A103422 -id:A103422 - cp's OEIS frontend

A103419 Number of compositions of n in which the least part is odd.

1, 1, 4, 6, 14, 28, 59, 117, 239, 484, 980, 1973, 3973, 7989, 16054, 32227, 64653, 129628, 259787, 520440, 1042305, 2086938, 4177680, 8361557, 16733221, 33482909, 66992641, 134028938, 268128902, 536373288, 1072934271, 2146173471, 4292842170, 8586488355

Offset: 1

Vladeta Jovovic, Feb 04 2005

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A103420-A103422, A027187, A027193, A026804, A026805.

b:= proc(n, i) option remember; `if`(n=0, irem(i, 2), add(
      (t-> b(t, min(i, j, `if`(t>0, t, j))))(n-j), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 26 2015

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n - 1)/((1 - x - x^(2n))*(1 - x - x^(2n - 1))), {n, 35}]], {x, 0, 35}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

G.f.: Sum((1-x)^2*x^(2*n-1)/((1-x-x^(2*n))*(1-x-x^(2*n-1))), n=1..infinity).
G.f.: Sum(x^k/((1-x)^k*(1+x^k)),k=1..infinity). - Vladeta Jovovic, Mar 02 2008
a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014

More terms from Robert G. Wilson v, Feb 05 2005

A103420 Number of compositions of n in which the least part is even.

Original entry on oeis.org

0, 1, 0, 2, 2, 4, 5, 11, 17, 28, 44, 75, 123, 203, 330, 541, 883, 1444, 2357, 3848, 6271, 10214, 16624, 27051, 43995, 71523, 116223, 188790, 306554, 497624, 807553, 1310177, 2125126, 3446237, 5587517, 9057611, 14680337, 23789891, 38546834, 62449682, 101163024

Offset: 1

Vladeta Jovovic, Feb 04 2005

Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 1000 terms from Robert Israel)

Cf. A103419, A103421, A103422, A027187, A027193, A026804, A026805.

N:= 50: # for a(1) .. a(N)
G:= add(x^(2*n)/((1-x)^n*(1+x^n)),n=1..N/2):
S:= series(G,x,N+1):
[seq(coeff(S,x,i),i=1..N)]; # Robert Israel, Oct 23 2024
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1-
      irem(m, 2), add(b(n-j, min(m, j)), j=1..n))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=1..42);  # Alois P. Heinz, Oct 23 2024

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n)/((1 - x - x^(2n))*(1 - x - x^(2n + 1))), {n, 40}]], {x, 0, 40}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

G.f.: Sum((1-x)^2*x^(2*n)/((1-x-x^(2*n))*(1-x-x^(2*n+1))), n=1..infinity).
G.f.: Sum(x^(2*n)/((1-x)^n*(1+x^n)),n=1..infinity). - Vladeta Jovovic, Mar 02 2008
a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-1). - Vaclav Kotesovec, May 01 2014

More terms from Robert G. Wilson v, Feb 05 2005

A103421 Number of compositions of n in which the greatest part is odd.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 30, 62, 129, 263, 534, 1076, 2160, 4318, 8612, 17145, 34097, 67764, 134638, 267506, 531606, 1056812, 2101854, 4182462, 8327263, 16588973, 33066080, 65945522, 131588128, 262702054, 524699094, 1048433468, 2095744336

Offset: 1

Vladeta Jovovic, Feb 04 2005

Cf. A103419, A103420, A103422, A027187, A027193, A026804, A026805.

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n - 1)/((1 - 2x + x^(2n - 1))*(1 - 2x + x^(2n))), {n, 35}]], {x, 0, 35}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

G.f.: Sum((1-x)^2*x^(2*n-1)/((1-2*x+x^(2*n-1))*(1-2*x+x^(2*n))), n=1..infinity).

a(n) + A103422(n) = 2^(n-1). - R. J. Mathar, Mar 24 2018

More terms from Robert G. Wilson v, Feb 05 2005

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A103419 Number of compositions of n in which the least part is odd.

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A103420 Number of compositions of n in which the least part is even.

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A103421 Number of compositions of n in which the greatest part is odd.

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