The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A208903

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A208903

The sum over all bitstrings b of length n with at least two runs of the number of runs in b not immediately followed by a longer run.
(history; published version)

#22 by N. J. A. Sloane at Mon Jul 09 23:21:30 EDT 2018

STATUS

reviewed

approved

#21 by Joerg Arndt at Sat Jul 07 12:35:20 EDT 2018

STATUS

proposed

reviewed

#20 by Jon E. Schoenfield at Sat Jul 07 11:05:10 EDT 2018

STATUS

editing

proposed

#19 by Jon E. Schoenfield at Sat Jul 07 11:05:01 EDT 2018

FORMULA

a(n) = 2^n * (2 + (n - 1)/2 - (1/2)^(n - 1) - 2 (1 - (1/2)^Floor[floor(n/2]) ) + (1/2)^(Floor[floor(n/2] ) + 1) (1 + (-1)^n)) - 2.

EXAMPLE

When n=3, : 101, 010 each have 3, ; 100, 011 each have 1, ; 001, 110 each have 2. (000, 111 do not have at least two runs so they do not contribute.) Summing these gives 6+2+4 = 12 so a(3) = 12. (000,111 do not have at least two runs so they do not contribute.

STATUS

reviewed

editing

Discussion

Sat Jul 07

11:05

Jon E. Schoenfield: (deleted repeated sentence)

#18 by Joerg Arndt at Sat Jul 07 04:01:43 EDT 2018

STATUS

proposed

reviewed

#17 by Michel Marcus at Sat Jul 07 03:58:43 EDT 2018

STATUS

editing

proposed

#16 by Michel Marcus at Sat Jul 07 03:58:36 EDT 2018

REFERENCES

A. Gabhe, Problems and Solutions: 11623, The Amer. Math. Monthly 119 (2012), no. 2, 161.

LINKS

Aruna Gabhe, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.119.02.161">Problem 11623</a>, Am. Math. Monthly 119 (2012) 161.

#15 by Michel Marcus at Sat Jul 07 03:57:55 EDT 2018

LINKS

<a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,-6,16,-8).

CROSSREFS

Cf. A056453, A208900-, A208901, A208902.

STATUS

approved

editing

#14 by Charles R Greathouse IV at Sat Jun 13 00:54:11 EDT 2015

LINKS

<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (5,-6,-6,16,-8).

Discussion

Sat Jun 13

00:54

OEIS Server: https://oeis.org/edit/global/2439

#13 by Charles R Greathouse IV at Fri Jun 12 15:33:20 EDT 2015

LINKS

<a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (5,-6,-6,16,-8).

Discussion

Fri Jun 12

15:33

OEIS Server: https://oeis.org/edit/global/2437