A131805 Row sums of triangular array T: T(j,k) = -(k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

0, -1, 1, 0, 4, 3, 9, 8, 16, 15, 25, 24, 36, 35, 49, 48, 64, 63, 81, 80, 100, 99, 121, 120, 144, 143, 169, 168, 196, 195, 225, 224, 256, 255, 289, 288, 324, 323, 361, 360, 400, 399, 441, 440, 484, 483, 529, 528, 576, 575, 625, 624, 676, 675, 729, 728, 784, 783, 841

Offset: 0

Author

Klaus Brockhaus, Jul 18 2007

Keywords

• sign
• easy

Comments

Interleaving of A000290 and A067998 (starting at second term).

First differences are -1, 2, -1, 4, -1, 6, -1, 8, -1, 10, ...: a(n+1) - a(n) = (-1)^(n+1)*A124625(n+2).

Main diagonal of T is in A001057, antidiagonal sums are in A131804.

Examples

			First seven rows of T are
[ 0 ],
[ 0, -1 ],
[ 0, -1, 2 ],
[ 0, -1, 3, -2 ],
[ 0, -1, 4, -2, 3 ],
[ 0, -1, 5, -2, 4, -3 ],
[ 0, -1, 6, -2, 5, -3, 4 ]
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000290 (n^2), A067998 (n^2-2*n), A124625, A001057, A131804.

Cf. A131118.

Programs

Magma

m:=59; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:= -k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j, k]: k in [1..j] ]: j in [1..m] ];

Magma

m:=29; &cat[ [ n^2, n^2-1 ]: n in [0..m] ];

Maxima

makelist((2*n*(n-1)+(2*n+3)*(-1)^n-3)/8,n,0,58); /* Bruno Berselli, Mar 27 2012 */

PARI

{m=58; for(n=0, m, r=n%2; print1(((n-r)/2)^2-r, ","))}

Formula

a(0) = 0; a(n) = a(n-1) - (n mod 2) + n*(1 - (n mod 2)) for n > 0.

G.f.: x*(-1+2*x+x^2)/((1-x)^3*(1+x)^2).

a(n) = -A131118(2n) = (2n(n-1)+(2n+3)(-1)^n-3)/8. - Bruno Berselli, Mar 27 2012