|
| |
|
|
A078057
|
|
Expansion of (1+x)/(1-2*x-x^2).
|
|
67
|
|
|
|
1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243, 275807, 665857, 1607521, 3880899, 9369319, 22619537, 54608393, 131836323, 318281039, 768398401, 1855077841, 4478554083, 10812186007, 26102926097, 63018038201, 152139002499, 367296043199, 886731088897
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Let x_n be the sequence 1,3,7,17,41,99,239,... (this sequence or A001333) and let y_n = 1,2,5,12,29,70,169,... (A000129). Then {+- x_n +- y_n*sqrt(2) } are the units in the ring of algebraic integers Z[ sqrt(2) ].
Consider a string of n red, blue and green beads (with start and end points distinct and not interchangeable). If one pairing is disallowed, so that a red bead cannot immediately follow a blue bead or vice versa, how many different strings exist of any given length? Answer is a(n). E.g., a(3)=17 because there are 17 strings of length 3: RRR, RRG, RGR, RGG, RGB, GRR, GRG, GGR, GGG, GGB, GBG, GBB, BGR, BGG, BGB, BBG, BBB - Wayne VanWeerthuizen, May 02 2004
The number of Khalimsky-continuous functions with one fixed endpoint. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
The sequence (-1)^C(n+1,2)*a(n) with g.f. (1-3x-x^2-x^3)/(1+6x^2+x^4) is the Hankel transform of the signed central binomial coefficients (-1)^C(n+1,2)*A001405(n). - Paul Barry, Jun 24 2008
An elephant sequence, see A175655. For the central square six A[5] vectors, with decimal values between 21 and 336, lead to this sequence. For the corner squares these vectors lead to the companion sequence A000129 (without the leading 0). - Johannes W. Meijer, Aug 15 2010
Sequence is related to rhombus substitution tilings showing 8-fold rotational symmetry (see A001333). - L. Edson Jeffery, Apr 04 2011
Number of length-n strings of 3 letters {0,1,2} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012
|
|
|
REFERENCES
|
A. Froehlich and M. J. Taylor, Algebraic Number Theory, Cambridge, 1991 (see p. 3).
Thomas Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = 2*a(n-1) + a(n-2); lim_{n->oo} a(n+1)/a(n) = 1 + sqrt(2) (i.e., the silver ratio). - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
G.f.: G(0)/(2*x) -1/x, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = T(n+1, i) / i^(n+1), where T(n, x) denotes the Chebyshev polynomial of the first kind. - Michael Somos, Jul 28 2018
E.g.f.: exp(x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Jan 31 2023
|
|
|
EXAMPLE
|
G.f. = 1 + 3*x + 7*x^2 + 17*x^3 + 41*x^4 + 99*x^5 + 239*x^6 + 577*x^7 + ... - Michael Somos, Jul 28 2018
|
|
|
MATHEMATICA
|
Expand[Table[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2, {n, 1, 30}]] (* Artur Jasinski, Dec 10 2006 *)
CoefficientList[Series[(1 + x)/(1 - 2 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2014 *)
a[ n_] := ChebyshevT[n+1, I] / I^(n+1); (* Michael Somos, Jul 28 2018 *)
|
|
|
PROG
|
(Haskell)
(PARI) {a(n) = polchebyshev(n+1, 1, I) / I^(n+1)}; /* Michael Somos, Jul 28 2018 */
|
|
|
CROSSREFS
|
Essentially the same as A001333, which has many more references.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
