A115359 - OEIS

%I #17 Jan 19 2025 14:16:14

%S 1,-1,1,0,0,1,0,-1,0,1,0,0,0,0,1,0,0,-1,0,0,1,0,0,0,0,0,0,1,0,0,0,-1,

%T 0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,

%U 0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1

%N Matrix (1,x)-(x,x^2) in Riordan array notation.

%H Antti Karttunen, <a href="/A115359/b115359.txt">Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle</a>

%F Number triangle T(n, k)=if(n=k, 1, 0) OR if(n=2k+1, -1, 0).

%F a(n) = A010054(n) - A379480(n). [As a flat sequence with starting offset 1] - _Antti Karttunen_, Jan 19 2025

%e Triangle begins:

%e n\k|  0   1   2   3   4   5   6   7   8   9

%e ---+-----------------------------------------

%e 0  |  1;

%e 1  | -1,  1;

%e 2  |  0,  0,  1;

%e 3  |  0, -1,  0,  1;

%e 4  |  0,  0,  0,  0,  1;

%e 5  |  0,  0, -1,  0,  0,  1;

%e 6  |  0,  0,  0,  0,  0,  0,  1;

%e 7  |  0,  0,  0, -1,  0,  0,  0,  1;

%e 8  |  0,  0,  0,  0,  0,  0,  0,  0,  1;

%e 9  |  0,  0,  0,  0, -1,  0,  0,  0,  0,  1;

%e etc. Row and column numbering added by _Antti Karttunen_, Jan 19 2025

%o (PARI) tabl(nn) = {T = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); for (n=1, nn, for (k=1, n, print1(T[n, k], ", ");); print(););} \\ _Michel Marcus_, Mar 28 2015

%o (PARI)

%o A115359off1(n) = (ispolygonal(n,3)-(!(n%2) && issquare(n/2))); \\ (This is one-based)

%o A115359(n) = A115359off1(1+n); \\ (zero-based) - _Antti Karttunen_, Jan 19 2025

%Y Row sums are 1,0,1,0,1,0... (A059841), Diagonal sums are A115360. Inverse is A115361.

%Y Cf. A000217, A001105, A010054, A379480.

%Y Cf. also A115356.

%K easy,sign,tabl

%O 0,1

%A _Paul Barry_, Jan 21 2006