0,4

Let u(k), v(k), w(k) be be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (A006356 with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (this sequence with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = A077998 with an extra initial 0. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002. Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary Adamson, Dec 23 2003.

Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006054 counts walks of length n between the vertex of degree 1 and the vertex of degree 3. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

Form the digraph with matrix [1,1,0;1,0,1;1,1,1]. A006054(n) counts walks of length n between the vertices with loops. - Paul Barry (pbarry(AT)wit.ie), Oct 15 2004

a(n), n>1 = round(k*A006356(n-1)), where k = .3568958678... = 1/(1+2*Cos Pi/7) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2008

A006054(n) = sum of (n-2)-th row terms of triangle A144159. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]

Nonzero terms = INVERT transform of (1, 1, 2, 2, 3, 3,...). Example: 56 = (1, 1, 2, 5, 11, 25) dot (3, 3, 2, 2, 1, 1) = (3 + 3 + 4 + 10 + 11 + 25). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 20 2009]

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.3.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 434

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Index entries for sequences related to linear recurrences with constant coefficients

G.f.: x^2/(1-2x-x^2+x^3).

Sum_{k, 0<=k<=n+2} a(k) = A077850(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 07 2006

Let M = the 3 X 3 matrix [1,1,0; 1,2,1; 0,1,2], then M^n*[1,0,0] = [A080937(n-1), A094790(n), A006054(n-1)]. E.g. M^3*[1,0,0] = [5,9,5] = [A080937(2), A094790(3), A006054(2)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 15 2006

A006054:=z**2/(1-2*z-z**2+z**3); [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A006356, A007583, A005578.

Cf. A080937, A094790.

A144159 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]

Sequence in context: A151529 A017919 A017920 this_sequence A106805 A094981 A097779

Adjacent sequences: A006051 A006052 A006053 this_sequence A006055 A006056 A006057

nonn

N. J. A. Sloane (njas(AT)research.att.com).