A143100 -id:A143100 - cp's OEIS frontend

A143098 First differences of A143097.

1, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2

Offset: 1

Gary W. Adamson, Jul 24 2008

Binomial transform gives A143100.

1 followed by {2, -1, 2} repeated. - Amiram Eldar, Jun 02 2025

Cf. A143097, A143100.

Join[{1}, Flatten[Table[{2, -1, 2}, {33}]]] (* Amiram Eldar, Jun 02 2025 *)

G.f.: x(1+2x-x^2+x^3)/((1-x)(1+x+x^2)). - R. J. Mathar, Sep 06 2008

A143099 A007318 * A143097.

Original entry on oeis.org

1, 3, 9, 22, 50, 113, 256, 576, 1281, 2818, 6146, 13313, 28672, 61440, 131073, 278530, 589826, 1245185, 2621440, 5505024, 11534337, 24117250, 50331650, 104857601, 218103808, 452984832, 939524097, 1946157058, 4026531842, 8321499137, 17179869184, 35433480192

Offset: 1

Gary W. Adamson, Jul 24 2008

A143100 = (1, 3, 4, 6, 13, 30, 64, 129, ...).

			a(4) = 22 = (1, 3, 3, 1) dot (1, 2, 4, 3) = (1 + 6 + 12 + 3).
a(4) = 22 = 2*a(3) + A143099(3) = 2*9 + 4, where 4 = A143100(3).

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Cf. A143097, A143098, A143100.

A143097 := proc(n) if(n<=1)then return n: elif(n mod 3 <= 1)then return n+1-2*(n mod 3): else return n: fi: end: A143099 := proc(n) return add(binomial(n-1,k-1)*A143097(k),k=1..n): end: seq(A143099(n),n=1..32); # Nathaniel Johnston, Apr 30 2011

Binomial transform of A143097: (1, 2, 4, 3, 5, 7, 6, 8, 10, 9, 11, ...). a(n) = 2*a(n-1) + A143100(n-1).

G.f.: x*(5*x^4-7*x^3+5*x^2-3*x+1)/((1-x)*(x^2-x+1)*(1-2*x)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009; [corrected by R. J. Mathar, Sep 16 2009]

cp's OEIS Frontend

A143098 First differences of A143097.

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