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A204162 Symmetric matrix based on f(i,j) = (floor((i+1)/2) if i=j and = 1 otherwise), by antidiagonals.

%I A204162 #14 Nov 06 2018 21:18:05
%S A204162 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,
%T A204162 1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,
%U A204162 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A204162 Symmetric matrix based on f(i,j) = (floor((i+1)/2) if i=j and = 1 otherwise), by antidiagonals.
%C A204162 A204162 represents the matrix M given by f(i,j) = (floor((i+1)/2) if i=j and 1 otherwise) for i >= 1 and j >= 1.  See A204163 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.
%H A204162 Antti Karttunen, <a href="/A204162/b204162.txt">Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)</a>
%e A204162 Northwest corner:
%e A204162   1 1 1 1 1 1
%e A204162   1 1 1 1 1 1
%e A204162   1 1 2 1 1 1
%e A204162   1 1 1 2 1 1
%e A204162   1 1 1 1 3 1
%e A204162   1 1 1 1 1 3
%t A204162 f[i_, j_] := 1; f[i_, i_] := Floor[(i + 1)/2];
%t A204162 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204162 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204162 Flatten[Table[f[i, n + 1 - i],
%t A204162   {n, 1, 15}, {i, 1, n}]]  (* A204162 *)
%t A204162 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204162 c[n_] := CoefficientList[p[n], x]
%t A204162 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204162 Table[c[n], {n, 1, 12}]
%t A204162 Flatten[%]                 (* A204163 *)
%t A204162 TableForm[Table[c[n], {n, 1, 10}]]
%o A204162 (PARI)
%o A204162 up_to = 65703; \\ = binomial(362+1,2)
%o A204162 A204162sq(row,col) = if(row==col,(row+1)\2,1);
%o A204162 A204162list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A204162sq((a-(col-1)),col))); (v); };
%o A204162 v204162 = A204162list(up_to);
%o A204162 A204162(n) = v204162[n]; \\ _Antti Karttunen_, Nov 06 2018
%Y A204162 Cf. A204163, A204016, A202453.
%K A204162 nonn,tabl
%O A204162 1,13
%A A204162 _Clark Kimberling_, Jan 12 2012
%E A204162 Definition corrected to match with terms by _Antti Karttunen_, Nov 06 2018