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A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

%I A271427 #50 Nov 17 2024 16:10:23
%S A271427 0,7,42,301,2100,14707,102942,720601,5044200,35309407,247165842,
%T A271427 1730160901,12111126300,84777884107,593445188742,4154116321201,
%U A271427 29078814248400,203551699738807,1424861898171642,9974033287201501,69818233010410500,488727631072873507,3421093417510114542
%N A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.
%C A271427 In general, the ordinary generating function for the recurrence b(n) = k^n - b(n-1), where n>0 and b(0)=0, is k*x/((1 + x)*(1 - k*x)). This recurrence gives the closed form b(n) = k*(k^n - (-1)^n)/(k + 1).
%H A271427 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,7).
%F A271427 O.g.f.: 7*x/(1 - 6*x - 7*x^2).
%F A271427 E.g.f.: (7/8)*(exp(7*x) - exp(-x)).
%F A271427 a(n) = 6*a(n-1) + 7*a(n-2).
%F A271427 a(n) = 7*(7^n - (-1)^n)/8.
%F A271427 a(n) = 7*A015552(n).
%F A271427 Sum_{n>0} 1/(a(n) + a(n-1)) = 1/6 = A020793.
%F A271427 Limit_{n->oo} a(n-1)/a(n) = 1/7 = A020806.
%e A271427 a(2) = 7^2 - a(2-1) = 49 - 7 = 42.
%e A271427 a(4) = 7^4 - a(4-1) = 2401 - 301 = 2100.
%t A271427 LinearRecurrence[{6, 7}, {0, 7}, 30]
%t A271427 Table[7 (7^n - (-1)^n)/8, {n, 0, 30}]
%o A271427 (PARI) vector(50, n, n--; 7*(7^n-(-1)^n)/8) \\ _Altug Alkan_, Apr 13 2016
%o A271427 (Python) for n in range(0,10**2):print((int)((7*(7**n-(-1)**n))/8))
%o A271427 # _Soumil Mandal_, Apr 14 2016
%Y A271427 Cf. A000420, A015552.
%Y A271427 Cf. similar sequences with the recurrence b(n) = k^n - b(n-1): A125122 (k=1), A078008 (k=2), A054878 (k=3), A109499 (k=4), A109500 (k=5), A109501 (k=6), this sequence (k=7), A093134 (k=8), A001099 (k=n).
%K A271427 nonn,easy
%O A271427 0,2
%A A271427 _Ilya Gutkovskiy_, Apr 13 2016