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A103632
Expansion of (1 - x + x^2)/(1 - x - x^4).
4
1, 0, 1, 1, 2, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 76, 105, 145, 200, 276, 381, 526, 726, 1002, 1383, 1909, 2635, 3637, 5020, 6929, 9564, 13201, 18221, 25150, 34714, 47915, 66136, 91286, 126000, 173915, 240051, 331337, 457337, 631252, 871303, 1202640
OFFSET
0,5
COMMENTS
Diagonal sums of A103631.
The Kn11 sums, see A180662, of triangle A065941 equal the terms of this sequence without a(0) and a(1). - Johannes W. Meijer, Aug 11 2011
For n >= 2, a(n) is the number of palindromic compositions of n-2 with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1,2 of the Hoggatt et al. reference. Example: a(9) = 8 because we have 7, 151, 11311, 232, 313, 12121, 21112, and 1111111. - Emeric Deutsch, Aug 17 2016
Essentially the same as A003411. - Georg Fischer, Oct 07 2018
LINKS
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
FORMULA
G.f.: (1 - x + x^2)/(1 - x - x^4).
a(n) = a(n-1) + a(n-4) with a(0)=1, a(1)=0, a(2)=1 and a(3)=1.
a(n) = Sum_{k=0..floor(n/2)} binomial(floor((2*n-3*k-1)/2), n-2*k).
a(n) = A003269(n+1) - A003269(n-4), n > 4.
MAPLE
A103632 := proc(n): add( binomial(floor((2*n-3*k-1)/2), n-2*k), k=0..floor(n/2)) end: seq(A103632(n), n=0..46); # Johannes W. Meijer, Aug 11 2011
MATHEMATICA
LinearRecurrence[{1, 0, 0, 1}, {1, 0, 1, 1}, 50] (* G. C. Greubel, Mar 10 2019 *)
PROG
(GAP) a:=[1, 0, 1, 1];; for n in [5..50] do a[n]:=a[n-1]+a[n-4]; od; a; # Muniru A Asiru, Oct 07 2018
(PARI) my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^4)) \\ G. C. Greubel, Mar 10 2019
(Magma) I:=[1, 0, 1, 1]; [n le 4 select I[n] else Self(n-1) + Self(n-4): n in [1..50]]; // G. C. Greubel, Mar 10 2019
(Sage) ((1-x+x^2)/(1-x-x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Mar 10 2019
CROSSREFS
Cf. A275446.
Sequence in context: A098492 A217314 A173508 * A214076 A067859 A006207
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 11 2005
EXTENSIONS
Formula corrected by Johannes W. Meijer, Aug 11 2011
STATUS
approved