A270109 a(n) = n^3 + (n+1)*(n+2).

2, 7, 20, 47, 94, 167, 272, 415, 602, 839, 1132, 1487, 1910, 2407, 2984, 3647, 4402, 5255, 6212, 7279, 8462, 9767, 11200, 12767, 14474, 16327, 18332, 20495, 22822, 25319, 27992, 30847, 33890, 37127, 40564, 44207, 48062, 52135, 56432, 60959, 65722, 70727, 75980, 81487, 87254

Offset: 0

Author

Bruno Berselli, Mar 11 2016, at the suggestion of Giuseppe Amoruso in BASE Cinque forum

Keywords

• nonn
• easy

Comments

For n>1, many consecutive terms of the sequence are generated by floor(sqrt(n^2 + 2)^3) + n^2 + 2.

It appears that this is a subsequence of A000037 (the nonsquares).

The primes in the sequence belong to A045326.

Inverse binomial transform is 2, 5, 8, 6, 0, 0, 0, ... (0 continued).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• MathsSmart, Number pattern and Puzzle - 7, 20, 47, 94, 167.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Subsequence of A001651, A047212.

Cf. A000037, A045326.

Cf. A027444: numbers of the form n^3+n*(n+1); A085490: numbers of the form n^3+(n-1)*n.

Cf. A008865: numbers of the form n+(n+1)*(n+2); A130883: numbers of the form n^2+(n+1)*(n+2).

Programs

Magma

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

Mathematica

Table[n^3 + (n + 1) (n + 2), {n, 0, 50}]

Maxima

makelist(n^3+(n+1)*(n+2), n, 0, 50);

PARI

vector(50, n, n--; n^3+(n+1)*(n+2))

Sage

[n^3+(n+1)*(n+2) for n in (0..50)]

Formula

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

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

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

a(n+h) - a(n) + a(n-h) = n^3 + n^2 + (6*h^2+3)*n + (2*h^2+2) for any h. This identity becomes a(n) = n^3 + n^2 + 3*n + 2 if h=0.

a(h*a(n) + n) = (h*a(n))^3 + (3*n+1)*(h*a(n))^2 + (3*n^2+2*n+3)*(h*a(n)) + a(n) for any h, therefore a(h*a(n) + n) is always a multiple of a(n).

a(n) + a(-n) = 2*A059100(n) = A255843(n).

a(n) - a(-n) = 4*A229183(n).