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

0,5

Comments

Row sums are {1, 2, 5, 20, 95, 430, 1815, 7274, 28105, 105752, 390111, ...}.

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)

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.

Example

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

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}], {n, 0, 10}]//Flatten

Prog

(Python) # see Indranil Ghosh link
(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
(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

Crossrefs

Cf. A095263.

Sequence in context: A152655 A144493 A118180 * A045912 A290554 A267264

Adjacent sequences: A176479 A176480 A176481 * A176483 A176484 A176485

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 18 2010

Extensions

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

Edited by G. C. Greubel, May 06 2019

Status

approved