A171834 - OEIS

%I #25 Sep 08 2022 08:45:50

%S 1,3,10,18,27,37,52,68,85,103,126,150,175,201,232,264,297,331,370,410,

%T 451,493,540,588,637,687,742,798,855,913,976,1040,1105,1171,1242,1314,

%U 1387,1461,1540,1620,1701,1783,1870,1958,2047,2137,2232,2328,2425,2523

%N Partial sums of numbers congruent to {0, 1, 2, 7} mod 8 (A047527).

%H G. C. Greubel, <a href="/A171834/b171834.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).

%F a(n) = Sum_{i=1..n} A047527(i).

%F From _G. C. Greubel_, Sep 04 2018: (Start)

%F a(n) = (4*n^2 + 2*n - 3 + 2*(1 + (-1)^n)*I^n - (-1)^n)/4, where I = sqrt(-1).

%F G.f.: x*(1+x+5*x^2+x^3)/((1-x)^2*(1-x^4)).

%F E.g.f.: (2*cos(x) +(2*x^2 +3*x -1)*sinh(x) +(2*x^2 +3*x -2)*cosh(x))/2. (End)

%t Accumulate[Select[Range[120],MemberQ[{0,1,2,7},Mod[#,8]]&]]  (* _Harvey P. Dale_, Mar 08 2011 *)

%t Table[(4*n^2 +2*n -3 +2*(1 +(-1)^n)*I^n -(-1)^n)/4, {n, 1, 100}] (* _G. C. Greubel_, Sep 04 2018 *)

%o (PARI) vector(100, n, (4*n^2 +2*n -3 +2*(1 +(-1)^n)*I^n -(-1)^n)/4) \\ _G. C. Greubel_, Sep 04 2018

%o (PARI) x='x+O('x^99); Vec(x*(1+x+5*x^2+x^3)/((1-x)^2*(1-x^4))) \\ _Altug Alkan_, Sep 05 2018

%o (Magma) C<I> := ComplexField(); [Round((4*n^2 +2*n -3 +2*(1 +(-1)^n)*I^n -(-1)^n)/4): n in [1..100]]; // _G. C. Greubel_, Sep 04 2018

%Y Cf. A047527.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Dec 19 2009