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
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
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
MAPLE
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
KEYWORD
nonn,easy
AUTHOR
Jason Earls, Nov 21 2000
EXTENSIONS
Name corrected by Andrey Zabolotskiy, Nov 21 2017
STATUS
approved