|
%I #15 Nov 20 2021 11:36:01
%S 1,2,4,8,13,20,31,44,61,84,111,144,186,234,291,360,438,528,634,752,
%T 886,1040,1210,1400,1615,1850,2110,2400,2715,3060,3441,3852,4299,4788,
%U 5313,5880,6496,7154,7861,8624,9436,10304,11236,12224,13276,14400,15588
%N Row sums of correlation triangle for floor((n+3)/3).
%C Row sums of number triangle A115265.
%C Sum of the smallest parts in all the partitions of k into 3 parts such that 3 <= k <= n+3. - _Wesley Ivan Hurt_, Nov 03 2021
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,-3,0,3,0,0,-2,1).
%F G.f.: (1+x+x^2)^2/((1-x^3)^4*(1-x^2)).
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} [j<=k]*floor((k-j+3)/3)*[j<=n-k]*floor((n-k-j+3)/3).
%F From _Wesley Ivan Hurt_, Nov 03 2021: (Start)
%F a(n) = Sum_{m=1..n+3} Sum_{k=1..floor(m/3)} Sum_{i=k..floor((m-k)/2)} k.
%F a(n) = 2*a(n-1)-3*a(n-4)+3*a(n-6)-2*a(n-9)+a(n-10). (End)
%t T[n_, k_] := Sum[Boole[j <= k] * Floor[(k - j + 3)/3] * Boole[j <= n - k] * Floor[(n - k - j + 3)/3], {j, 0, n}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 46}] (* _Jean-François Alcover_, Jul 15 2017 *)
%t LinearRecurrence[{2,0,0,-3,0,3,0,0,-2,1},{1,2,4,8,13,20,31,44,61,84},50] (* _Harvey P. Dale_, Nov 20 2021 *)
%Y Cf. A115265.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jan 18 2006
|
