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

0,5

Links

Table of n, a(n) for n=0..65.

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

Example

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.

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);

Crossrefs

T(1, n) gives A000225(n+1), T(2, n) for n>0 gives A006589.

Other diagonals: A063396, A063397, A063398, A063395.

Sequence in context: A176331 A157836 A205497 * A344527 A193871 A108470

Adjacent sequences: A063391 A063392 A063393 * A063395 A063396 A063397

Keyword

easy,nonn,tabl

Author

Floor van Lamoen, Jul 16 2001

Extensions

Entry revised by N. J. A. Sloane, Jun 15 2005

a(51)=2111 corrected by Georg Fischer, Jul 29 2020

Status

approved