A101292 a(n) = n! + Sum_{i=1..n} i.

1, 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, 3628855, 39916866, 479001678, 6227020891, 87178291305, 1307674368120, 20922789888136, 355687428096153, 6402373705728171, 121645100408832190, 2432902008176640210, 51090942171709440231, 1124000727777607680253

Offset

0,2

Comments

Called "factoriangular" numbers by Castillo. - N. J. A. Sloane, Aug 30 2016

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Eric F. Bravo, Carlos A. Gómez, and Florian Luca, Product of Consecutive Tribonacci Numbers With Only One Distinct Digit, J. Int. Seq., Vol. 22 (2019), Article 19.6.3.

Romer C. Castillo, On the Sum of Corresponding Factorials and Triangular Numbers: Some Preliminary Results, Asia Pacific Journal of Multidisciplinary Research, Vol. 3, No. 4, November 2015 Part I.

Romer C. Castillo, On the Sum of Corresponding Factorials and Triangular Numbers: Runsums, Trapezoids and Politeness, Asia Pacific Journal of Multidisciplinary Research, 3 (2015), 95-101.

Romer C. Castillo, Generalized Factoriangular Numbers and Factoriangular Triangles, International Journal of Advanced Research and Publications, 2017.

Romer C. Castillo, On the Generalization of Factoriangular Numbers, Asian Res. J. Math. (2022) Vol. 18, No. 5, 1-21, Art. No. ARJOM.86060.

Carlos Alexis Gómez Ruiz and Florian Luca, Fibonacci factoriangular numbers, Indagationes Mathematicae, Volume 28, Issue 4, August 2017, p. 796-804.

Formula

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

Example

a(3) = 3! + (1 + 2 + 3) = 12.
a(5) = 5! + (1 + 2 + 3 + 4 + 5) = 135.

Maple

seq(n!+n*(n+1)/2, n=0..22); # Emeric Deutsch, Mar 12 2005
a:= proc(n) option remember; `if`(n<3, [1, 2, 5][n+1],
      ((11*n^2+10*n-70)*a(n-1)-(34*n^2-81*n+60)*a(n-2)
       +(23*n-10)*(n-2)*a(n-3))/(11*n-24))
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Aug 30 2016

Mathematica

Array[#! + # (# + 1)/2 &, 23, 0] (* Michael De Vlieger, Nov 12 2019 *)

Crossrefs

Cf. A275928.

Sequence in context: A176638 A131467 A000103 * A350441 A181899 A131267

Adjacent sequences: A101289 A101290 A101291 * A101293 A101294 A101295

Keyword

base,nonn,easy

Author

Parthasarathy Nambi, Dec 21 2004

Extensions

More terms from Emeric Deutsch, Mar 12 2005

a(0)=1 prepended by Alois P. Heinz, Aug 30 2016

Status

approved