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]

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

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.

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

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.

Index entries for sequences related to linear recurrences with constant coefficients

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 434

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).