A133457 - OEIS

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

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OFFSET

1,5

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

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)

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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

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