A177342 a(n) = (4*n^3-3*n^2+5*n-3)/3.

1, 9, 31, 75, 149, 261, 419, 631, 905, 1249, 1671, 2179, 2781, 3485, 4299, 5231, 6289, 7481, 8815, 10299, 11941, 13749, 15731, 17895, 20249, 22801, 25559, 28531, 31725, 35149, 38811, 42719, 46881, 51305, 55999, 60971, 66229, 71781, 77635

Offset: 1

Author

Bruno Berselli, May 06 2010 - Nov 27 2010

Keywords

• nonn
• easy

Comments

This sequence is related to the fourth powers (A000583) by n^4 = n*a(n) - Sum_{i=1..n-1} a(i) - (n-1), with n>1.

Also, n*a(n) - Sum_{i=1..n-1} a(i) provides the first column of A162624 and the second column of A162622 (or A162623). - Bruno Berselli, revised Dec 14 2012

Links

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

Crossrefs

First differences: 2*A084849.

Partial sums: A178073.

Cf. A174723, A162622-A162624.

Programs

Magma

[(4*n^3-3*n^2+5*n-3)/3: n in [1..39]]; // Bruno Berselli, Aug 24 2011

Magma

I:=[1,9,31,75]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 19 2013

Mathematica

CoefficientList[Series[(1 + 5 x + x^2 + x^3) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
Table[(4 n^3 - 3 n^2 + 5 n - 3)/3, {n, 1, 40}] (* Bruno Berselli, Feb 17 2015 *)
LinearRecurrence[{4,-6,4,-1},{1,9,31,75},40] (* Harvey P. Dale, Jul 31 2021 *)

PARI

a(n)=(4*n^3-3*n^2+5*n-3)/3 \\ Charles R Greathouse IV, Jun 23 2011

Formula

G.f.: x*(1 + 5*x + x^2 + x^3)/(1 - x)^4.

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

a(n) - a(-n) = 2*A004006(2n).

a(n) + a(-n) = -A002522(n).

a(n) = 1 + (n-1)*(4*n^2+n+6)/3 = 2*A174723(n)-1.

Extensions

Formulae added and revised by Bruno Berselli, Feb 17 2015