OFFSET
0,2
COMMENTS
Numbers a(n) such that (a(n) + B)^(1/3) + (a(n) - B)^(1/3) = n, where B = sqrt(a(n)^2 + 1).
4*a(n) is the sum of two cubes. In fact: 2*n*(n^2 + 3) = (n-1)^3 + (n+1)^3. - Bruno Berselli, Apr 11 2016
From Olivier Gérard, Aug 07 2016 (Start)
Row sums of n consecutive integers, starting at 2, seen as a triangle:
.
2 | 2
7 | 3 4
18 | 5 6 7
38 | 8 9 10 11
70 | 12 13 14 15 16
117 | 17 18 19 20 21 22
(End)
Take a long horizontal strip of paper 1 unit high and mark two points on the top edge, n/2 and n units from the top left corner. Then fold over the top left corner so that the fold line passes through the bottom left corner and the point n units along the top edge. Then draw a line from the bottom left corner of the strip through the new position of the n/2 point. The point at which that shallow diagonal line meets the top edge of the strip of paper will be a(n) from the top left corner. - Elliott Line, Jul 09 2018
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).
FORMULA
G.f.: x*(2 - x + 2*x^2) / (x-1)^4. - R. J. Mathar, Sep 22 2013
a(n)^2 + 1 = (n^2 + 1)^2 * ((n/2)^2 + 1). - Joerg Arndt, Jan 22 2015
E.g.f.: exp(x)*x*(4 + 3*x + x^2)/2. - Stefano Spezia, Jul 04 2021
MAPLE
MATHEMATICA
Table[(n^3 + 3n)/2, {n, 0, 100}] (* T. D. Noe, Sep 16 2013 *)
CoefficientList[Series[x (2 - x + 2 x^2)/(x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 23 2013 *)
PROG
(Python)
{print((n**3+3*n)/2, end=', ') for n in range(0, 100)} # Simplified by Derek Orr, Mar 12 2015
(PARI) vector(100, n, ((n-1)^3+3*n-3)/2) \\ Derek Orr, Mar 12 2015
(Magma) [n*(n^2 + 3) div 2: n in [0..50]]; // Vincenzo Librandi, Sep 23 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Derek Orr, Sep 15 2013
STATUS
approved