A095802 - OEIS

%I #19 Jan 09 2020 03:57:01

%S 1,-2,9,3,-6,25,-4,15,-10,49,5,-12,35,-14,81,-6,21,-20,63,-18,121,7,

%T -18,45,-28,99,-22,169,-8,27,-30,77,-36,143,-26,225,9,-24,55,-42,117,

%U -44,195,-30,289,-10,33,-40,91,-54,165,-52,255,-34,361,11,-30,65,-56,135,-66,221,-60,323,-38,441,-12,39,-50,105,-72,187,-78,285,-68,399,-42,529

%N Upper right triangular matrix T^2, where T(i,j) = (-1)^i*(1-2*i) for 1 <= i <= j.

%C Equivalently, (lower left) triangle M^2 = transpose(T)^2. The following description refers to the lower triangular version, but OEIS's "TABL" link displays the values more appropriately as an upper right triangle. - _M. F. Hasler_, Apr 18 2009

%C For n rows, use matrices in each row from the sequence 1, -3, 5, -7, ... (filling in with zeros except for the n-th row). Let the matrix = M, then square and delete the zeros. For example, the 3-row generator would be [1 0 0 / 1 -3 0 / 1 -3 5] = M. The nonzero elements of M^2 give the first 6 terms of the sequence.

%F Diagonal elements are the odd squares: a(k(k+1)/2)=(2k+1)^2. First element in row k is (-1)^k*k. - _M. F. Hasler_, Apr 18 2009

%e The matrix

%e   [  1   0   0   0 ...]

%e   [  1  -3   0   0 ...]

%e   [  1  -3   5   0 ...]

%e   [  1  -3   5  -7 ...]

%e squared yields

%e   [ +1   0   0   0 ...]

%e   [ -2  +9   0   0 ...]

%e   [ +3  -6  25   0 ...]

%e   [ -4  15 -10  49 ...]; the lower left triangle gives this sequence: 1; -2, 9; 3, -6, 25; ...

%o (PARI) T=matrix(12,12,i,j,if(j>=i,(-1)^i*(1-2*i)))^2; concat(vector(#T,i,vecextract(T[,i],2^i-1))) \\ _M. F. Hasler_, Apr 18 2009

%Y Row sums with signs as shown = A002412, Hexagonal pyramidal numbers: (1, 7, 22, 50, 95, ...).

%Y Cf. A002412.

%K sign,tabl

%O 1,2

%A _Gary W. Adamson_, Jun 07 2004

%E Edited and extended by _M. F. Hasler_, Apr 18 2009