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%I #39 Dec 22 2021 02:20:12
%S 0,1,0,1,2,0,2,1,2,0,1,2,3,0,3,1,3,0,1,3,2,3,0,2,3,1,2,3,0,1,2,3,4,0,
%T 4,1,4,0,1,4,2,4,0,2,4,1,2,4,0,1,2,4,3,4,0,3,4,1,3,4,0,1,3,4,2,3,4,0,
%U 2,3,4,1,2,3,4,0,1,2,3,4,5,0,5,1,5,0,1,5,2,5,0,2,5,1,2,5,0,1,2,5,3,5,0,3,5
%N Irregular triangle read by rows: row n gives exponents in expression for n as a sum of powers of 2.
%C This sequence contains every increasing finite sequence. For example, the finite sequence {0,2,3,5} arises from n = 45.
%C Essentially A030308(n,k)*k, then entries removed where A030308(n,k)=0. - _R. J. Mathar_, Nov 30 2007
%C In the corresponding irregular triangle {a(n)+1}, the m-th row gives all positive integer roots m_i of polynomial {m,k}. - see link [Shevelev]; see also A264613. - _Vladimir Shevelev_, Dec 13 2015
%H Reinhard Zumkeller, <a href="/A133457/b133457.txt">Rows n = 1..1024 of triangle, flattened</a>
%H Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/m1/m1.Abstract.html">The number of permutations with prescribed up-down structure as a function of two variables</a>, INTEGERS, 12 (2012), #A1. (See Section 3, Theorem 21 and Section 8, Theorem 50)
%F a(n) = A048793(n) - 1.
%e 1 = 2^0.
%e 2 = 2^1.
%e 3 = 2^0 + 2^1.
%e 4 = 2^2.
%e 5 = 2^0 + 2^2.
%e etc. and reading the exponents gives the rows of the triangle.
%p A133457 := proc(n) local a,bdigs,i ; a := [] ; bdigs := convert(n,base,2) ; for i from 1 to nops(bdigs) do if op(i,bdigs) <> 0 then a := [op(a),i-1] ; fi ; od: a ; end: seq(op(A133457(n)),n=1..80) ; # _R. J. Mathar_, Nov 30 2007
%t Array[Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[#, 2] &, 41] // Flatten (* _Michael De Vlieger_, Oct 08 2017 *)
%o (Haskell)
%o a133457 n k = a133457_tabf !! (n-1) !! n
%o a133457_row n = a133457_tabf !! (n-1)
%o a133457_tabf = map (fst . unzip . filter ((> 0) . snd) . zip [0..]) $
%o tail a030308_tabf
%o -- _Reinhard Zumkeller_, Oct 28 2013, Feb 06 2013
%Y Cf. A003071, A030308, A048793, A067255, A264613.
%Y Cf. A073642 (row sums), A272011 (rows reversed).
%K base,tabf,easy,nonn,look
%O 1,5
%A _Masahiko Shin_, Nov 27 2007
%E More terms from _R. J. Mathar_, Nov 30 2007
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