A003411 -id:A003411 - cp's OEIS frontend

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

Paul Barry, Feb 11 2005

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

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).

Cf. A275446.

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

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

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

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

my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^4)) \\ G. C. Greubel, Mar 10 2019

((1-x+x^2)/(1-x-x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Mar 10 2019

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.

Formula corrected by Johannes W. Meijer, Aug 11 2011

A048590 Pisot sequence L(8,9).

Original entry on oeis.org

8, 9, 11, 14, 18, 24, 32, 43, 58, 79, 108, 148, 203, 279, 384, 529, 729, 1005, 1386, 1912, 2638, 3640, 5023, 6932, 9567, 13204, 18224, 25153, 34717, 47918, 66139, 91289, 126003, 173918, 240054, 331340, 457340, 631255, 871306, 1202643, 1659980, 2291232, 3162535

Offset: 0

David W. Wilson

nonn

Colin Barker, Table of n, a(n) for n = 0..1000

See A008776 for definitions of Pisot sequences.

Cf. A003411.

Lxy:=[8,9]; [n le 2 select Lxy[n] else Ceiling(Self(n-1)^2/Self(n-2)): n in [1..50]]; // Bruno Berselli, Feb 05 2016

RecurrenceTable[{a[0] == 8, a[1] == 9, a[n] == Ceiling[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 50}] (* Bruno Berselli, Feb 05 2016 *)

pisotL(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
  a
}
pisotL(50, 8, 9) \\ Colin Barker, Aug 07 2016

a(n) = 2*a(n-1) - a(n-2) + a(n-4) - a(n-5) for n > 5 (holds at least up to n = 1000 but is not known to hold in general).

cp's OEIS Frontend

A103632 Expansion of (1 - x + x^2)/(1 - x - x^4).

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