A176482 - OEIS

%I #24 May 20 2021 08:19:01

%S 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,

%T 304,1,1,983,1284,1369,1369,1284,983,1,1,3179,4159,4454,4519,4454,

%U 4159,3179,1,1,10281,13457,14431,14706,14706,14431,13457,10281,1

%N 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)).

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

%H Indranil Ghosh, <a href="/A176482/b176482.txt">Rows 0..120, flattened</a>

%H B. Adamczewski, Ch. Frougny, A. Siegel and W. Steiner, <a href="http://arxiv.org/abs/0907.0206">Rational numbers with purely periodic beta-expansion</a>,  Bull. Lond. Math. Soc. 42:3 (2010), pp. 538-552; also arXiv:0907.0206 [math.NT], 2009-2010.

%H Indranil Ghosh, <a href="/A176482/a176482.txt">Python Program to generate the b-file</a>

%H Roger L. Bagula, <a href="/A176482/a176482_1.txt">Three methods to generate the sequence b(n)</a>

%F 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.

%e Triangle begins as:

%e   1;

%e   1,     1;

%e   1,     3,     1;

%e   1,     9,     9,     1;

%e   1,    29,    35,    29,     1;

%e   1,    94,   120,   120,    94,     1;

%e   1,   304,   395,   415,   395,   304,     1;

%e   1,   983,  1284,  1369,  1369,  1284,   983,     1;

%e   1,  3179,  4159,  4454,  4519,  4454,  4159,  3179,     1;

%e   1, 10281, 13457, 14431, 14706, 14706, 14431, 13457, 10281,     1;

%e   1, 33249, 43527, 46697, 47651, 47861, 47651, 46697, 43527, 33249, 1;

%e ...

%e T(4,3) = a(4) - a(3) - a(4 - 3) + 1 = 42 - 13 - 1 + 1 = 29. - _Indranil Ghosh_, Feb 18 2017

%t 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

%o (Python) # see Indranil Ghosh link

%o (PARI)

%o {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)))))};

%o {T(n,k) = b(n) -b(k) -b(n-k) +1};

%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 06 2019

%o (Sage)

%o def b(n):

%o     if (n==0): return 0

%o     elif (n==1): return 1

%o     elif (n==2): return 4

%o     elif (n==3): return 13

%o     else: return 4*b(n-1) -3*b(n-2) +2*b(n-3) -b(n-4)

%o def T(n, k): return b(n) - b(k) - b(n-k) + 1

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 06 2019

%Y Cf. A095263.

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Apr 18 2010

%E Name and formula sections edited by _Indranil Ghosh_, Feb 18 2017

%E Edited by _G. C. Greubel_, May 06 2019