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

1, 5, 5, 25, 25, 125, 125, 625, 625, 3125, 3125, 15625, 15625, 78125, 78125, 390625, 390625, 1953125, 1953125, 9765625, 9765625, 48828125, 48828125, 244140625, 244140625, 1220703125, 1220703125, 6103515625, 6103515625, 30517578125, 30517578125, 152587890625, 152587890625

Offset: 0

Author

Marks R. Nester

Keywords

• nonn
• easy

Comments

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

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]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (0,5).

Crossrefs

Column k=5 of A321391.

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

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

Programs

Magma

[5^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Mathematica

LinearRecurrence[{0,5},{1,5},30] (* or *) Riffle[5^Range[0, 20], 5^Range[20]] (* Harvey P. Dale, Jul 28 2018 *)
Table[5^Ceiling[n/2], {n,0,40}] (* Robert A. Russell, Nov 07 2018 *)

PARI

vector(40, n, n--; 5^floor((n+1)/2)) \\ G. C. Greubel, Nov 07 2018

Formula

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

a(n) = 5*a(n-2). - Colin Barker, May 06 2012

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]

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

E.g.f.: cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x). - Stefano Spezia, Jun 06 2023

Extensions

a(0)=1 prepended by Robert A. Russell, Nov 07 2018