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A176508
Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 1, b(n) = A003269(n).
1
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 3, 5, 6, 7, 7, 7, 6, 5, 3, 1
OFFSET
0,24
COMMENTS
Row sums are: {1, 2, 2, 2, 2, 6, 10, 14, 18, 30, 51,...}.
LINKS
FORMULA
T(n,m) = A003269(n) - A003269(m) - A003269(n-m) + 1.
EXAMPLE
Triangle:
1;
1, 1;
1, 0, 1;
1, 0, 0, 1;
1, 0, 0, 0, 1;
1, 1, 1, 1, 1, 1;
1, 1, 2, 2, 2, 1, 1;
1, 1, 2, 3, 3, 2, 1, 1;
1, 1, 2, 3, 4, 3, 2, 1, 1;
1, 2, 3, 4, 5, 5, 4, 3, 2, 1;
1, 3, 5, 6, 7, 7, 7, 6, 5, 3, 1;
...
T(6,3) = A003269(6) - A003269(3) - A003269(6-3) + 1 = 3 - 1 - 1 + 1 = 2. - Indranil Ghosh, Feb 17 2017
MATHEMATICA
b[0]:= 0; b[1]:= 1; b[2]:= 1; b[3]:= 1; b[n_]:= b[n] = b[n-1] + b[n-4]; T[n_, m_]:= b[n] - b[m] - b[n-m] + 1; Table[T[n, m], {n, 0, 10}, {m, 0, n} ]//Flatten
PROG
(Python) # See Indranil Ghosh link
(PARI)
{b(n) = if(n==0, 0, if(n==1, 1, if(n==2, 1, if(n==3, 1, b(n-1) +b(n-4)))) )};
{T(n, k) = b(n) -b(k) -b(n-k) +1}; \\ G. C. Greubel, May 06 2019
(Magma) b:= func< n | n eq 0 select 0 else (&+[Binomial(n-1-3*j, j): j in [0..Floor((n-1)/3)]]) >; [[b(n)-b(k)-b(n-k)+1: k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 06 2019
(Sage)
def b(n):
if (n==0): return 0
elif (n==1): return 1
elif (n==2): return 1
elif (n==3): return 1
else: return b(n-1) + b(n-4)
def T(n, k): return b(n) - b(k) - b(n-k) + 1
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 06 2019
CROSSREFS
Cf. A003269.
Sequence in context: A362933 A081503 A351206 * A241492 A227739 A047971
KEYWORD
nonn,easy,tabl
AUTHOR
Roger L. Bagula, Apr 19 2010
EXTENSIONS
Name and formula sections were edited and corrected by Indranil Ghosh, Feb 17 2017
Edited by G. C. Greubel, May 06 2019
STATUS
approved