This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A176482 #29 Jun 11 2025 02:51:31
%S A176482 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 A176482 304,1,1,983,1284,1369,1369,1284,983,1,1,3179,4159,4454,4519,4454,
%U A176482 4159,3179,1,1,10281,13457,14431,14706,14706,14431,13457,10281,1
%N 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)).
%H A176482 Indranil Ghosh, <a href="/A176482/b176482.txt">Rows 0..120, flattened</a>
%H A176482 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 A176482 Indranil Ghosh, <a href="/A176482/a176482.txt">Python Program to generate the b-file</a>
%H A176482 Roger L. Bagula, <a href="/A176482/a176482_1.txt">Three methods to generate the sequence b(n)</a>
%F A176482 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 A176482 Triangle begins as:
%e A176482 1;
%e A176482 1, 1;
%e A176482 1, 3, 1;
%e A176482 1, 9, 9, 1;
%e A176482 1, 29, 35, 29, 1;
%e A176482 1, 94, 120, 120, 94, 1;
%e A176482 1, 304, 395, 415, 395, 304, 1;
%e A176482 1, 983, 1284, 1369, 1369, 1284, 983, 1;
%e A176482 1, 3179, 4159, 4454, 4519, 4454, 4159, 3179, 1;
%e A176482 1, 10281, 13457, 14431, 14706, 14706, 14431, 13457, 10281, 1;
%e A176482 1, 33249, 43527, 46697, 47651, 47861, 47651, 46697, 43527, 33249, 1;
%e A176482 ...
%e A176482 T(4,3) = a(4) - a(3) - a(4 - 3) + 1 = 42 - 13 - 1 + 1 = 29. - _Indranil Ghosh_, Feb 18 2017
%t A176482 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
%o A176482 (Python) # see Indranil Ghosh link
%o A176482 (PARI)
%o A176482 {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 A176482 {T(n,k) = b(n) -b(k) -b(n-k) +1};
%o A176482 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 06 2019
%o A176482 (Sage)
%o A176482 def b(n):
%o A176482 if (n==0): return 0
%o A176482 elif (n==1): return 1
%o A176482 elif (n==2): return 4
%o A176482 elif (n==3): return 13
%o A176482 else: return 4*b(n-1) -3*b(n-2) +2*b(n-3) -b(n-4)
%o A176482 def T(n, k): return b(n) - b(k) - b(n-k) + 1
%o A176482 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 06 2019
%Y A176482 Cf. A095263.
%K A176482 nonn,tabl,easy
%O A176482 0,5
%A A176482 _Roger L. Bagula_, Apr 18 2010
%E A176482 Name and formula sections edited by _Indranil Ghosh_, Feb 18 2017
%E A176482 Edited by _G. C. Greubel_, May 06 2019