

A056453


Number of palindromes of length n using exactly two different symbols.


15



0, 0, 2, 2, 6, 6, 14, 14, 30, 30, 62, 62, 126, 126, 254, 254, 510, 510, 1022, 1022, 2046, 2046, 4094, 4094, 8190, 8190, 16382, 16382, 32766, 32766, 65534, 65534, 131070, 131070, 262142, 262142, 524286, 524286, 1048574, 1048574, 2097150, 2097150, 4194302
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OFFSET

1,3


COMMENTS

Also the number of bitstrings of length n+2 where the last two runs have the same length. (A run is a maximal subsequence of (possibly just one) identical bits.)  David Nacin, Mar 03 2012
Also, the decimal representation of the diagonal from the corner to the origin of the nth stage of growth of the twodimensional cellular automaton defined by "Rule 62", based on the 5celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero.  Robert Price, Apr 22 2017


REFERENCES

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.


LINKS



FORMULA

a(n) = 2^floor((n+1)/2)  2.
a(n) = a(n1) + 2*a(n2)  2*a(n3).  David Nacin, Mar 03 2012
G.f.: 2*x^3/((1x)*(12*x^2)).  David Nacin, Mar 03 2012
G.f.: k!(x^(2k1)+x^(2k))/Product_{i=1..k}(1ix^2), where k=2 is the number of symbols.  Robert A. Russell, Sep 25 2018
a(n) = k! S2(ceiling(n/2),k), where k=2 is the number of symbols and S2 is the Stirling subset number.  Robert A. Russell, Sep 25 2018
E.g.f.: 1  2*cosh(x) + cosh(sqrt(2)*x)  2*sinh(x) + sqrt(2)*sinh(sqrt(2)*x).  Stefano Spezia, Jun 06 2023


EXAMPLE

The palindromes in two symbols of length three take the forms 000, 111, 010, 101. Of these only two have exactly two symbols. Thus a(3) = 2.  David Nacin, Mar 03 2012


MATHEMATICA

Table[2^(Floor[n/2] + 1)  2, {n, 0, 40}] (* David Nacin, Mar 03 2012 *)
LinearRecurrence[{1, 2, 2}, {0, 0, 2}, 40] (* David Nacin, Mar 03 2012 *)
k=2; Table[k! StirlingS2[Ceiling[n/2], k], {n, 1, 30}] (* Robert A. Russell, Sep 25 2018 *)


PROG

(PARI) a(n) = 2^((n+1)\2)2; \\ Altug Alkan, Sep 25 2018


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



