A103632 Expansion of (1 - x + x^2)/(1 - x - x^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

Author

Paul Barry, Feb 11 2005

Keywords

• easy
• nonn

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

• Muniru A Asiru, Table of n, a(n) for n = 0..1000
• V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Crossrefs

Cf. A275446.

Programs

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

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

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 *)

PARI

my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^4)) \\ 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

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.

Extensions

Formula corrected by Johannes W. Meijer, Aug 11 2011