A029460 -id:A029460 - cp's OEIS frontend

A146205 Number of paths of the simple random walk on condition that the median applied to the partial sums S_0=0, S_1,...,S_n, n odd (n=15 in this example), is equal to half-integer values k+1/2, -[n/2]-1<=k<=[n/2].

35, 35, 245, 245, 735, 735, 1225, 1225, 1225, 1225, 735, 735, 245, 245, 35, 35

Offset: 0

Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Oct 28 2008, May 04 2010

1) Closed-form expressions for sequences see Pfeifer (2010).
2) The median taken on partial sums of the simple random walk represents the market price in a simulation model wherein a single security among non-cooperating and asymetrically informed traders is traded (Pfeifer et al. 2009).
3) A146207=A146205+(0,A146206) see lemma 2 in Pfeifer (2010).

			All possible different paths (sequences of partial sums) in case of n=3:
{0,-1,-2,-3}; median=-1.5
{0,-1,-2,-1}; median=-1
{0,-1,0,-1}; median=-0.5
{0,-1,0,1}; median=0
{0,1,0,-1}; median=0
{0,1,0,1}; median=0.5
{0,1,2,1}; median=1
{0,1,2,3}; median=1.5
sequence of integers in case of n=3: 1,1,1,1

Pfeifer, C. (2010) Probability distribution of the median taken on partial sums of the simple random walk, Submitted to Stochastic Analysis and Applications

C. Pfeifer, K. Schredelseker, G. U. H. Seeber, On the negative value of information in informationally inefficient markets. Calculations for large number of traders, Eur. J. Operat. Res., 195 (1) (2009) 117-126.

A117692, A108347, A029460, A029466, A135553, A137272, A146206, A146207

A029484 Numbers k that divide the (left) concatenation of all numbers <= k written in base 15 (most significant digit on left).

Original entry on oeis.org

1, 7, 32, 49, 61, 91, 169, 224, 791, 1568, 10304, 34112, 160832, 733376, 966721, 1127392, 4197571, 10914848, 13250272, 15000608, 62776133, 70412363, 82053664, 138391456, 198795233, 211659392, 272510336, 484441216, 1448538133, 1846451173, 2444373281, 2681439341, 11942145152, 22206078181, 25210297984

Offset: 1

Olivier Gérard

No other terms below 3*10^10.

No multiple of 3 or 5 can be in this sequence, since the numbers resulting from these concatenations are all congruent to 1 mod 15. - Alonso del Arte, Sep 16 2016

			In base 15, 7654321 is 84557956 in decimal, and we verify that this is a multiple of 7, as 84557956/7 = 12079708. Hence 7 is in the sequence.
87654321 base 15 is 1451432956 and 1451432956/8 = 181429119.5. Hence 8 is not in the sequence.

Index entries for related sequences

Cf. A029447-A029470 (in particular, A029460), A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Select[Range[10^3], Divisible[FromDigits[#, 15] &@ Flatten@ Reverse@ IntegerDigits[Range@ #, 15], #] &] (* Michael De Vlieger, Sep 16 2016 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2001
Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002
a(14)-a(23) from Max Alekseyev, May 15 2011
a(24)-a(35) from Jason Yuen, Jun 05 2024

cp's OEIS Frontend

A146205 Number of paths of the simple random walk on condition that the median applied to the partial sums S_0=0, S_1,...,S_n, n odd (n=15 in this example), is equal to half-integer values k+1/2, -[n/2]-1<=k<=[n/2].

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