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%I #105 Sep 08 2022 08:45:02
%S 1,1,9,49,169,441,961,1849,3249,5329,8281,12321,17689,24649,33489,
%T 44521,58081,74529,94249,117649,145161,177241,214369,257049,305809,
%U 361201,423801,494209,573049,660969,758641,866761,986049,1117249,1261129,1418481,1590121
%N a(n) = n^4 - 2*n^3 + 3*n^2 - 2*n + 1, the Alexander polynomial for reef and granny knots.
%C "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.
%C First differences are in A105374. - _Wesley Ivan Hurt_, Apr 18 2016
%D Richard Courant and Herbert Robbins, What Is Mathematics?, 2nd Ed. 1996, pp. 501-505.
%H Vincenzo Librandi, <a href="/A058031/b058031.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (1-4*x+14*x^2+4*x^3+9*x^4)/(1-x)^5. - _Colin Barker_, Jan 17 2012
%F a(n) = (n^2-n+1)^2. - _Carmine Suriano_, Feb 16 2012
%F 3*a(n+3) = A062938(n) + A062938(n+1) + A062938(n+2). - _Bruno Berselli_, Feb 16 2012
%F a(n) = (n-2)*(n-1)*n*(n+1) + (2*n-1)^2. - _Charlie Marion_, Apr 11 2013
%F a(n) = A002061(n)^2. - _Richard R. Forberg_, Sep 03 2013
%F a(n) = (n*(n-1))^2 + (n-1)^2 + n^2, sum of three squares. - _Carmine Suriano_, Jun 16 2014
%F a(n) = A002378(A002378(n-1))+A002378(n-1)+1, where A002378(-1)=0. [_Bruno Berselli_, May 28 2015]
%F E.g.f.: exp(x)*(1 + 4*x^2 + 4*x^3 + x^4). - _Ilya Gutkovskiy_, Apr 16 2016
%F a(n) = (n-1)^4 + 2*(n-1)^3 + 3*(n-1)^2 + 2*(n-1) + 1. - _Bruce J. Nicholson_, Apr 07 2017
%F For n>0 a(n) = A002522(n)*A002522(n-1) - 1. - _Bruce J. Nicholson_, Jul 02 2017
%p A058031:=n->(n^2 - n + 1)^2; seq(A058031(n), n=0..50); # _Wesley Ivan Hurt_, Jun 19 2014
%t Table[(n^2 - n + 1)^2, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jun 19 2014 *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {1, 1, 9, 49, 169}, 50] (* _Vincenzo Librandi_, Apr 11 2017 *)
%o (Magma) [(n^2 - n + 1)^2 : n in [0..50]]; // _Wesley Ivan Hurt_, Jun 19 2014
%o (PARI) a(n)=n^4-2*n^3+3*n^2-2*n+1 \\ _Charles R Greathouse IV_, Jun 19 2014
%o (PARI) lista(nn) = for(n=0, nn, print1((n^2-n+1)^2, ", ")); \\ _Altug Alkan_, Apr 16 2016
%o (Python) def a(n): return n**4 - 2*n**3 + 3*n**2 - 2*n + 1 # _Indranil Ghosh_, Apr 06 2017
%Y Cf. A000290, A002061, A002378, A002522, A062938, A099254, A105374.
%K nonn,easy
%O 0,3
%A _Jason Earls_, Nov 21 2000
%E Name corrected by _Andrey Zabolotskiy_, Nov 21 2017
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