A133457 Irregular triangle read by rows: row n gives exponents in expression for n as a sum of powers of 2.

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, 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, 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

Offset: 1

Author

Masahiko Shin, Nov 27 2007

Keywords

• base
• tabf
• easy
• nonn
• look

Comments

This sequence contains every increasing finite sequence. For example, the finite sequence {0,2,3,5} arises from n = 45.

Essentially A030308(n,k)*k, then entries removed where A030308(n,k)=0. - R. J. Mathar, Nov 30 2007

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

Examples

			1 = 2^0.
2 = 2^1.
3 = 2^0 + 2^1.
4 = 2^2.
5 = 2^0 + 2^2.
etc. and reading the exponents gives the rows of the triangle.
		

Links

• Reinhard Zumkeller, Rows n = 1..1024 of triangle, flattened
• Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. (See Section 3, Theorem 21 and Section 8, Theorem 50)

Crossrefs

Cf. A003071, A030308, A048793, A067255, A264613.

Cf. A073642 (row sums), A272011 (rows reversed).

Programs

Haskell

a133457 n k = a133457_tabf !! (n-1) !! n
a133457_row n = a133457_tabf !! (n-1)
a133457_tabf = map (fst . unzip . filter ((> 0) . snd) . zip [0..]) $
                   tail a030308_tabf
-- Reinhard Zumkeller, Oct 28 2013, Feb 06 2013

Maple

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

Mathematica

Array[Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[#, 2] &, 41] // Flatten (* Michael De Vlieger, Oct 08 2017 *)

Formula

a(n) = A048793(n) - 1.

Extensions

More terms from R. J. Mathar, Nov 30 2007