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 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). 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified September 12 20:35 EDT 2024. Contains 375854 sequences. (Running on oeis4.)