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A036557 Number of multiples of 3 in 0..2^n-1 with an even sum of base-2 digits.

%I A036557 #30 Sep 08 2022 08:44:52
%S A036557 1,1,2,3,6,10,20,35,70,126,252,463,926,1730,3460,6555,13110,25126,
%T A036557 50252,97223,194446,379050,758100,1486675,2973350,5858126,11716252,
%U A036557 23166783,46333566,91869970,183739940,365088395,730176790
%N A036557 Number of multiples of 3 in 0..2^n-1 with an even sum of base-2 digits.
%H A036557 G. C. Greubel, <a href="/A036557/b036557.txt">Table of n, a(n) for n = 0..1000</a>
%H A036557 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-8,-3,6).
%F A036557 From _Ralf Stephan_, Aug 29 2004: (Start)
%F A036557 a(n) = (1/12)*(3^((n+1)/2) + 3^((n+2)/2) + 2^(n+1) + (-1)^n + 3), n > 0.
%F A036557 G.f.: (1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5)/((1-x^2)*(1-2*x)*(1-3*x^2)). (End)
%F A036557 a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) - 3*a(n-4) + 6*a(n-5). - _Wesley Ivan Hurt_, Apr 13 2021
%t A036557 Sum[ Sum[ Binomial[ Floor[ n/2 ], i ], {i, r, n, 6} ]*Sum[ Binomial[ Ceiling[ n/2 ], i ], {i, r, n, 6} ], {r, 0, 5} ]
%t A036557 Join[{1}, LinearRecurrence[{2, 4, -8, -3, 6}, {1, 2, 3, 6, 10}, 50]] (* _G. C. Greubel_, Dec 31 2017 *)
%o A036557 (PARI) x='x+O('x^30); Vec((1-x-4*x^2+3*x^3+3*x^4-x^5)/((1-x^2)*(1-2*x)*(1-3*x^2))) \\ _G. C. Greubel_, Dec 31 2017
%o A036557 (Magma) I:=[1,2,3,6,10]; [1] cat [n le 5 select I[n] else 2*Self(n-1) + 4*Self(n-2) - 8*Self(n-3) - 3*Self(n-4) + 6*Self(n-5): n in [1..30]]; // _G. C. Greubel_, Dec 31 2017
%Y A036557 Cf. A036555, A036556.
%K A036557 nonn,base,easy
%O A036557 0,3
%A A036557 _Dan Hoey_