cp's OEIS Frontend

A063394 Border sum triangle, read by rows: Let T(n,0)=T(n,n)=1. In general T(n,m) is the sum of the elements (apart from T(n,m) itself) in the border of the rectangle with vertices T(0,0), T(n-m,0), T(n,m) and T(m,m).

%I A063394 #13 Jul 29 2020 08:37:42
%S A063394 1,1,1,1,3,1,1,7,7,1,1,15,19,15,1,1,31,47,47,31,1,1,63,111,131,111,63,
%T A063394 1,1,127,255,343,343,255,127,1,1,255,575,863,979,863,575,255,1,1,511,
%U A063394 1279,2111,2655,2655,2111,1279,511,1,1,1023,2815,5055,6943,7683,6943,5055,2815,1023,1
%N A063394 Border sum triangle, read by rows: Let T(n,0)=T(n,n)=1. In general T(n,m) is the sum of the elements (apart from T(n,m) itself) in the border of the rectangle with vertices T(0,0), T(n-m,0), T(n,m) and T(m,m).
%F A063394 If m < 0 or m > n then T(n, m) = 0; if m = 0 or m = n then T(n, m) = 1; otherwise T(n, m) = Sum( T(n-i, m-i), i=1..m) + Sum( T(n-i, m), i=1..n-m) + Sum( T(n-m-i, 0), i=1..n-m) + Sum( T(i, i), i=1..m-1).
%F A063394 The U-coordinates are nicer. Label the elements U(0, 0), U(1, 0), U(0, 1), U(2, 0), U(1, 1), U(0, 2), ...
%F A063394 Then U(n, 0) = U(0, m) = 1; for n>=1, m>=1, U(n, m) = Sum_{i=0..n-1} U(i, 0) + Sum_{j=0..m-1} U(0, j) - U(0, 0) + Sum_{j=0..m-1} U(n, j) + Sum_{i=0..n-1} U(i, m). Hence U(z, w) = Sum U(n, m) z^n w^m = (1-2*z-2*w+5*z*w-2*z^2*w^2)/((1-z)*(1-w)*(1-2*z-2*w+3*z*w)). - _N. J. A. Sloane_, Jun 16 2005
%e A063394 The triangle begins:
%e A063394 ..........1
%e A063394 ........1...1
%e A063394 ......1...3...1
%e A063394 ....1...7...7...1
%e A063394 ..1..15..19...15..1
%e A063394 E.g. 19 = 7 + 1 + 1 + 1 + 1 + 1 + 7.
%p A063394 T:=proc(n,m) option remember; local i,j,k,t1,t2,t3; if m < 0 or m > n then RETURN(0); fi; if m = 0 or m = n then RETURN(1); fi; add( T(n-i,m-i),i=1..m) + add( T(n-i,m),i=1..n-m) + add( T(n-m-i,0),i=1..n-m) + add( T(i,i),i=1..m-1); end;
%p A063394 U:=(1-2*z-2*w+5*z*w-2*z^2*w^2)/(1-z)/(1-w)/(1-2*z-2*w+3*z*w);
%Y A063394 T(1, n) gives A000225(n+1), T(2, n) for n>0 gives A006589.
%Y A063394 Other diagonals: A063396, A063397, A063398, A063395.
%K A063394 easy,nonn,tabl
%O A063394 0,5
%A A063394 _Floor van Lamoen_, Jul 16 2001
%E A063394 Entry revised by _N. J. A. Sloane_, Jun 15 2005
%E A063394 a(51)=2111 corrected by _Georg Fischer_, Jul 29 2020