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 A155822 Number of compositions of n with no part greater than 3 such that no two adjacent parts are equal. 2
 1, 1, 1, 3, 3, 4, 8, 9, 12, 21, 27, 37, 58, 78, 109, 164, 227, 319, 467, 656, 928, 1341, 1896, 2689, 3859, 5477, 7782, 11126, 15817, 22496, 32103, 45679, 65003, 92668, 131912, 187777, 267556, 380941, 542363, 772581, 1100098, 1566414, 2230997 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Carlitz compositions with no part greater than 3. LINKS D. I. Bevan, Table of n, a(n) for n = 0..5000 Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 206. Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,2). FORMULA From David Bevan, Feb 02 2009: (Start) For n>5, a(n) = a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5). For n>6, a(n) = a(n-3) + a(n-4) + a(n-5) + 2*a(n-6). (End) G.f.: -(x+1)*(x^2-x+1)*(x^2+1) / (2*x^5-x^4+2*x^3-x^2+x-1). - Colin Barker, Feb 13 2013 G.f.: 1/(1 - Sum_{j=1..3} x^j/(1 + x^j) ) and generally for Carlitz compositions with no part greater than r the o.g.f. is 1/(1 - Sum_{j=1..r} x^j/(1 + x^j) ). - Geoffrey Critzer, Nov 21 2013 EXAMPLE a(5) = 4 because we have 5 = 1 + 3 + 1 = 2 + 1 + 2 = 2 + 3 = 3+2. MAPLE From David Bevan, Feb 02 2009: (Start) a := proc(k) if k=0 then 1 else b(1, k)+b(2, k)+b(3, k) fi end; b := proc(r, k) option remember; if k

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Last modified August 14 17:12 EDT 2022. Contains 356122 sequences. (Running on oeis4.)