cp's OEIS Frontend

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

%I A034262 #77 Jun 20 2024 14:54:16
%S A034262 0,2,10,30,68,130,222,350,520,738,1010,1342,1740,2210,2758,3390,4112,
%T A034262 4930,5850,6878,8020,9282,10670,12190,13848,15650,17602,19710,21980,
%U A034262 24418,27030,29822,32800,35970,39338,42910,46692,50690,54910,59358,64040,68962,74130
%N A034262 a(n) = n^3 + n.
%C A034262 k such that x^3 + x + k factors over the integers. - _James R. Buddenhagen_, Apr 19 2005
%C A034262 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
%C A034262 From _Bruno Berselli_, Sep 06 2018: (Start)
%C A034262 After 0, sum of next n even numbers:
%C A034262 ... 2,                           2
%C A034262 ... 4,  6,                      10
%C A034262 ... 8, 10, 12,                  30
%C A034262 .. 14, 16, 18, 20,              68
%C A034262 .. 22, 24, 26, 28, 30,         130
%C A034262 .. 32, 34, 36, 38, 40, 42,     222 etc. (End)
%C A034262 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
%C A034262 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
%H A034262 Reinhard Zumkeller, <a href="/A034262/b034262.txt">Table of n, a(n) for n = 0..10000</a>
%H A034262 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A034262 a(n) = 2*A006003(n).
%F A034262 a(n) = A002522(n)*A001477(n). - _Zerinvary Lajos_, Apr 20 2008
%F A034262 For n>1, a(n) = floor(n^5/(n^2-1)). - _Gary Detlefs_, Feb 10 2010
%F A034262 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
%F A034262 a(n) = -a(-n) for all n in Z. - _Michael Somos_, Jul 11 2017
%F A034262 G.f.: 2*x*(x^2+x+1)/(x-1)^4. - _Alois P. Heinz_, Oct 08 2022
%F A034262 E.g.f.: x*(2 + 3*x + x^2)*exp(x). - _Stefano Spezia_, Jun 20 2024
%t A034262 Table[n^3 + n, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 06 2010 *)
%o A034262 (Haskell)
%o A034262 a034262 n = a000578 n + n  -- _Reinhard Zumkeller_, Sep 26 2014
%o A034262 (PARI) {a(n) = n^3 + n}; /* _Michael Somos_, Jul 11 2017 */
%o A034262 (Sage) [n**3+n for n in (0..38)] # _Stefano Spezia_, Oct 08 2022
%Y A034262 Cf. A006003.
%Y A034262 Cf. A000290, A000578, A001477, A002522.
%Y A034262 Cf. A071253, A092181, A155977.
%Y A034262 Cf. A001621.
%Y A034262 Cf. A001620, A248177.
%K A034262 nonn,easy
%O A034262 0,2
%A A034262 Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 21 2000