A229183 a(n) = n*(n^2 + 3)/2.

0, 2, 7, 18, 38, 70, 117, 182, 268, 378, 515, 682, 882, 1118, 1393, 1710, 2072, 2482, 2943, 3458, 4030, 4662, 5357, 6118, 6948, 7850, 8827, 9882, 11018, 12238, 13545, 14942, 16432, 18018, 19703, 21490, 23382, 25382, 27493, 29718, 32060, 34522, 37107, 39818

Offset: 0

Author

Derek Orr, Sep 15 2013

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A006003 (row sums of integers, starting with 1).

Cf. A027480 (row sums of integers, starting with 0).

Programs

Magma

[n*(n^2 + 3) div 2: n in [0..50]]; // Vincenzo Librandi, Sep 23 2013

Maple

A229183 := proc(n)
    n*(n^2+3) /2;
end proc:
[seq(A229183(n),n=0..30)] ; # R. J. Mathar, Aug 16 2019

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 *)

PARI

vector(100,n,((n-1)^3+3*n-3)/2) \\ Derek Orr, Mar 12 2015

Python

{print((n**3+3*n)/2,end=', ') for n in range(0,100)} # Simplified by Derek Orr, Mar 12 2015

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