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%I #32 Sep 08 2022 08:46:07
%S 1,3,5,7,11,15,18,24,30,34,42,50,55,65,75,81,93,105,112,126,140,148,
%T 164,180,189,207,225,235,255,275,286,308,330,342,366,390,403,429,455,
%U 469,497,525,540,570,600,616,648,680,697,731,765,783,819,855,874
%N Expansion of (1 + 2*x + 2*x^2)/(1 - x - 2*x^3 + 2*x^4 + x^6 - x^7).
%C Subsequence of A008732: a(n) = A008732(A047212(n+1)).
%C See also Deléham's example in A008732: these numbers are in the first (A000566), third (A005475) and fifth (A028895) column.
%H Bruno Berselli, <a href="/A238738/b238738.txt">Table of n, a(n) for n = 0..1000</a>
%H Bruno Berselli, <a href="/A238738/a238738.jpg">Illustration of the initial terms</a>.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F G.f.: (1 + 2*x + 2*x^2) / ((1 - x)^3*(1 + x + x^2)^2).
%F a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7), with n>6.
%F a(3k) = k*(5*k + 7)/2 + 1 (A000566);
%F a(3k+1) = k*(5*k + 11)/2 + 3 (A005475);
%F a(3k+2) = k*(5*k + 15)/2 + 5 (A028895).
%F a(n) = (floor(n/3)+1)*(4*n-7*floor(n/3)+2)/2. [_Luce ETIENNE_, Jun 14 2014]
%e G.f.: 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 15*x^5 + 18*x^6 + 24*x^7 + ...
%t CoefficientList[Series[(1 + 2 x + 2 x^2)/(1 - x - 2 x^3 + 2 x^4 + x^6 - x^7), {x, 0, 60}], x]
%o (PARI) Vec((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)+O(x^60))
%o (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)));
%o (Maxima) makelist(coeff(taylor((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7), x, 0, n), x, n), n, 0, 60);
%o (Sage) m = 60; L.<x> = PowerSeriesRing(ZZ, m); f = (1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7); print(f.coefficients())
%Y Cf. A000212 (see illustration above), A000217, A008732, A211538.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Mar 04 2014
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