A034262 a(n) = n^3 + n.

0, 2, 10, 30, 68, 130, 222, 350, 520, 738, 1010, 1342, 1740, 2210, 2758, 3390, 4112, 4930, 5850, 6878, 8020, 9282, 10670, 12190, 13848, 15650, 17602, 19710, 21980, 24418, 27030, 29822, 32800, 35970, 39338, 42910, 46692, 50690, 54910, 59358, 64040, 68962, 74130

Offset: 0

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 21 2000

Keywords

• nonn
• easy

Comments

k such that x^3 + x + k factors over the integers. - James R. Buddenhagen, Apr 19 2005

If a(n)=X [A155977], Y=b(n) [A071253], Z=c(n) [A034262], then X^2+Y^2 = n*Z^3; e.g., if n=3, a(3)=270, b(3)=90, c(3)=30, then 270^2+90^2=3*30^3. - Vincenzo Librandi, Nov 24 2010

From Bruno Berselli, Sep 06 2018: (Start)

After 0, sum of next n even numbers:

... 2, 2

... 4, 6, 10

... 8, 10, 12, 30

.. 14, 16, 18, 20, 68

.. 22, 24, 26, 28, 30, 130

.. 32, 34, 36, 38, 40, 42, 222 etc. (End)

Sequence occurs in the binomial identity Sum_{k = 0..n} a(k)* binomial(n,k)/binomial(n+k,k) = n*(n + 1)/2. Cf. A092181 and A155977. - Peter Bala, Feb 12 2019

For n >= 2, a(n) is the sum of the numbers in the 1st and last columns of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A006003.

Cf. A000290, A000578, A001477, A002522.

Cf. A071253, A092181, A155977.

Cf. A001621.

Cf. A001620, A248177.

Programs

Haskell

a034262 n = a000578 n + n  -- Reinhard Zumkeller, Sep 26 2014

Mathematica

Table[n^3 + n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)

PARI

{a(n) = n^3 + n}; /* Michael Somos, Jul 11 2017 */

Sage

[n**3+n for n in (0..38)] # Stefano Spezia, Oct 08 2022

Formula

a(n) = 2*A006003(n).

a(n) = A002522(n)*A001477(n). - Zerinvary Lajos, Apr 20 2008

For n>1, a(n) = floor(n^5/(n^2-1)). - Gary Detlefs, Feb 10 2010

Sum_{n>=1} 1/a(n) = 0.6718659855... = gamma + Re psi(1+i) = A001620+A248177. [Borwein et al., J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Jul 17 2012

a(n) = -a(-n) for all n in Z. - Michael Somos, Jul 11 2017

G.f.: 2*x*(x^2+x+1)/(x-1)^4. - Alois P. Heinz, Oct 08 2022

E.g.f.: x*(2 + 3*x + x^2)*exp(x). - Stefano Spezia, Jun 20 2024