A176482 Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 1 (see formula section for recurrence for b(n)).

1, 1, 1, 1, 3, 1, 1, 9, 9, 1, 1, 29, 35, 29, 1, 1, 94, 120, 120, 94, 1, 1, 304, 395, 415, 395, 304, 1, 1, 983, 1284, 1369, 1369, 1284, 983, 1, 1, 3179, 4159, 4454, 4519, 4454, 4159, 3179, 1, 1, 10281, 13457, 14431, 14706, 14706, 14431, 13457, 10281, 1

Offset: 0

Author

Roger L. Bagula, Apr 18 2010

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     3,     1;
  1,     9,     9,     1;
  1,    29,    35,    29,     1;
  1,    94,   120,   120,    94,     1;
  1,   304,   395,   415,   395,   304,     1;
  1,   983,  1284,  1369,  1369,  1284,   983,     1;
  1,  3179,  4159,  4454,  4519,  4454,  4159,  3179,     1;
  1, 10281, 13457, 14431, 14706, 14706, 14431, 13457, 10281,     1;
  1, 33249, 43527, 46697, 47651, 47861, 47651, 46697, 43527, 33249, 1;
...
T(4,3) = a(4) - a(3) - a(4 - 3) + 1 = 42 - 13 - 1 + 1 = 29. - _Indranil Ghosh_, Feb 18 2017
		

Links

• Indranil Ghosh, Rows 0..120, flattened
• B. Adamczewski, Ch. Frougny, A. Siegel and W. Steiner, Rational numbers with purely periodic beta-expansion, Bull. Lond. Math. Soc. 42:3 (2010), pp. 538-552; also arXiv:0907.0206 [math.NT], 2009-2010.
• Indranil Ghosh, Python Program to generate the b-file
• Roger L. Bagula, Three methods to generate the sequence b(n)

Crossrefs

Cf. A095263.

Programs

Mathematica

b[0]:=0; b[1]:=1; b[2]:=4; b[3]=13; b[n_]:= b[n]= 4*b[n-1] -3*b[n-2] + 2*b[n-3] -b[n-4]; T[n_, m_]:=b[n]-b[m]-b[n-m]+1; Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

PARI

{b(n) = if(n==0, 0, if(n==1, 1, if(n==2, 4, if(n==3, 13, 4*b(n-1) -3*b(n-2) + 2*b(n-3) -b(n-4)))))};
{T(n,k) = b(n) -b(k) -b(n-k) +1};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, May 06 2019

Python

# see Indranil Ghosh link

Sage

def b(n):
    if (n==0): return 0
    elif (n==1): return 1
    elif (n==2): return 4
    elif (n==3): return 13
    else: return 4*b(n-1) -3*b(n-2) +2*b(n-3) -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

Formula

With b(n) = 4*b(n-1) - 3*b(n-2) + 2*b(n-3) - b(n-4), with b(0) = 0, b(1) = 1, b(2) = 4 and b(3) = 13, then the triangle is generated by T(n, k) = b(n) - b(k) - b(n-k) + 1.

Extensions

Name and formula sections edited by Indranil Ghosh, Feb 18 2017

Edited by G. C. Greubel, May 06 2019