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 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified June 19 11:32 EDT 2021. Contains 345127 sequences. (Running on oeis4.)