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 A062200 Number of compositions of n such that two adjacent parts are not equal modulo 2. 10
 1, 1, 1, 3, 2, 6, 6, 11, 16, 22, 37, 49, 80, 113, 172, 257, 377, 573, 839, 1266, 1874, 2798, 4175, 6204, 9274, 13785, 20577, 30640, 45665, 68072, 101393, 151169, 225193, 335659, 500162, 745342, 1110790, 1655187, 2466760, 3675822, 5477917, 8163217, 12164896, 18128529, 27015092 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also (0,1)-strings such that all maximal blocks of 1's have even length and all maximal blocks of 0's have odd length. REFERENCES I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(Problems 2.4.3, 2.4.13). LINKS Table of n, a(n) for n=0..44. Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 19. Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1). FORMULA a(n) = Sum_{j=0..n+1} binomial(n-j+1, 3*j-n+1). a(n) = 2*a(n-2) + a(n-3) - a(n-4). G.f.: -(x^2-x-1)/(x^4-x^3-2*x^2+1). More generally, g.f. for the number of compositions of n such that two adjacent parts are not equal modulo p is 1/(1-Sum_{i=1..p} x^i/(1+x^i-x^p)). G.f.: W(0)/(2*x^2) -1/x^2, where W(k) = 1 + 1/(1 - x*(k - x)/( x*(k+1 - x) - 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013 EXAMPLE From Joerg Arndt, Oct 27 2012: (Start) The 11 such compositions of n=7 are [ 1] 1 2 1 2 1 [ 2] 1 6 [ 3] 2 1 4 [ 4] 2 3 2 [ 5] 2 5 [ 6] 3 4 [ 7] 4 1 2 [ 8] 4 3 [ 9] 5 2 [10] 6 1 [11] 7 The 16 such compositions of n=8 are [ 1] 1 2 1 4 [ 2] 1 2 3 2 [ 3] 1 2 5 [ 4] 1 4 1 2 [ 5] 1 4 3 [ 6] 1 6 1 [ 7] 2 1 2 1 2 [ 8] 2 1 2 3 [ 9] 2 1 4 1 [10] 2 3 2 1 [11] 3 2 1 2 [12] 3 2 3 [13] 3 4 1 [14] 4 1 2 1 [15] 5 2 1 [16] 8 (End) MATHEMATICA LinearRecurrence[{0, 2, 1, -1}, {1, 1, 1, 3}, 50] (* Harvey P. Dale, Feb 26 2012 *) Join[{1}, Table[Sum[ Binomial[n-j+1, 3j-n+1], {j, 0, n-1}], {n, 50}]] (* Harvey P. Dale, Feb 26 2012 *) PROG (PARI) x='x+O('x^66); Vec(-(x^2-x-1)/(x^4-x^3-2*x^2+1)) \\ Joerg Arndt, May 13 2013 CROSSREFS Cf. A003242, A062201, A062202, A062203. Sequence in context: A154028 A157793 A096375 * A114208 A014686 A053090 Adjacent sequences: A062197 A062198 A062199 * A062201 A062202 A062203 KEYWORD nonn,easy AUTHOR Vladeta Jovovic, Jun 13 2001 STATUS approved

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