A188947 a(n) = n^3 - 2*n^2 + 2*n + 1.

2, 5, 16, 41, 86, 157, 260, 401, 586, 821, 1112, 1465, 1886, 2381, 2956, 3617, 4370, 5221, 6176, 7241, 8422, 9725, 11156, 12721, 14426, 16277, 18280, 20441, 22766, 25261, 27932, 30785, 33826, 37061, 40496, 44137, 47990, 52061, 56356, 60881, 65642, 70645

Offset: 1

Author

Adeniji, Adenike, Apr 14 2011

Keywords

• nonn
• easy

Comments

The original definition was "Identity difference partial one - one transformation semigroup is a semigroup having the property that the difference between max im(alpha) and min im(alpha) is not greater than 1. This is denoted by S = IDI_n for each n." [Needs editing.]

For all n >= 3, a(n) expressed in base n has the three digits n-2, 2, and 1; for example, a(16) in hexadecimal is "E21". For all n >= 3, a(n+1) expressed in base n is "1112". For all n >= 7, a(n+2) expressed in base n is "1465". - Mathew Englander, Jan 07 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A027444, A053698, A056106 (first differences), A060354, A162607, A188377, A188716.

Programs

Magma

[n^3 - 2*n^2 + 2*n + 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

Mathematica

A188947[n_] := n^3 - 2*n^2 + 2*n + 1; Table[A188947[n], {n, 1, 42}] (* Robert P. P. McKone, Jan 31 2021 *)

PARI

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

Formula

a(n) = (n+1) + n*(n-1)^2 = n^3 - 2*n^2 + 2*n + 1 = 1 + A053698(n-1).

G.f.: ( -x*(-2 + 3*x - 8*x^2 + x^3) ) / ( (x-1)^4 ). - R. J. Mathar, Apr 14 2011

a(n) = A060354(n) + A162607(n+1). - Lechoslaw Ratajczak, Sep 24 2020

E.g.f.: exp(x)*(1 + x)*(1 + x^2) - 1. - Stefano Spezia, Apr 10 2022

Extensions

Edited by N. J. A. Sloane, Apr 23 2011