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).
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, 1, 1, 127, 255, 343, 343, 255, 127, 1, 1, 255, 575, 863, 979, 863, 575, 255, 1, 1, 511, 1279, 2111, 2655, 2655, 2111, 1279, 511, 1, 1, 1023, 2815, 5055, 6943, 7683, 6943, 5055, 2815, 1023, 1
Offset: 0
Examples
The triangle begins: ..........1 ........1...1 ......1...3...1 ....1...7...7...1 ..1..15..19...15..1 E.g. 19 = 7 + 1 + 1 + 1 + 1 + 1 + 7.
Crossrefs
Programs
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Maple
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; 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);
Formula
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).
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), ...
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
Extensions
Entry revised by N. J. A. Sloane, Jun 15 2005
a(51)=2111 corrected by Georg Fischer, Jul 29 2020