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A056451 Number of palindromes using a maximum of five different symbols.

%I A056451 #42 Jun 07 2023 08:31:39
%S A056451 1,5,5,25,25,125,125,625,625,3125,3125,15625,15625,78125,78125,390625,
%T A056451 390625,1953125,1953125,9765625,9765625,48828125,48828125,244140625,
%U A056451 244140625,1220703125,1220703125,6103515625,6103515625,30517578125,30517578125,152587890625,152587890625
%N A056451 Number of palindromes using a maximum of five different symbols.
%C A056451 Number of achiral rows of n colors using up to five colors. For a(3) = 25, the rows are AAA, ABA, ACA, ADA, AEA, BAB, BBB, BCB, BDB, BEB, CAC, CBC, CCC, CDC, CEC, DAD, DBD, DCD, DDD, DED, EAE, EBE, ECE, EDE, and EEE. - _Robert A. Russell_, Nov 09 2018
%D A056451 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]
%H A056451 Vincenzo Librandi, <a href="/A056451/b056451.txt">Table of n, a(n) for n = 0..2000</a>
%H A056451 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,5).
%F A056451 a(n) = 5^floor((n+1)/2).
%F A056451 a(n) = 5*a(n-2). - _Colin Barker_, May 06 2012
%F A056451 G.f.: (1+5*x) / (1-5*x^2). - _Colin Barker_, May 06 2012 [Adapted to offset 0 by _Robert A. Russell_, Nov 07 2018]
%F A056451 a(n) = C(5,0)*A000007(n) + C(5,1)*A057427(n) + C(5,2)*A056453(n) + C(5,3)*A056454(n) + C(5,4)*A056455(n) + C(5,5)*A056456(n). - _Robert A. Russell_, Nov 08 2018
%F A056451 E.g.f.: cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x). - _Stefano Spezia_, Jun 06 2023
%t A056451 LinearRecurrence[{0,5},{1,5},30] (* or *) Riffle[5^Range[0, 20], 5^Range[20]] (* _Harvey P. Dale_, Jul 28 2018 *)
%t A056451 Table[5^Ceiling[n/2], {n,0,40}] (* _Robert A. Russell_, Nov 07 2018 *)
%o A056451 (Magma) [5^Floor((n+1)/2): n in [0..40]]; // _Vincenzo Librandi_, Aug 16 2011
%o A056451 (PARI) vector(40, n, n--; 5^floor((n+1)/2)) \\ _G. C. Greubel_, Nov 07 2018
%Y A056451 Column k=5 of A321391.
%Y A056451 Cf. A000007, A016116, A056391, A056453, A056454, A056455, A056456, A057427,
%Y A056451 Cf. A000351 (oriented), A032122 (unoriented), A032088(n>1) (chiral).
%K A056451 nonn,easy
%O A056451 0,2
%A A056451 _Marks R. Nester_
%E A056451 a(0)=1 prepended by _Robert A. Russell_, Nov 07 2018