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 A344236 Number of n-step walks from a universal vertex to the other on the diamond graph. 2
 0, 1, 2, 5, 14, 33, 90, 221, 582, 1465, 3794, 9653, 24830, 63441, 162762, 416525, 1067574, 2733673, 7003970, 17938661, 45954542, 117709185, 301527354, 772364093, 1978473510, 5067929881, 12981823922, 33253543445, 85180839134, 218195012913, 558918369450 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is the number of n-step walks from vertex A to vertex C on the graph below. B--C | /| |/ | A--D LINKS Index entries for linear recurrences with constant coefficients, signature (0,5,4). FORMULA a(n) = a(n-1) + 4*a(n-2) + (-1)^n for n > 1. a(n) = A344261(n-1) + 2*a(n-2) + 2*A344261(n-2) for n > 1. a(n) = A344261(n) - (-1)^n. a(n) = A006131(n) - A344261(n). a(n) = (A006131(n) - (-1)^n)/2. a(n) = ((sqrt(17)-1)/(4*sqrt(17)))*((1-sqrt(17))/2)^n + ((sqrt(17)+1)/(4*sqrt(17)))*((1+sqrt(17))/2)^n - (1/2)*(-1)^n. G.f.: (2*x^2 + x)/(-4*x^3 - 5*x^2 + 1). a(n) = 5*a(n-2) + 4*a(n-3) for n > 2. - Stefano Spezia, May 13 2021 EXAMPLE Let A, B, C and D be the vertices of the diamond graph, where A and C are the universal vertices. Then, a(3) = 5 walks from A to C are: (A, B, A, C), (A, C, A, C), (A, C, B, C), (A, C, D, C), and (A, D, A, C). MAPLE f := proc(n) option remember; if n <= 2 then n; else 5*f(n - 2) + 4*f(n - 3); end if; end proc MATHEMATICA LinearRecurrence[{0, 5, 4}, {0, 1, 2}, 30] PROG (Python) def A344236_list(n):     list = [0, 1, 2] + [0] * (n - 3)     for i in range(3, n):         list[i] = 5 * list[i - 2] + 4 * list[i - 3]     return list print(A344236_list(31)) # M. Eren Kesim, Jul 19 2021 (PARI) my(p=Mod('x, 'x^2-'x-4)); a(n) = (vecsum(Vec(lift(p^n))) + n%2) >> 1; \\ Kevin Ryde, May 13 2021 CROSSREFS Cf. A006131, A344261. Sequence in context: A090803 A018015 A080039 * A265226 A299164 A131408 Adjacent sequences:  A344233 A344234 A344235 * A344237 A344238 A344239 KEYWORD nonn,easy,walk AUTHOR M. Eren Kesim, May 12 2021 STATUS approved

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Last modified September 26 13:21 EDT 2021. Contains 347668 sequences. (Running on oeis4.)