A008747 - OEIS

%I #52 Sep 08 2022 08:44:36

%S 1,1,2,3,5,6,9,11,14,17,21,24,29,33,38,43,49,54,61,67,74,81,89,96,105,

%T 113,122,131,141,150,161,171,182,193,205,216,229,241,254,267,281,294,

%U 309,323,338,353,369,384,401,417,434,451,469,486,505,523,542,561

%N Expansion of (1+x^4)/((1-x)*(1-x^2)*(1-x^3)).

%C For n>=1, the set {A008747(6n+-1)} is the set of numbers of the form a^2 + 5*(a+1)^2 for -inf < a < inf. Furthermore the set A008747(6n) is A033581(n). - _Kieren MacMillan_, Dec 19 2007

%C For n>1, a(n-1) is the number of aperiodic necklaces (Lyndon words) with k<=3 black beads and n-k white beads. For n=4 we have for example a(3)=3 aperiodic necklaces: BWWW, BBWW and BBBW. BWBW is periodic and is not counted. - _Herbert Kociemba_, Oct 23 2016

%H Seiichi Manyama, <a href="/A008747/b008747.txt">Table of n, a(n) for n = 0..10000</a>

%H Robert Morris, <a href="https://arxiv.org/abs/math/0702370">Minimal percolating sets in bootstrap percolation</a>, arXiv:math/0702370 [math.CO], 2007-2008.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

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

%F a(n) = ceiling((n+1)^2/6).

%F a(n) = (12*n + 23 + 6*n^2 + 9*(-1)^n + 4*A061347(n))/36. - _R. J. Mathar_, Mar 15 2011

%F a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=6, a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 5. - _Harvey P. Dale_, Sep 05 2012

%F From _Michael Somos_, Oct 25 2016: (Start)

%F Euler transform of length 8 sequence [ 1, 1, 1, 1, 0, 0, 0, -1].

%F a(n) = a(-2-n) for all n in Z.

%F a(2*n-1) = A071619(n).

%F a(3*n-1) = 2*A077043(n).

%F a(n) - a(n-1) = A051274(n). (End)

%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 11*x^7 + 14*x^8 + ...

%p A008747:=n->ceil((n+1)^2/6): seq(A008747(n), n=0..100); # _Wesley Ivan Hurt_, Oct 25 2016

%t CoefficientList[Series[(1+x^4)/((1-x)(1-x^2)(1-x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,5,6},60] (* _Harvey P. Dale_, Sep 05 2012 *)

%o (PARI) Vec((1+x^4)/((1-x)*(1-x^2)*(1-x^3))+O(x^60)) \\ _Charles R Greathouse IV_, Sep 25 2012

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^4)/((1-x)*(1-x^2)*(1-x^3)) )); // _G. C. Greubel_, Aug 03 2019

%o (Sage) ((1+x^4)/((1-x)*(1-x^2)*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 03 2019

%o (GAP) a:=[1,1,2,3,5,6];; for n in [7..60] do a[n]:=a[n-1]+a[n-2]-a[n-4] -a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Aug 03 2019

%Y Cf. A008747, A033581, A051274, A061347, A071619, A077043.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_