0,2

Number of walks of length 2n+5 in the path graph P_6 from one end to the other one. Example: a(1)=5 because in the path ABCDEF we have ABABCDEF, ABCBCDEF, ABCDCDEF, ABCDEDEF and ABCDEFEF. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004

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

Everett, C. J.; Stein, P. R.; The combinatorics of random walk with absorbing barriers. Discrete Math. 17 (1977), no. 1, 27-45.

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.

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.

N. J. A. Sloane, Transforms

G.f.: 1/(1-5x+6x^2-x^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004

a(n)=5a(n-1)-6a(n-2)+a(n-3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004

a(n)=sum(binomial(5+2k, 7j+k-2)-binomial(5+2k, 7j+k-1), j=-infinity..infinity) (a finite sum).

a:=k->sum(binomial(5+2*k, 7*j+k-2), j=ceil((2-k)/7)..floor((7+k)/7))-sum(binomial(5+2*k, 7*j+k-1), j=ceil((1-k)/7)..floor((6+k)/7)): seq(a(k), k=0..25);

A005021:=-(z-1)*(z-5)/(-1+5*z-6*z**2+z**3); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence apart from the initial 1.]

Double partial sums of A060557. Bisection of A052547.

Cf. A094789, A094790, A080937.

Sequence in context: A124806 A059509 A137745 this_sequence A067325 A121525 A163872

Adjacent sequences: A005018 A005019 A005020 this_sequence A005022 A005023 A005024

nonn,walk

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