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A130330 Triangle read by rows, the matrix product A130321 * A000012, both taken as infinite lower triangular matrices. 5
1, 3, 1, 7, 3, 1, 15, 7, 3, 1, 31, 15, 7, 3, 1, 63, 31, 15, 7, 3, 1, 127, 63, 31, 15, 7, 3, 1, 255, 127, 63, 31, 15, 7, 3, 1, 511, 255, 127, 63, 31, 15, 7, 3, 1, 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1, 2047, 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums are A000295: (1, 4, 11, 26, 57, 120, 247,...), the Eulerian numbers.

T(n,k) is the number of length n+1 binary words containing at least two 1's such that the first 1 is preceded by exactly (k-1) 0's. T(3,2) = 3 because we have: 0101, 0110, 0111. - Geoffrey Critzer, Dec 31 2013

Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/

having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A110441. - Peter Bala, Jul 22 2014

From Wolfdieter Lang, Oct 28 2019:(Start)

This triangle gives the solution of the following problem. Iterate the function f(x) = (x - 1)/2 to obtain f^{[k]}(x) = (x - (2^(k+1) - 1))/2^(k+1), for k >= 0. Find the positive integer x values for which the iterations stay integer and reach 1. Only odd integers x qualify, and the answer is x = x(n) = 2*T(n, 0) = 2*(2^(n+1) - 1), with the iterations T(n,0), ..., T(n,n) = 1.

This iteration is motivated by a problem posed by Johann Peter Hebel (1760 - 1826) in "Zweites Rechnungsexempel" from 1804, with the solution x = 31 corresponding to row n = 3 [15 7 3 1]. The egg selling woman started with 31 = T(4, 0) eggs and after four customers obtained, one after the other, always a number of eggs which was one half of the woman's remaining number of eggs plus 1/2 (selling only whole eggs, of course) she had one egg left. See the link and reference. [For Hebel's first problem see a comment in A000225.]

(End)

REFERENCES

Johann Peter Hebel, Gesammelte Werke in sechs Bänden, Herausgeber: Jan Knopf, Franz Littmann und Hansgeorg Schmidt-Bergmann unter Mitarbeit von Ester Stern, Wallstein Verlag, 2019. Band 3, S. 36-37, Solution, S. 40-41. See also the link below.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..11324 [first 150 rows; offset shifted by Georg Fischer, Oct 29 2019]

Johann Peter Hebel, Zweites Rechnungsexempel., 1804; Solution: Auflösung des zweiten Rechnungsexempels. , 1805.

FORMULA

A130321 * A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1;...) and A000012 = (1; 1,1; 1,1,1;...).

In every column k with offset n = k: 2^(m+1) - 1 = A000225(m+1) = (1, 3, 7, 15, ...), for m >= 0.

G.f.: 1/((1-y*x)*(1-x)*(1-2x)). - Geoffrey Critzer, Dec 31 2013

T(n, k) = 2^((n - k) + 1) - 1, n >= 0, k = 0..n. - Wolfdieter Lang, Oct 28 2019

EXAMPLE

First few rows of the triangle T(n, k) are:

n\k     0    1    2    3   4   5   6  7  8  9 10 11 12 ...

0:      1

1:      3    1

2:      7    3    1

3      15    7    3    1

4:     31   15    7    3   1

5:     63   31   15    7   3   1

6:    127   63   31   15   7   3   1

7:    255  127   63   31  15   7   3  1

8:    511  255  127   63  31  15   7  3  1

9:   1023  511  255  127  63  31  15  7  3  1

10:  2047 1023  511  255 127  63  31 15  7  3  1

11:  4095 2047 1023  511 255 127  63 31 15  7  3  1

12:  8191 4095 2047 1023 511 255 127 63 31 15  7  3  1

... reformatted and extended. - Wolfdieter Lang, Oct 28 2019

MATHEMATICA

nn=12; a=1/(1- x); b=1/(1-2x); Map[Select[#, #>0&]&, Drop[CoefficientList[Series[a x^2 b/(1-y x), {x, 0, nn}], {x, y}], 2]]//Grid  (* Geoffrey Critzer, Dec 31 2013 *)

PROG

(Haskell)

a130330 n k = a130330_row n !! (k-1)

a130330_row n = a130330_tabl !! (n-1)

a130330_tabl = iterate (\xs -> (2 * head xs + 1) : xs) [1]

-- Reinhard Zumkeller, Mar 31 2012

CROSSREFS

Cf. A130321, A000012, A000225. A110441.

Sequence in context: A249755 A249248 A104797 * A050227 A135540 A161942

Adjacent sequences:  A130327 A130328 A130329 * A130331 A130332 A130333

KEYWORD

nonn,easy,tabl

AUTHOR

Gary W. Adamson, May 24 2007

EXTENSIONS

More terms from Geoffrey Critzer, Dec 31 2013

Edited by Wolfdieter Lang, Oct 28 2019

STATUS

approved

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Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.)