A270867 a(n) = n^3 + 2*n^2 + 4*n + 1.

1, 8, 25, 58, 113, 196, 313, 470, 673, 928, 1241, 1618, 2065, 2588, 3193, 3886, 4673, 5560, 6553, 7658, 8881, 10228, 11705, 13318, 15073, 16976, 19033, 21250, 23633, 26188, 28921, 31838, 34945, 38248, 41753, 45466, 49393, 53540, 57913, 62518, 67361, 72448

Offset: 0

Author

Vincenzo Librandi, Apr 01 2016

Keywords

• nonn
• easy

Comments

Numbers of the type (m+1)^3 - (m-1)*m. Similar sequences are: A069778 with the closed form (m+1)^3 - m*(m+1), A152015 with (m+1)^3 - (m+1)*(m+2).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015 (page 19, 4th row; page 21, 3rd row).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A069778, A016957, A152015, A270109.

Programs

Magma

[n^3+2*n^2+4*n+1: n in [0..50]];

Maple

A270867:=n->n^3+2*n^2+4*n+1: seq(A270867(n), n=0..100); # Wesley Ivan Hurt, Apr 01 2016

Mathematica

Table[n^3 + 2 n^2 + 4 n + 1, {n, 0, 40}]

PARI

x='x+O('x^99); Vec((1+4*x-x^2+2*x^3)/(1-x)^4) \\ Altug Alkan, Apr 01 2016

Python

for i in range(0,100):print(i**3+2*i**2+4*i+1) # Soumil Mandal, Apr 02 2016

Formula

O.g.f.: (1 + 4*x - x^2 + 2*x^3)/(1 - x)^4.

E.g.f.: (1 + 7*x + 5*x^2 + x^3)*exp(x).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = -A270109(-n-1). - Bruno Berselli, Apr 01 2016

a(n+2) - 2*a(n+1) + a(n) = A016957(n+1). - Wesley Ivan Hurt, Apr 02 2016