|
%I #74 May 12 2022 17:48:58
%S 1,2,5,12,34,135,741,5068,40356,362925,3628855,39916866,479001678,
%T 6227020891,87178291305,1307674368120,20922789888136,355687428096153,
%U 6402373705728171,121645100408832190,2432902008176640210,51090942171709440231,1124000727777607680253
%N a(n) = n! + Sum_{i=1..n} i.
%C Called "factoriangular" numbers by Castillo. - _N. J. A. Sloane_, Aug 30 2016
%C The only Fibonacci numbers in this sequence are 1, 2, 5, 34. [Ruiz and Luca, verifying a conjecture of Castillo] - _Eric M. Schmidt_, Nov 07 2017
%H Alois P. Heinz, <a href="/A101292/b101292.txt">Table of n, a(n) for n = 0..450</a>
%H Eric F. Bravo, Carlos A. Gómez, and Florian Luca, <a href="https://www.emis.de/journals/JIS/VOL22/Gomez/gomez3.html">Product of Consecutive Tribonacci Numbers With Only One Distinct Digit</a>, J. Int. Seq., Vol. 22 (2019), Article 19.6.3.
%H Romer C. Castillo, <a href="http://oaji.net/articles/2015/1543-1445023442.pdf">On the Sum of Corresponding Factorials and Triangular Numbers: Some Preliminary Results</a>, Asia Pacific Journal of Multidisciplinary Research, Vol. 3, No. 4, November 2015 Part I.
%H Romer 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 Romer C. Castillo, <a href="http://www.ijarp.org/published-research-papers/nov2017/Generalized-Factoriangular-Numbers-And-Factoriangular-Triangles.pdf">Generalized Factoriangular Numbers and Factoriangular Triangles</a>, International Journal of Advanced Research and Publications, 2017.
%H Romer C. Castillo, <a href="https://doi.org/10.9734/ARJOM/2022/v18i530374">On the Generalization of Factoriangular Numbers</a>, Asian Res. J. Math. (2022) Vol. 18, No. 5, 1-21, Art. No. ARJOM.86060.
%H Carlos Alexis Gómez Ruiz and Florian Luca, <a href="https://dx.doi.org/10.1016/j.indag.2017.05.002">Fibonacci factoriangular numbers</a>, Indagationes Mathematicae, Volume 28, Issue 4, August 2017, p. 796-804.
%F a(n) = n! + n*(n+1)/2.
%e a(3) = 3! + (1 + 2 + 3) = 12.
%e a(5) = 5! + (1 + 2 + 3 + 4 + 5) = 135.
%p seq(n!+n*(n+1)/2,n=0..22); # _Emeric Deutsch_, Mar 12 2005
%p a:= proc(n) option remember; `if`(n<3, [1,2,5][n+1],
%p ((11*n^2+10*n-70)*a(n-1)-(34*n^2-81*n+60)*a(n-2)
%p +(23*n-10)*(n-2)*a(n-3))/(11*n-24))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 30 2016
%t Array[#! + # (# + 1)/2 &, 23, 0] (* _Michael De Vlieger_, Nov 12 2019 *)
%Y Cf. A275928.
%K base,nonn,easy
%O 0,2
%A _Parthasarathy Nambi_, Dec 21 2004
%E More terms from _Emeric Deutsch_, Mar 12 2005
%E a(0)=1 prepended by _Alois P. Heinz_, Aug 30 2016
|
