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A133457 Irregular triangle read by rows: row n gives exponents in expression for n as a sum of powers of 2. 11
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 (list; graph; refs; listen; history; text; internal format)
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.

Sequence in context: A227696 A033687 A263452 * A324120 A218857 A260803

Adjacent sequences:  A133454 A133455 A133456 * A133458 A133459 A133460

KEYWORD

base,tabf,easy,nonn,look

AUTHOR

Masahiko Shin, Nov 27 2007

EXTENSIONS

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

STATUS

approved

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Last modified September 21 07:08 EDT 2019. Contains 327253 sequences. (Running on oeis4.)