A152046 a(n) = Product_{k=1..floor((n-1)/2)} (1 + 8*cos(k*Pi/n)^2) for n >= 0.

1, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531, 5726623061, 11453246123

Offset: 0

Author

Roger L. Bagula, Nov 21 2008

Keywords

• nonn
• easy

Comments

Apparently the same as A001045 after the first term. - R. J. Mathar, Nov 27 2008 [This conjecture is true. - Peter Luschny, Nov 28 2019]

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Forrest M. Hilton, Finite Dynamical Laminations, arXiv:2408.01353 [math.DS], 2024. See p. 16.
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Crossrefs

Cf. A001045.

Programs

Mathematica

a[n_] := Product[(1 + 8 Cos[k Pi/n]^2), {k, 1, Floor[(n - 1)/2]}];
Table[Round[a[n]], {n, 0, 20}]

PARI

Vec((1 - 2*x^2) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 28 2019

PARI

apply( {A152046(n)=2^n\/3+!n}, [0..40]) \\ M. F. Hasler, Feb 13 2020

Formula

From Sergei N. Gladkovskii, May 22 2013 and Sep 09 2013: (Start)

G.f.: 1 + A(x) where A(x) is the g.f. of A001045.

G.f.: 1 + Q(0)/3, where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction).

G.f.: 1+ Q(0)*x/2 , where Q(k) = 1 + 1/(1 - x*(2*k+1 + 2*x)/( x*(2*k+2 + 2*x) + 1/Q(k+1) )); (continued fraction). (End)

From Colin Barker, Nov 28 2019: (Start)

a(n) = a(n-1) + 2*a(n-2) for n>2.

a(n) = ((-1)^(1 + n) + 2^n)/ 3 for n>0. (End)

E.g.f.: (3 - exp(-x) + exp(2*x))/3. - Stefano Spezia, Feb 13 2020

Extensions

Edited by Peter Luschny, Nov 28 2019