A100705 a(n) = n^3 + (n+1)^2.

1, 5, 17, 43, 89, 161, 265, 407, 593, 829, 1121, 1475, 1897, 2393, 2969, 3631, 4385, 5237, 6193, 7259, 8441, 9745, 11177, 12743, 14449, 16301, 18305, 20467, 22793, 25289, 27961, 30815, 33857, 37093, 40529, 44171, 48025, 52097, 56393, 60919, 65681, 70685, 75937

Offset: 0

Author

Giovanni Teofilatto, Jan 03 2005

Keywords

• nonn
• easy

Comments

The finite simple continued fraction [1;n-1,n,n+1] has a numerator in the resulting rational number (n^3+2*n+n^2+1)/(n*(n^2+1)) that is the same as a(n). - J. M. Bergot, Sep 29 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A053698.

Programs

Magma

[n^3 + (n+1)^2: n in [0..50]]; // Vincenzo Librandi, Sep 30 2011

Mathematica

Table[n^3 + (n + 1)^2, {n, 0, 45}]
LinearRecurrence[{4,-6,4,-1},{1,5,17,43},50] (* Harvey P. Dale, Dec 03 2014 *)

PARI

a(n)=n^3+(n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1+x+3*x^2+x^3)/(x-1)^4. - R. J. Mathar, Sep 29 2011

a(n) = A053698(n) + n. - Bruno Berselli, Sep 30 2011

From Elmo R. Oliveira, Apr 19 2025: (Start)

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

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

Extensions

More terms from Mark Hudson and Farideh Firoozbakht, Jan 04 2005