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 A060945 Number of compositions (ordered partitions) of n into 1's, 2's and 4's. 12
 1, 1, 2, 3, 6, 10, 18, 31, 55, 96, 169, 296, 520, 912, 1601, 2809, 4930, 8651, 15182, 26642, 46754, 82047, 143983, 252672, 443409, 778128, 1365520, 2396320, 4205249, 7379697, 12950466, 22726483, 39882198, 69988378, 122821042, 215535903, 378239143, 663763424, 1164823609 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Diagonal sums of A038137. - Paul Barry, Oct 24 2005 From Gary W. Adamson, Oct 28 2010: (Start) INVERT transform of the aerated Fibonacci sequence (1, 0, 1, 0, 2, 0, 3, 0, 5, ...). a(n) = term (4,4) in the n-th power of the matrix [0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,1,1]. (End) Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=3, I={2}. - Vladimir Baltic, Mar 07 2012 Number of compositions of n if the summand 2 is frozen in place or equivalently, if the ordering of the summand 2 does not count. - Gregory L. Simay, Jul 18 2016 a(n) - a(n-2) = number of compositions of n with no 2's = A005251(n+1). - Gregory L. Simay, Jul 18, 2016 In general, the number of compositions of n with summand k frozen in place is equal to the number of compositions of n with only summands 1,...,k,2k. - Gregory L. Simay, May 10 2017 LINKS Harry J. Smith, Table of n, a(n) for n = 0..500 Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135 Index entries for linear recurrences with constant coefficients, signature (1,1,0,1). FORMULA a(n) = a(n-1) + a(n-2) + a(n-4). G.f.: 1 / (1 - x - x^2 - x^4). a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..n-k} C(i, n-k-i)*C(2*i-n+k, 3*k-2*n+2*i). - Paul Barry, Oct 24 2005 a(2n) = A238236(n), a(2n+1) = A097472(n). - Philippe Deléham, Feb 20 2014 a(n) + a(n+1) = A005314(n+2). - R. J. Mathar, Jun 17 2020 EXAMPLE There are 18=a(6) compositions of 6 with the summand 2 frozen in place: (6), (51), (15), (4,[2]), (3,3) (411), (141), (114), (3[2]1), (1[2]3)), ([222]), (3111), (1311), (1131), (1113), ([22]11), ([2]1111), (111111). Equivalently, the position of the summand 2 does not affect the composition count. For example, (321)=(231)=(312) and (123)=(213)=(132). MAPLE m:= 40; S:= series( 1/(1-x-x^2-x^4), x, m+1); seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Apr 09 2021 MATHEMATICA LinearRecurrence[{1, 1, 0, 1}, {1, 1, 2, 3}, 39] (* or *) CoefficientList[Series[1/(1-x-x^2-x^4), {x, 0, 38}], x] (* Michael De Vlieger, May 10 2017 *) PROG (Haskell) a060945 n = a060945_list !! (n-1) a060945_list = 1 : 1 : 2 : 3 : 6 : zipWith (+) a060945_list    (zipWith (+) (drop 2 a060945_list) (drop 3 a060945_list)) -- Reinhard Zumkeller, Mar 23 2012 (PARI) N=66; my(x='x+O('x^N)); Vec(1/(1-x-x^2-x^4)) /* Joerg Arndt, Oct 21 2012 */ (Magma) R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-x-x^2-x^4) )); // G. C. Greubel, Apr 09 2021 (Sage) def A060945_list(prec):     P. = PowerSeriesRing(ZZ, prec)     return P( 1/(1-x-x^2-x^4) ).list() A060945_list(40) # G. C. Greubel, Apr 09 2021 CROSSREFS Cf. A000045 (1's and 2's only), A023359 (all powers of 2) Same as unsigned version of A077930. All of A060945, A077930, A181532 are variations of the same sequence. - N. J. A. Sloane, Mar 04 2012 Sequence in context: A102702 A181532 A077930 * A023359 A082482 A066000 Adjacent sequences:  A060942 A060943 A060944 * A060946 A060947 A060948 KEYWORD nonn,easy,changed AUTHOR Len Smiley, May 07 2001 EXTENSIONS a(0) = 1 prepended by Joerg Arndt, Oct 21 2012 STATUS approved

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Last modified April 17 08:40 EDT 2021. Contains 343064 sequences. (Running on oeis4.)