A206806 - OEIS

%I #29 Jan 05 2025 12:31:17

%S 1,4,13,30,62,112,190,300,455,660,931,1274,1708,2240,2892,3672,4605,

%T 5700,6985,8470,10186,12144,14378,16900,19747,22932,26495,30450,34840,

%U 39680,45016,50864,57273,64260,71877,80142,89110,98800,109270,120540,132671,145684

%N Sum_{0<j<k<=n} s(k)-s(j), where s(j)=A002620(j) is the j-th quarter-square.

%C Partial sums of A049774. For a guide to related sequences, see A206817.

%H Vincenzo Librandi, <a href="/A206806/b206806.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).

%F From _Wesley Ivan Hurt_, Jul 10 2014: (Start)

%F a(n) = Sum_{i=1..n} i * (n-i) * (i-ceiling((i-1)/2)).

%F a(n) = (108 - 36n - n^2 + n^4 + (70n - 266) * ceiling((3 - n)/2) - (42n - 234) * ceiling((3 - n)/2)^2 + (8n - 88) * ceiling((3 - n)/2)^3 + 12 * ceiling((3 - n)/2)^4 - 4n * floor(n/2) - (12n - 12) * floor(n/2)^2 - (8n - 24) * floor(n/2)^3 + 12 * floor(n/2)^4) / 12. (End)

%F a(n) = (n*(1+3*(-1)^n-2*n+2*n^2+2*n^3))/48. - _Colin Barker_, Jul 10 2014

%F G.f.: -x^2*(2*x^2+x+1) / ((x-1)^5*(x+1)^2). - _Colin Barker_, Jul 10 2014

%p A206806:=n->add(i*(n-i)*(i-ceil((i-1)/2)), i=1..n): seq(A206806(n), n=2..50); # _Wesley Ivan Hurt_, Jul 10 2014

%t s[k_] := Floor[k/2]*Ceiling[k/2]; t[1] = 0;

%t Table[s[k], {k, 1, 20}]    (* A002620 *)

%t p[n_] := Sum[s[k], {k, 1, n}];

%t c[n_] := n*s[n] - p[n];

%t t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]

%t Table[c[n], {n, 2, 50}]    (* A049774 *)

%t f = Flatten[Table[t[n], {n, 2, 50}]]  (* A206806 *)

%t Table[Sum[i (n - i) (i - Ceiling[(i - 1)/2]), {i, n}], {n, 2, 50}] (* _Wesley Ivan Hurt_, Jul 10 2014 *)

%t CoefficientList[Series[-(2 x^2 + x + 1)/((x - 1)^5 (x + 1)^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 10 2014 *)

%o (Magma) [(108-36*n-n^2+n^4+(70*n-266)*Ceiling((3-n)/2)-(42*n-234)*Ceiling((3-n)/2)^2+(8*n-88)*Ceiling((3-n)/2)^3+12*Ceiling((3-n)/2)^4-4*n*Floor(n/2)-(12*n-12)*Floor(n/2)^2-(8*n-24)*Floor(n/2)^3+12*Floor(n/2)^4)/12: n in [2..50]]; // _Wesley Ivan Hurt_, Jul 10 2014

%o (PARI) vector(100, n, ((n+1)*(1+3*(-1)^(n+1)-2*(n+1)+2*(n+1)^2+2*(n+1)^3))/48) \\ _Colin Barker_, Jul 10 2014

%o (PARI) Vec(-x^2*(2*x^2+x+1)/((x-1)^5*(x+1)^2) + O(x^100)) \\ _Colin Barker_, Jul 10 2014

%o (Sage) [sum([sum([floor(k^2/4)-floor(j^2/4) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,44)] # _Danny Rorabaugh_, Apr 18 2015

%Y Cf. A206817, A002620.

%K nonn,easy

%O 2,2

%A _Clark Kimberling_, Feb 15 2012