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A251861 Number of non-palindromic words (length n>0) over the alphabet of 26 letters.

%I A251861 #49 Jun 15 2025 01:33:28
%S A251861 0,650,16900,456300,11863800,308898200,8031353200,208826607600,
%T A251861 5429491797600,141167083772000,3670344178072000,95428956352766400,
%U A251861 2481152865171926400,64509974695265340800,1677259342076898860800,43608742899220046995200,1133827315379721221875200,29479510200008489360729600,766467265200220723378969600,19928148895209267985244544000
%N A251861 Number of non-palindromic words (length n>0) over the alphabet of 26 letters.
%C A251861 Example: the acronyms 'OEIS' and 'SIEO' are two distinct non-palindromic words of length 4 among all possible such 456300 words (over 26 letters of the Latin alphabet).
%H A251861 Indranil Ghosh, <a href="/A251861/b251861.txt">Table of n, a(n) for n = 1..705</a>
%H A251861 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (26,26,-676).
%F A251861 a(n) = 2^(n/2-1)*13^(n/2)*((-1)^n*(sqrt(26)-1)-sqrt(26)-1)+26^n.
%F A251861 a(n) = 26^n - 26^ceiling(n/2).
%F A251861 G.f.: 650*x^2/((1 - 26*x)*(1 - 26*x^2)).
%F A251861 a(n+3) = 26*a(n+2) + 26*a(n+1) - 676*a(n). - _Robert Israel_, Dec 11 2014
%e A251861 For n=2, the a(2)=650 solutions are {ab,ac,...,az,...,yz}, but not, e.g., 'aa' or 'zz'.
%p A251861 seq(26^n - 26^ceil(n/2), n = 1 .. 50); # _Robert Israel_, Dec 11 2014
%t A251861 f[n_, b_] := b^n - b^Ceiling[n/2]; Array[ f[#, 26] &, 50] (* _Robert G. Wilson v_, Dec 10 2014 *)
%t A251861 Table[2^(n/2-1)*13^(n/2)*((-1)^n*(Sqrt[26]-1)-Sqrt[26]-1)+26^n, {n, 50}]
%o A251861 (PARI) a(n)=26^n-26^ceil(n/2) \\ _Charles R Greathouse IV_, Dec 10 2014
%Y A251861 Analogs for other numbers of elements: (1) A000004, (2) A233411, (3) A242278, (4) A242026, (5) A240437.
%Y A251861 Cf. A056450.
%K A251861 nonn,easy
%O A251861 1,2
%A A251861 _Mikk Heidemaa_, Dec 10 2014