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A236377 Real part of Sum_{k=0..n} (k + i^k)^2, where i=sqrt(-1).

%I A236377 #17 Sep 08 2022 08:46:06
%S A236377 1,1,2,10,35,59,84,132,213,293,374,494,663,831,1000,1224,1513,1801,
%T A236377 2090,2450,2891,3331,3772,4300,4925,5549,6174,6902,7743,8583,9424,
%U A236377 10384,11473,12561,13650,14874,16243,17611,18980,20500,22181,23861,25542,27390
%N A236377 Real part of Sum_{k=0..n} (k + i^k)^2, where i=sqrt(-1).
%C A236377 Corresponding imaginary parts: -i^(n*(n+1))*A052928(n+1).
%H A236377 Bruno Berselli, <a href="/A236377/b236377.txt">Table of n, a(n) for n = 0..1000</a>
%H A236377 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-2,-2,4,-4,3,-1).
%F A236377 G.f.: (1 - 2*x + 3*x^2 + 4*x^3 + 11*x^4 - 10*x^5 + 9*x^6)/((1 + x)*(1 + x^2)^2*(1 - x)^4).
%F A236377 a(n) = 3*a(n-1) -4*a(n-2) +4*a(n-3) -2*a(n-4) -2*a(n-5) +4*a(n-6) -4*a(n-7) +3*a(n-8) -a(n-9).
%F A236377 a(n) = A000330(n) + A127630(n) - A000035(n).
%e A236377 For n=6, sum_(k=0)^6 (k + i^k)^2 = 84 + 6*i, therefore a(6) = 84.
%t A236377 LinearRecurrence[{3, -4, 4, -2, -2, 4, -4, 3, -1}, {1, 1, 2, 10, 35, 59, 84, 132, 213}, 50]
%o A236377 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+3*x^2+4*x^3+11*x^4-10*x^5+9*x^6)/((1+x)*(1+x^2)^2*(1-x)^4)));
%Y A236377 Cf. A058373: real part of Sum_{k=0..n} (k + i)^2.
%K A236377 nonn,easy
%O A236377 0,3
%A A236377 _Bruno Berselli_, Jan 24 2014