be used to detect triangular tables whose rows sum to n! when decreased by 1.

%H Vincenzo Librandi, <a href="/A213169/b213169.txt">Table of n, a(n) for n = 0..200</a>

%H R. C. Castillo, <a href="http://www.apjmr.com/wp-content/uploads/2015/10/APJMR-2015-3.4.2.15.pdf">On the Sum of Corresponding Factorials and Triangular Numbers: Runsums, Trapezoids and Politeness</a>, Asia Pacific Journal of Multidisciplinary Research, 3 (2015), 95-101.

%H Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, <a href="https://arxiv.org/abs/2405.21007">Card Tricks and Information</a>, arXiv:2405.21007 [math.HO], 2024. See p. 5.

%t Table[n! + n + 1, {n, 0, 20}]

%o (Maxima) A213169(n):=n!+n+1$

%o makelist(A213169(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */

%Y Cf. A000142, A038507, A005095.

%K nonn,easy

%O 0,1

%A _Olivier GĂ©rard_, Nov 02 2012