A292625 Triangle read by rows: row n gives y transposed, where y is the solution to the matrix equation M*y=b, where the matrix M and vector b are defined by M(i,j) = ((2^(i+1) + 1)^(j-1) + 1)/2 and b(i) = ((2^(i+1)+1)^n + 1)/2 for 1 <= i,j <= n.

3, -29, 14, 509, -283, 31, -17053, 10104, -1306, 64, 1116637, -682005, 94994, -5466, 129, -144570461, 89619570, -12936231, 800108, -22107, 258, 37221717341, -23243908815, 3414230937, -218563987, 6481607, -88413, 515

Offset: 1

Author

Ahmad J. Masad, Sep 21 2017

Keywords

• sign
• tabl

Comments

The matrix M is given by A266577.

The solution is unique and has an explicit formula as shown by Max Alekseyev, see the MathOverflow link.

Conjecture: as m approaches infinity, the point continuation of the inverse hyperbolic sine scatterplot of the first m*(m+1)/2 terms of this sequence approaches a perfect circular sector with an angle equal to 2*Pi/9. See the last scatterplot in the graph section. - Ahmad J. Masad, Jun 02 2022

Examples

			The first row contains a single term, the solution x=3; the second row contains the solution of the system { x+3y=13, x+5y=41 }, which is x=-29 and y=14; the third row contains the solution of the system { x+3y+13z=63, x+5y+41z=365, x+9y+145z=2457 }, which is x=509, y=-283 and z=31; and so on.
The first seven rows in the triangular array are:
            3;
          -29,           14;
          509,         -283,         31;
       -17053,        10104,      -1306,         64;
      1116637,      -682005,      94994,      -5466,     129;
   -144570461,     89619570,  -12936231,     800108,  -22107,    258;
  37221717341, -23243908815, 3414230937, -218563987, 6481607, -88413, 515;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..50, flattened
• Ahmad J. Masad, Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets, MathOverflow, Sep 2017.

Crossrefs

Cf. A266577, A365450.

Programs

PARI

tblRow(k)=matsolve(matrix(k,k,i,j,((2^(i+1)+1)^(j-1) + 1)/2),vector(k,l,((2^(l+1)+1)^k + 1)/2)~)~;
firstTerms(r)={my(ans=[],t);while(t++<=r,ans=concat(ans,tblRow(t)));return(ans)}
a(n)={my(u);while(binomial(u+1,2)R. J. Cano, Oct 01 2017

Sage

def A292625row(n): return tuple([(-1)^(n+1) * ( product(2^(i+2)+1 for i in range(n)) - 2^(n*(n+3)/2-1) )]) + tuple( (-1)^(n+k) * SymmetricFunctions(QQ).e()[n+1-k].expand(n)( tuple(2^(i+2)+1 for i in range(n)) ) for k in range(2,n+1) ) # Max Alekseyev, Mar 20 2019

Extensions

Edited by Max Alekseyev, Mar 20 2019