A386359 a(n) = (1/4)*(9^n - 2*4^n + 3), n > 0; a(0) = 0.

0, 1, 13, 151, 1513, 14251, 130813, 1187551, 10728913, 96724051, 871171813, 7843167751, 70598995513, 635432902651, 5719063896013, 51472246152751, 463252899729313, 4169286834982051, 37523624464511413, 337712791979294551, 3039415815008418313, 27354745083854834251

Offset

0,3

Comments

a(n) is the number of ternary strings of length 2*n with an even number of 1's and 2's (possibly zero) and a positive even number of 0's.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Index entries for linear recurrences with constant coefficients, signature (14,-49,36).

Formula

a(n) = 14*a(n-1) - 49*a(n-2) + 36*a(n-3), n > 3.

G.f.: (18*x^3 - x^2 + x)/((1 - 9*x)*(1 - 4*x)*(1 - x)).

E.g.f.: (1/4)*(exp(9*x) - 2*exp(4*x) + 3*exp(x) - 2).

Example

For n=2, a(2)=13 since the strings of length 4 are (number of permutations in parentheses): 0022 (6), 0011 (6), 0000 (1).
For n=3, a(3)=151 since the strings of length 6 are (number of permutations in parentheses): 000000 (1), 000022 (15), 002222(15), 000011 (15), 001111 (15), 001122 (90).

Mathematica

LinearRecurrence[{14, -49, 36}, {0, 1, 13, 151}, 23] (* Stefano Spezia, Jul 19 2025 *)

Prog

(Magma) I:=[0, 1, 13, 151]; [n le 4 select I[n] else 14*Self(n-1)-49*Self(n-2)+36*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 24 2025

Crossrefs

Cf. A386227, A386293.

Sequence in context: A382898 A242576 A173448 * A396245 A097168 A382516

Adjacent sequences: A386356 A386357 A386358 * A386360 A386361 A386362

Keyword

nonn,easy

Author

Enrique Navarrete, Jul 19 2025

Status

approved