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, 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

Indranil Ghosh, Rows 0..100, flattened

Indranil Ghosh, Python Program to generate the b-file

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

Adjacent sequences: A176505 A176506 A176507 * A176509 A176510 A176511

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