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
In the same way that the sum of any two alternating terms of A006498 produces a term from A000045 (the Fibonacci sequence), so it could be thought of as a "meta-Fibonacci," and the sum of any two alternating terms of A013979 produces a term from A000930 (Narayana's cows), so it could analogously be called "meta-Narayana's cows," this sequence embeds (can generate) A000931 (the Padovan sequence), as the odd terms of A000931 are generated by the sum of successive elements (e.g. 1+2=3, 2+3=5, 3+6=9, 6+10=16) and its even terms are generated by the difference of "supersuccessive" (second-order successive or "alternating," separated by a single other term) terms (e.g. 10-3=7, 18-6=12, 31-10=21, 55-18=37) — or, equivalently, adding "supersupersuccessive" terms (separated by 2 other terms, e.g. 1+6=7, 2+10=12, 3+18=21, 6+31=37) — so it could be dubbed the "metaPadovan." - Michael Cohen and Yasuyuki Kachi, Jun 13 2024
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..500
Vladimir Baltić, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.
Michael Cohen and Yasuyuki Kachi, Recurrence Relations Rhythm. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol. 14639. Springer, Cham.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Disc. Appl. Math. 187 (2015) 82-90, Sect. 4.3.
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(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]), (33), (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<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( 1/(1-x-x^2-x^4) )); // G. C. Greubel, Apr 09 2021
(SageMath)
def A060945_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/(1-x-x^2-x^4) ).list()
A060945_list(40) # G. C. Greubel, Apr 09 2021
CROSSREFS
Same as unsigned version of A077930.
All of A060945, A077930, A181532 are variations of the same sequence. - N. J. A. Sloane, Mar 04 2012
KEYWORD
nonn,easy
AUTHOR
Len Smiley, May 07 2001
EXTENSIONS
a(0) = 1 prepended by Joerg Arndt, Oct 21 2012
STATUS
approved