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%I #44 Aug 02 2018 19:40:00
%S 2,1,5,5,6,7,10,9,12,12,14,15,17,16,20,20,21,22,25,24,27,27,29,30,32,
%T 31,35,35,36,37,40,39,42,42,44,45,47,46,50,50,51,52,55,54,57,57,59,60,
%U 62,61,65,65,66,67,70,69,72,72,74,75,77,76,80,80,81,82,85
%N Expansion of (2 + x + 5*x^2 + 3*x^3 + 3*x^4 + x^5) / ((1 - x^3) * (1 - x^4)) in powers of x.
%H G. C. Greubel, <a href="/A248699/b248699.txt">Table of n, a(n) for n = 0..2500</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,0,0,-1).
%F G.f.: (2 + x + 5*x^2 + 3*x^3 + 3*x^4 + x^5) / ((1 - x^3) * (1 - x^4)).
%F a(n) = - A071047(-2-n) for all n in Z.
%F a(n+7) = a(n+4) + a(n+3) - a(n) for all n in Z.
%F a(n+3) = a(n) + [3, 5, 2, 5][mod(n, 4) + 1] for all n in Z.
%F a(n+4) = a(n) + [4, 6, 5][mod(n, 3) + 1] for all n in Z.
%F a(n+12) = a(n) + 15 for all n in Z.
%F a(n) = a(n-3) + a(n-4) - a(n-7). - _Muniru A Asiru_, Aug 01 2018
%e G.f. = 2 + x + 5*x^2 + 5*x^3 + 6*x^4 + 7*x^5 + 10*x^6 + 9*x^7 + 12*x^8 + ...
%p seq(coeff(series((2+x+5*x^2+3*x^3+3*x^4+x^5)/((1-x^3)*(1-x^4)), x,n+1),x,n),n=0..70); # _Muniru A Asiru_, Aug 01 2018
%t a[ n_] := QUotient[ 5 n, 4] + { 0, 3, 2, 1, 1, 3, 1, 2, 1, 2, 2, 2}[[Mod[n, 12, 1]]];
%t CoefficientList[Series[(2 + x + 5*x^2 + 3*x^3 + 3*x^4 + x^5)/((1 - x^3)*(1 - x^4)), {x, 0, 50}], x]] (* _G. C. Greubel_, Aug 01 2018 *)
%t LinearRecurrence[{0, 0, 1, 1, 0, 0, -1}, {2, 1, 5, 5, 6, 7, 10}, 70] (* _Robert G. Wilson v_, Aug 01 2018 *)
%o (PARI) {a(n) = (5*n\4) + [ 2, 0, 3, 2, 1, 1, 3, 1, 2, 1, 2, 2][n%12 + 1]};
%o (PARI) {a(n) = polcoeff( if( n<0, n = -2-n; -(1 + 3*x + 3*x^2 + 5*x^3 + x^4 + 2*x^5), 2 + x + 5*x^2 + 3*x^3 + 3*x^4 + x^5) / ((1 - x^3) * (1 - x^4)) + x * O(x^n), n)};
%o (GAP) a:=[2,1,5,5,6,7,10];; for n in [8..70] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # _Muniru A Asiru_, Aug 01 2018
%Y Cf. A071047.
%K nonn,easy
%O 0,1
%A _Michael Somos_, Aug 01 2015
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