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A155020 a(0)=1, a(1)=1, a(2)=3; a(n) = 2*a(n-1) + 2*a(n-2) for n>2. 6
1, 1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equals 1 followed by A028859. - Klaus Brockhaus, Jul 18 2009

a(n) is the number of ways to arrange 1- and 2-cent postage stamps (totaling n cents) in a row so that the first stamp is correctly placed and any subsequent stamp may (or not) be placed upside down.

Number of compositions of n into parts k>=1 where there are F(k+1)=A000045(k+1) sorts of part k. - Joerg Arndt, Sep 30 2012

a(n) is the top-left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 1, 1].

From Tom Copeland, Nov 08 2014: (Start)

(Setting a(0)=0.)

This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t)=x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf. A091867.)

O.g.f.: G(x) =  -P[P[Cinv(-x),1],1] = -P[Cinv(-x),2] = x(1+x)/[1-2x(1+x)] = (x+x^2)/[1-2(x+x^2)] = x + 3 x^2 + 8 x^3 + ... = A155020(x) with a(0)=0.

Ginv(x) = -C[P[P(-x,-1),-1]] = -C[P(-x,-2)] = [-1+sqrt(1+4*x/(1+2x)]/2 = x*A064613(-x).

G(x) = x*(1+x) + 2*[x*(1+x)]^2 + 2^2*[x*(1+x)]^3 - ... , and so this array contains the row sums of A030528 * Diag(1,2^1,2^2,2^3,..).  (End)

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..2287

I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015.

Index entries for linear recurrences with constant coefficients, signature (2,2).

FORMULA

G.f.: (1 - x - x^2)/(1 - 2*x - 2*x^2).

G.f.: 1/( 1 - sum(k>=1, (x+x^2)^k ) ) - 1/( 1 - sum(k>=1, F(k+1)*x^k ) ) where F(k)=A000045(k). - Joerg Arndt, Sep 30 2012

a(n+1) = Sum_{k, 0<=k<=n} A154929(n,k) = A028859(n).

a(n) = (1/3)*sqrt(3)*{[1+sqrt(3)]^(n-1)-[1-sqrt(3)]^(n-1)}+(1/2) *{[1+sqrt(3)]^(n-1)+[1-sqrt(3)]^(n-1)} +(1/2)*[C(2*n,n) mod 2], with n>=0. - Paolo P. Lava, Jan 26 2009

a(n) = sum(bin(n-k,k)*2^(n-k-1),k=0..floor(n/2)) for n > 0. - Emanuele Munarini, Feb 04 2014

EXAMPLE

a(2) = 3 because we have {1,1}, {1,_1} and {2}.

a(3) = 8 because we can order the stamps in eight ways: {1,1,1}  {1,1,_1}  {1,_1,1}  {1,_1,_1}  {2,1}   {2,_1}  {1,2}   {1,_2}, where _1 and _2 are upside down stamps.

a(4) = 22 = 2*3 + 2*8 because we can append 2 or _2 to the a(2) examples and 1 or _1 to the a(3) examples. - Jon Perry, Nov 10 2014

MATHEMATICA

CoefficientList[Series[(1 - x - x^2)/(1 - 2 x - 2 x^2), {x, 0, 20}], x]

PROG

(PARI) Vec( (1-x-x^2)/(1-2*x-2*x^2) + O(x^66) )  /* Joerg Arndt, Sep 30 2012 */

(Maxima) makelist(sum(binomial(n-k, k)*2^(n-k-1), k, 0, floor(n/2)), n, 1, 12); /* Emanuele Munarini, Feb 04 2014 */

(MAGMA) I:=[1, 1, 3, 8]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 10 2014

CROSSREFS

Cf. A028859 (essentially the same sequence). - Klaus Brockhaus, Jul 18 2009

Cf. A000108, A064613, A030528, A091867.

Sequence in context: A278612 A024581 A028859 * A318902 A318903 A318904

Adjacent sequences:  A155017 A155018 A155019 * A155021 A155022 A155023

KEYWORD

nonn,easy

AUTHOR

Philippe Deléham, Jan 19 2009

STATUS

approved

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Last modified October 17 07:05 EDT 2018. Contains 316276 sequences. (Running on oeis4.)