A053698 a(n) = n^3 + n^2 + n + 1.

1, 4, 15, 40, 85, 156, 259, 400, 585, 820, 1111, 1464, 1885, 2380, 2955, 3616, 4369, 5220, 6175, 7240, 8421, 9724, 11155, 12720, 14425, 16276, 18279, 20440, 22765, 25260, 27931, 30784, 33825, 37060, 40495, 44136, 47989, 52060, 56355, 60880

Offset: 0

Author

Henry Bottomley, Mar 23 2000

Keywords

• nonn
• easy

Comments

a(n) = 1111 in base n.

n^3 + n^2 + n + 1 = (n^2 + 1)*(n + 1), therefore a(n) is never prime. - Alonso del Arte, Apr 22 2014

Examples

			a(2) = 15 because 2^3 + 2^2 + 2 + 1 = 8 + 4 + 2 + 1 = 15.
a(3) = 40 because 3^3 + 3^2 + 3 + 1 = 27 + 9 + 3 + 1 = 40.
a(4) = 85 because 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85.
From _Bruno Berselli_, Jan 02 2017: (Start)
The terms of the sequence are provided by the row sums of the following triangle (see the seventh formula in the previous section):
.   1;
.   3,   1;
.   9,   5,   1;
.  19,  13,   7,   1;
.  33,  25,  17,   9,   1;
.  51,  41,  31,  21,  11,   1;
.  73,  61,  49,  37,  25,  13,  1;
.  99,  85,  71,  57,  43,  29, 15,  1;
. 129, 113,  97,  81,  65,  49, 33, 17,  1;
. 163, 145, 127, 109,  91,  73, 55, 37, 19,  1;
. 201, 181, 161, 141, 121, 101, 81, 61, 41, 21, 1;
...
Columns from the first to the fifth, respectively: A058331, A001844, A056220 (after -1), A059993, A161532. Also, eighth column is A161549.
(End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A237627 (subset of semiprimes).

Cf. A056106 (first differences).

Cf. A062158, A027444, A104878, A055129.

Programs

Magma

[n^3+n^2+n+1: n in [0..50]]; // Vincenzo Librandi, May 01 2011

Maple

A053698:=n->n^3 + n^2 + n + 1; seq(A053698(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2014

Mathematica

Table[n^3 + n^2 + n + 1, {n, 0, 39}] (* Alonso del Arte, Apr 22 2014 *)
FromDigits["1111", Range[0, 50]] (* Paolo Xausa, May 11 2024 *)

PARI

Vec((1 + 5*x^2) / (1 - x)^4 + O(x^50)) \\ Colin Barker, Jan 02 2017

Python

def a(n): return (n**3+n**2+n+1) # Torlach Rush, May 08 2024

Formula

For n >= 2, a(n) = (n^4-1)/(n-1) = A024002(n)/A024000(n) = A002522(n)*(n+1) = A002061(n+1) + A000578(n).

G.f.: (1+5*x^2) / (1-x)^4. - Colin Barker, Jan 06 2012

a(n) = -A062158(-n). - Bruno Berselli, Jan 26 2016

a(n) = Sum_{i=0..n} 2*n*(n-i)+1. - Bruno Berselli, Jan 02 2017

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Jan 02 2017

a(n) = A104878(n+3,n) = A055129(4,n) for n > 0. - Mathew Englander, Jan 06 2021

E.g.f.: exp(x)*(x^3+4*x^2+3*x+1). - Nikolaos Pantelidis, Feb 06 2023