A059780 A generalized Pascal triangle of order 3: T(n,m), n >= 0, 2*n >= m >= 0.

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 3, 2, 3, 1, 1, 4, 4, 4, 5, 4, 4, 4, 1, 1, 5, 7, 6, 9, 7, 9, 6, 7, 5, 1, 1, 6, 11, 10, 14, 14, 15, 14, 14, 10, 11, 6, 1, 1, 7, 16, 17, 21, 26, 25, 29, 25, 26, 21, 17, 16, 7, 1, 1, 8, 22, 28, 32, 44, 44, 52, 49, 52, 44, 44, 32, 28, 22, 8, 1, 1, 9, 29, 44, 50

Offset

0,6

References

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 44.

Links

Table of n, a(n) for n=0..85.

Formula

T(n, m) = T(n-1, m-2) + T(n-1, m-1) + T(n-1, m) - 2*T(n-2, m-2). In words, each entry is the sum of 3 entries above it (i.e., in the previous row) minus twice the entry two rows above it.

Example

Triangle begins
. . . 1
. . 1 1 1
. 1 2 1 2 1
1 3 2 3 2 3 1

Maple

T := proc(n, m) option remember: if m=0 then RETURN(1) fi: if m=2*n then RETURN(1) fi: if n=1 and m=1 then RETURN(1) fi: if n=2 and m mod 2 = 0 and m >= 0 and m<=2*n then RETURN(1) fi: if n=2 and m mod 2 = 1 and m >= 0 and m<=2*n then RETURN(2) fi: if m>2*n then RETURN(0) fi: if m<0 then RETURN(0) fi: T(n-1, m-2)+T(n-1, m-1)+T(n-1, m)-2*T(n-2, m-2): end:for n from 0 to 10 do for m from 0 to 2*n do printf(`%d, `, T(n, m)) od: od: # James A. Sellers, Feb 24 2001

Crossrefs

Sequence in context: A322530 A303364 A131332 * A075119 A224076 A137278

Adjacent sequences: A059777 A059778 A059779 * A059781 A059782 A059783

Keyword

nonn,easy,tabf

Author

N. J. A. Sloane, Feb 22 2001

Extensions

More terms from James A. Sellers, Feb 24 2001

Status

approved