|
| |
|
|
A078057
|
|
Expansion of (1+x)/(1-2*x-x^2).
|
|
25
|
|
|
|
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 (waynemv(AT)yahoo.com), 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 tiings showing 8-fold rotational symmetry (see A001333). - L. Edson Jeffery, April 4, 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).
Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.
Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63).
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 0..219
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to linear recurrences with constant coefficients
Shiva Samieinia, Home Page.
Index to sequences with linear recurrences with constant coefficients, signature (2,1).
|
|
|
FORMULA
|
a(0)=1; a(1)=3; a(n) = 2*a(n-1) + a(n-2) - Wayne VanWeerthuizen (waynemv(AT)yahoo.com), May 02 2004
a(n) = 2*a(n-1) + a(n-2); a(n+1)/a(n) tends to silver ratio 1+sqrt(2) as n tends to infinity. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
a(n)=Sum_{k, 0<=k<=n}A147720(n,k)*3^k*(-1/3)^(n-k). [From Philippe DELEHAM, Nov 15 2008]
a(n)=(1/2)*[1+sqrt(2)]^n-(1/2)*sqrt(2)*[1-sqrt(2)]^n+(1/2)*[1-sqrt(2)]^n+(1/2)*[1+sqrt(2)]^n *sqrt(2), with n>=0 [From Paolo P. Lava, Nov 20 2008] a(n) = Pell(n)+Pell(n+1) with Pell(n) = A000129(n). - Johannes W. Meijer, Aug 15 2010
|
|
|
MATHEMATICA
|
Expand[Table[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2, {n, 1, 30}]] - Artur Jasinski, Dec 10 2006
|
|
|
CROSSREFS
|
Essentially the same as A001333, which has many more references.
Cf. A131887, A131935, A000129.
Sequence in context: A089737 A001333 A123335 * A089742 A187258 A131721
Adjacent sequences: A078054 A078055 A078056 * A078058 A078059 A078060
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane, Nov 17 2002
|
|
|
STATUS
|
approved
|
| |
|
|
