(MAGMAMagma) 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

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V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

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LinearRecurrence[{1, 0, 0, 1}, {1, 0, 1, 1}, 50] (* G. C. Greubel, Mar 10 2019 *)

(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

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Expansion of (1 - x + x^2)/(1 - x - x^4).

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

G.f.: (1 - x + x^2)/(1 - x - x^4).

a(n) = sumSum_{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.

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]

Formula corrected by _Johannes W. Meijer, _, Aug 11 2011

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