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A058031 a(n) = n^4 - 2*n^3 + 3*n^2 - 2*n + 1, the Alexander polynomial for reef and granny knots. 6
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 (list; graph; refs; listen; history; text; internal format)
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 * A228212 A027608 A003297

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

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Last modified February 19 06:46 EST 2018. Contains 299330 sequences. (Running on oeis4.)