

A078057


Expansion of (1+x)/(12*xx^2).


65



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
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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 Khalimskycontinuous 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. (13xx^2x^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 8fold rotational symmetry (see A001333).  L. Edson Jeffery, Apr 04 2011
Number of lengthn 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(L1)*xx^2).  Joerg Arndt, Oct 11 2012
Row sums of A035607, when seen as a triangle read by rows.  Reinhard Zumkeller, Jul 20 2013


REFERENCES

A. Froehlich and M. J. Taylor, Algebraic Number Theory, Cambridge, 1991 (see p. 3).
Thomas Koshy, Pell and PellLucas Numbers with Applications, Springer, New York, 2014.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. BautistaRamos and C. GuillenGalvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8
Tanya Khovanova, Recursive Sequences
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353374.
Shiva Samieinia, Digital straight line segments and curves, Licentiate Thesis, Stockholm University, Department of Mathematics, Report 2007:6.
S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 7589
Gy. Tasi and F. Mizukami, Quantum algebraiccombinatoric study of the conformational properties of nalkanes, J. Math. Chemistry, 25, 1999, 5564 (see p. 63).
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

a(n) = 2*a(n1) + a(n2); a(0)=1; a(1)=3.  Wayne VanWeerthuizen, May 02 2004
a(n) = 2*a(n1) + a(n2); 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)^(nk).  Philippe Deléham, Nov 15 2008
a(n) = (1/2)*[1+sqrt(2)]^n(1/2)*sqrt(2)*[1sqrt(2)]^n+(1/2)*[1sqrt(2)]^n+(1/2)*[1+sqrt(2)]^n *sqrt(2), with n>=0.  Paolo P. Lava, Nov 20 2008
a(n) = Pell(n)+Pell(n+1) with Pell(n) = A000129(n).  Johannes W. Meijer, Aug 15 2010
G.f.: G(0)/(2*x) 1/x, where G(k)= 1 + 1/(1  x*(2*k1)/(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


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)
a078057 = sum . a035607_row  Reinhard Zumkeller, Jul 20 2013
(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.
Cf. A131887, A131935, A000129.
Sequence in context: A089737 A123335 A001333 * A089742 A187258 A131721
Adjacent sequences: A078054 A078055 A078056 * A078058 A078059 A078060


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Nov 17 2002


STATUS

approved



