A058031 a(n) = n^4 - 2*n^3 + 3*n^2 - 2*n + 1, the Alexander polynomial for reef and granny knots.

1, 1, 9, 49, 169, 441, 961, 1849, 3249, 5329, 8281, 12321, 17689, 24649, 33489, 44521, 58081, 74529, 94249, 117649, 145161, 177241, 214369, 257049, 305809, 361201, 423801, 494209, 573049, 660969, 758641, 866761, 986049, 1117249, 1261129, 1418481, 1590121

Offset

0,3

Comments

"The standard knot invariant, in the pre-Jones era of knot theory, was the Alexander polynomial, invented in 1926. This assigns to each knot a polynomial in a variable t, which can be calculated by following a standard procedure." See Courant and Robbins, p. 503.

First differences are in A105374. - Wesley Ivan Hurt, Apr 18 2016

References

Richard Courant and Herbert Robbins, What Is Mathematics?, 2nd Ed. 1996, pp. 501-505.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: (1-4*x+14*x^2+4*x^3+9*x^4)/(1-x)^5. - Colin Barker, Jan 17 2012

a(n) = (n^2-n+1)^2. - Carmine Suriano, Feb 16 2012

3*a(n+3) = A062938(n) + A062938(n+1) + A062938(n+2). - Bruno Berselli, Feb 16 2012

a(n) = (n-2)*(n-1)*n*(n+1) + (2*n-1)^2. - Charlie Marion, Apr 11 2013

a(n) = A002061(n)^2. - Richard R. Forberg, Sep 03 2013

a(n) = (n*(n-1))^2 + (n-1)^2 + n^2, sum of three squares. - Carmine Suriano, Jun 16 2014

a(n) = A002378(A002378(n-1))+A002378(n-1)+1, where A002378(-1)=0. [Bruno Berselli, May 28 2015]

E.g.f.: exp(x)*(1 + 4*x^2 + 4*x^3 + x^4). - Ilya Gutkovskiy, Apr 16 2016

a(n) = (n-1)^4 + 2*(n-1)^3 + 3*(n-1)^2 + 2*(n-1) + 1. - Bruce J. Nicholson, Apr 07 2017

For n>0 a(n) = A002522(n)*A002522(n-1) - 1. - Bruce J. Nicholson, Jul 02 2017

Maple

A058031:=n->(n^2 - n + 1)^2; seq(A058031(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

Mathematica

Table[(n^2 - n + 1)^2, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 19 2014 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 1, 9, 49, 169}, 50] (* Vincenzo Librandi, Apr 11 2017 *)

Prog

(Magma) [(n^2 - n + 1)^2 : n in [0..50]]; // Wesley Ivan Hurt, Jun 19 2014
(PARI) a(n)=n^4-2*n^3+3*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 19 2014
(PARI) lista(nn) = for(n=0, nn, print1((n^2-n+1)^2, ", ")); \\ Altug Alkan, Apr 16 2016
(Python) def a(n): return n**4 - 2*n**3 + 3*n**2 - 2*n + 1 # Indranil Ghosh, Apr 06 2017

Crossrefs

Cf. A000290, A002061, A002378, A002522, A062938, A099254, A105374.

Sequence in context: A273374 A020245 A082608 * A359726 A228212 A027608

Adjacent sequences: A058028 A058029 A058030 * A058032 A058033 A058034

Keyword

nonn,easy

Author

Jason Earls, Nov 21 2000

Extensions

Name corrected by Andrey Zabolotskiy, Nov 21 2017

Status

approved