OFFSET
0,2
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).
FORMULA
a(n) = 2*A006003(n).
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
MATHEMATICA
Table[n^3 + n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
PROG
(Haskell)
a034262 n = a000578 n + n -- Reinhard Zumkeller, Sep 26 2014
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 21 2000
STATUS
approved