A285998 - OEIS

%I #29 Dec 23 2022 16:22:56

%S 1,4,7,16,22,40,50,80,95,140,161,224,252,336,372,480,525,660,715,880,

%T 946,1144,1222,1456,1547,1820,1925,2240,2360,2720,2856,3264,3417,3876,

%U 4047,4560,4750,5320,5530,6160,6391,7084,7337,8096,8372,9200,9500,10400,10725

%N a(n) = Sum_{k=0..floor(n/2)} (n-k)*(k+1).

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

%F G.f.: x * (1+3*x)/((1-x)^4*(1+x)^3). - _Joerg Arndt_, Jun 25 2017

%F a(n) = 1/16 + 13*n/24 + 7*n^2/16 + n^3/12 + (-1)^n*(-1/16 + n/8 + n^2/16). - _Vaclav Kotesovec_, Jun 25 2017

%F E.g.f.: (x*(21 + 18*x + 2*x^2)*cosh(x) + (3 + 30*x + 15*x^2 + 2*x^3)*sinh(x))/24. - _Stefano Spezia_, Dec 23 2022

%e a(2) = (2*1)+(1*2) = 4.

%e a(3) = (3*1)+(2*2) = 7.

%e a(4) = (4*1)+(3*2)+(2*3) = 16.

%e a(5) = (5*1)+(4*2)+(3*3) = 22.

%t Table[Sum[(n - k) (k + 1), {k, 0, Floor[n/2]}], {n, 49}] (* _Michael De Vlieger_, Jun 23 2017 *)

%o (PARI) a(n) = sum(k=0, n\2, (n-k)*(k+1)); \\ _Michel Marcus_, Jun 15 2017

%o (PARI) a(n) = my(r = n%2, n = (n + 4)>>1); 4 * binomial(n, 3) + r * binomial(n, 2); \\ _David A. Corneth_, Jun 23 2017

%Y Cf. A000292 (with n instead of n/2).

%K nonn,easy

%O 1,2

%A _Zhandos Mambetaliyev_, Jun 15 2017

%E More terms from _Michel Marcus_, Jun 15 2017