A131804 -id:A131804 - cp's OEIS frontend

A131118 a(4n) = -n^2, a(4n+1) = n^2, a(4n+2) = 1-n^2, a(4n+3) = n*(n+1).

0, 0, 1, 0, -1, 1, 0, 2, -4, 4, -3, 6, -9, 9, -8, 12, -16, 16, -15, 20, -25, 25, -24, 30, -36, 36, -35, 42, -49, 49, -48, 56, -64, 64, -63, 72, -81, 81, -80, 90, -100, 100, -99, 110, -121, 121, -120, 132, -144, 144, -143, 156, -169, 169, -168

Offset: 0

Paul Curtz, Sep 24 2007

Up to signs, the first differences are in A131804. - R. J. Mathar, Mar 17 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

List([0..60], n-> ((7+4*n-2*n^2)*(-1)^n -2*((n+4)+(n+2)*(-1)^n)*(-1)^Binomial(n+1,2) +5)/32 ); # G. C. Greubel, Nov 18 2019

[((7+4*n-2*n^2)*(-1)^n -2*((n+4)+(n+2)*(-1)^n)*(-1)^Binomial(n+1,2) +5)/32: n in [0..60]]; // G. C. Greubel, Nov 18 2019

seq(((7+4*n-2*n^2)*(-1)^n -2*((n+4)+(n+2)*(-1)^n)*(-1)^binomial(n+1,2) +5)/32, n=0..60); # G. C. Greubel, Nov 18 2019

Table[((7+4*n-2*n^2)*(-1)^n -2*((n+4)+(n+2)*(-1)^n)*(-1)^Binomial[n+1,2] +5)/32, {n,0,60}] (* G. C. Greubel, Nov 18 2019 *)

a(n) = ((7+4*n-2*n^2)*(-1)^n -2*((n+4)+(n+2)*(-1)^n)*(-1)^binomial(n+1,2) +5)/32; \\ G. C. Greubel, Nov 18 2019

[((7+4*n-2*n^2)*(-1)^n -2*((n+4)+(n+2)*(-1)^n)*(-1)^binomial(n+1,2) +5)/32 for n in (0..60)] # G. C. Greubel, Nov 18 2019

From R. J. Mathar, Mar 17 2009: (Start)
a(n) = -2*a(n-1) -2*a(n-2) -2*a(n-3) +2*a(n-5) +2*a(n-6) +2*a(n-7) +a(n-8).
G.f.: x^2*(1+x^2+x^3+2*x)/((1-x)*(1+x^2)^2*(1+x)^3). (End)
a(n) = ((-2*n^2+4*n+7)*(-1)^n - 2*((n+4)+(n+2)*(-1)^n)*i^(n*(n+1))+5)/32, where i=sqrt(-1). - Bruno Berselli, Mar 27 2012

More terms from Sean A. Irvine, Mar 13 2011

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.

Original entry on oeis.org

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

Klaus Brockhaus, Jul 18 2007

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.

			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 ]

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).

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

Cf. A131118.

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] ];

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

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

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

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

cp's OEIS Frontend

A131118 a(4n) = -n^2, a(4n+1) = n^2, a(4n+2) = 1-n^2, a(4n+3) = n*(n+1).

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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.

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