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A380463 Partial sums of floor(n^2/13).

%I A380463 #24 Jun 25 2025 07:58:37
%S A380463 0,0,0,0,1,2,4,7,11,17,24,33,44,57,72,89,108,130,154,181,211,244,281,
%T A380463 321,365,413,465,521,581,645,714,787,865,948,1036,1130,1229,1334,1445,
%U A380463 1562,1685,1814,1949,2091,2239,2394,2556,2725,2902,3086,3278,3478
%N A380463 Partial sums of floor(n^2/13).
%H A380463 Hoang Xuan Thanh, <a href="/A380463/b380463.txt">Table of n, a(n) for n = 0..10000</a>
%H A380463 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,0,0,0,0,1,-3,3,-1).
%F A380463 G.f.: ((x^4+x^9)*(1-x+x^2))/((1-x)^3*(1-x^13)).
%F A380463 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-13) - 3*a(n-14) + 3*a(n-15) - a(n-16).
%F A380463 a(n) = floor((2*n^3 + 3*n^2 - 35*n + 48)/78) - [(n+6 mod 13)<6].
%e A380463 a(9) = 0+0+0+0+1+1+2+3+4+6 = 17.
%t A380463 LinearRecurrence[{3, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 3,-1}, {0, 0, 0, 0, 1, 2, 4, 7, 11, 17, 24, 33, 44, 57, 72, 89}, 60]
%o A380463 (PARI) a(n)=(2*n^3+3*n^2-35*n+48)\78 - ((n+6)%13<6)
%o A380463 (SageMath) (((x^4+x^9)*(1-x+x^2))/((1-x)^3*(1-x^13))).series(x, 52).coefficients(x, sparse=False) # _Stefano Spezia_, Jun 23 2025
%Y A380463 Cf. A173653, A173645, A175724.
%K A380463 nonn,easy
%O A380463 0,6
%A A380463 _Hoang Xuan Thanh_, Jun 22 2025