A187277 Let S denote the palindromes in the language {0,1,2,...,n-1}*; a(n) = number of words of length 4 in the language SS.

1, 16, 57, 136, 265, 456, 721, 1072, 1521, 2080, 2761, 3576, 4537, 5656, 6945, 8416, 10081, 11952, 14041, 16360, 18921, 21736, 24817, 28176, 31825, 35776, 40041, 44632, 49561, 54840, 60481, 66496, 72897, 79696, 86905, 94536, 102601, 111112, 120081, 129520, 139441, 149856, 160777, 172216, 184185

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Table 1.

Formula

From Colin Barker, Jul 24 2013: (Start) (Conjectured formulas; later proven)

a(n) = n*(2*n^2 +n -2).

G.f.: x*(1 +12*x - x^2)/(x-1)^4. (End)

The above conjecture is true: A284873(4, n) evaluates to the same polynomial. - Andrew Howroyd, Oct 10 2017

Maple

Using the Maple code from A007055: [seq(F(b, 4), b=1..50)];

Mathematica

Array[# (2 #^2 + # - 2) &, 45] (* or *)
Rest@ CoefficientList[Series[-x (x^2 - 12 x - 1)/(x - 1)^4, {x, 0, 45}], x] (* Michael De Vlieger, Oct 10 2017 *)

Prog

(PARI) a(n) = 2*n^3 + n^2 - 2*n; \\ Andrew Howroyd, Oct 10 2017
(Magma) [2*n^3 + n^2 - 2*n: n in [1..50]]; // G. C. Greubel, Jul 25 2018
(Python)
def A187277(n): return n*(n*((n<<1)|1)-2) # Chai Wah Wu, Feb 19 2024

Crossrefs

Row 4 of A284873.

Sequence in context: A169882 A202993 A221068 * A316443 A223313 A235679

Adjacent sequences: A187274 A187275 A187276 * A187278 A187279 A187280

Keyword

nonn

Author

N. J. A. Sloane, Mar 07 2011

Status

approved