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Number of palindromes using a maximum of five different symbols.
11

%I #42 Jun 07 2023 08:31:39

%S 1,5,5,25,25,125,125,625,625,3125,3125,15625,15625,78125,78125,390625,

%T 390625,1953125,1953125,9765625,9765625,48828125,48828125,244140625,

%U 244140625,1220703125,1220703125,6103515625,6103515625,30517578125,30517578125,152587890625,152587890625

%N Number of palindromes using a maximum of five different symbols.

%C 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 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 Vincenzo Librandi, <a href="/A056451/b056451.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,5).

%F a(n) = 5^floor((n+1)/2).

%F a(n) = 5*a(n-2). - _Colin Barker_, May 06 2012

%F 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 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 E.g.f.: cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x). - _Stefano Spezia_, Jun 06 2023

%t LinearRecurrence[{0,5},{1,5},30] (* or *) Riffle[5^Range[0, 20], 5^Range[20]] (* _Harvey P. Dale_, Jul 28 2018 *)

%t Table[5^Ceiling[n/2], {n,0,40}] (* _Robert A. Russell_, Nov 07 2018 *)

%o (Magma) [5^Floor((n+1)/2): n in [0..40]]; // _Vincenzo Librandi_, Aug 16 2011

%o (PARI) vector(40, n, n--; 5^floor((n+1)/2)) \\ _G. C. Greubel_, Nov 07 2018

%Y Column k=5 of A321391.

%Y Cf. A000007, A016116, A056391, A056453, A056454, A056455, A056456, A057427,

%Y Cf. A000351 (oriented), A032122 (unoriented), A032088(n>1) (chiral).

%K nonn,easy

%O 0,2

%A _Marks R. Nester_

%E a(0)=1 prepended by _Robert A. Russell_, Nov 07 2018