A086445 -id:A086445 - cp's OEIS frontend

A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.

0, 0, 1, 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485

Offset: 0

Paul Curtz, Oct 11 2019

nonn

OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000129, A001045, A054977, A139756, A141413, A166444, A183190.

Cf. A000079, A005578, A063524, A086445, A139818, A257113.

a[n_] := If[n>3, (2^(n-3) + (-1)^n)/3, If[n == 2, 1, 0]]; (* Jean-François Alcover, Oct 16 2019 *)

a(n) is the fourth row of the following array:
  0,  0,  0,  0, 0, 1, 3, 7, 14, 27, 51, 97, ...
  0,  0,  0,  0, 1, 2, 4, 7, 13, 24, 46, 89, ... = A086445
  0,  0,  0,  1, 1, 2, 3, 6, 11, 22, 43, 86, ... = 0, 0, 0, A005578(n)
  0,  0,  1,  0, 1, 1, 3, 5, 11, 21, 43, 85, ... = a(n)
  0,  1, -1,  1, 0, 2, 2, 6, 10, 22, 42, 86, ...
  1, -2,  2, -1, 2, 0, 4, 4, 12, 20, 44, 84, ...
From the main diagonal onward, every row is an autosequence of the first kind.
From Stefano Spezia, Oct 16 2019: (Start)
O.g.f.: x^2*(-1 + x + x^2)/(-1 + x + 2*x^2).
E.g.f.: (1/24)*exp(-x)*(8 - 9*exp(x) + exp(3*x) + 6*exp(x)*x + 6*exp(x)*x^2).
a(n) = a(n-1) + 2*a(n-2) for n > 4. (End)
a(n) = Sum_{k=0..n-1} A183190(n-k-2, n-2*k-2). - Jean-François Alcover, Nov 10 2019

Partially edited by Peter Luschny, Nov 12 2019

A340660 A000079 is the first row. For the second row, subtract A001045. For the third row, subtract A001045 from the second one, etc. The resulting array is read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 1, 0, 3, 8, 1, -1, 2, 5, 16, 1, -2, 1, 2, 11, 32, 1, -3, 0, -1, 6, 21, 64, 1, -4, -1, -4, 1, 10, 43, 128, 1, -5, -2, -7, -4, -1, 22, 85, 256, 1, -6, -3, -10, -9, -12, 1, 42, 171, 512, 1, -7, -4, -13, -14, -23, -20, -1, 86, 341, 1024

Offset: 0

Paul Curtz, Jan 15 2021

Every row has the signature (1,2).
(Among consequences: a(n) read by antidiagonals is
1,
1,  2,
1,  1, 4,
1,  0, 3,  8,
1, -1, 2,  5, 16
1, -2, 1,  2, 11, 32,
1, -3, 0, -1,  6, 21, 64,
... .
The row sums and their first two difference table terms are
1, 3, 6, 12, 23, 45, 88, ... = A086445(n+1) - 1
2, 3, 6, 11, 22, 43, 86, ... = A005578(n+2)
1, 3, 5, 11, 21, 43, 85, ... = A001045(n+2).
The antidiagonal sums are
b(n) = 1, 1, 3, 2, 5, 3, 9, 4, 15, 5, 27, 6, 49, 7, ... .)

			Square array:
1,  2,  4,   8,  16,  32,  64,  128, ... = A000079(n)
1,  1,  3,   5,  11,  21,  43,   85, ... = A001045(n+1)
1,  0,  2,   2,   6,  10,  22,   42, ... = A078008(n)
1, -1,  1,  -1,   1,  -1,   1,   -1, ... = A033999(n)
1, -2,  0,  -4,  -4, -12, -20,  -44, ... = -A084247(n)
1, -3, -1,  -7,  -9, -23, -41,  -87, ... = (-1)^n*A140966(n+1)
1, -4, -2, -10, -14, -34, -62, -130, ... = -A135440(n)
1, -5, -3, -13, -19, -45, -83, -173, ... = -A155980(n+3) or -A171382(n+1)
...

Cf. A000079, A001045, A033999, A078008, A084247, A135440, A140966, A155980, A171382.

Cf. A005578, A086445.

A:= (n, k)-> (<<0|1>, <2|1>>^k. <<1, 2-n>>)[1$2]:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 21 2021

A340660[m_, n_] := LinearRecurrence[{1, 2}, {1, m}, {n}]; Table[Reverse[Table[A340660[m, n + m - 2] // First, {m, 2, -n + 3, -1}]], {n, 1, 11}] // Flatten (* Robert P. P. McKone, Jan 28 2021 *)

T(n, k) = 2^k - n*(2^k - (-1)^k)/3;
matrix(10,10,n,k,T(n-1,k-1)) \\ Michel Marcus, Jan 19 2021

A(n,k) = 2^k - n*round(2^k/3).

A340849 a(n) = A001045(n) + A052928(n).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 27, 49, 93, 179, 351, 693, 1377, 2743, 5475, 10937, 21861, 43707, 87399, 174781, 349545, 699071, 1398123, 2796225, 5592429, 11184835, 22369647, 44739269, 89478513, 178956999, 357913971, 715827913

Offset: 0

Paul Curtz, Jan 24 2021

nonn

a(2*n) is divisible by 3.
a(3*n+2) is divisible by 3.
a(n) is the minimum number of moves to solve a Towers of Hanoi puzzle with 4 pegs and n disks where a disk cannot move away from the destination peg (or symmetrically, a disk cannot return to the initial peg). - Woosuk Kwak, Jan 25 2024

Woosuk Kwak, 1+3 Towers of Hanoi, Puzzling Stack Exchange.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A001045, A052928, A100314, A109613, A128209.

LinearRecurrence[{3, -1, -3, 2}, {0, 1, 3, 5}, 32] (* Robert P. P. McKone, Jan 28 2021 *)

a(n+1) - 2*a(n) = -A109613(n-2), for a(0)=0, a(1)=1. a(n) + a(n+1) = A100314(n).
a(n+1) - a(n) = A128209(n) for n >= 0.
a(n+2) = 1 + 2*A086445(n). - Hugo Pfoertner, Jan 24 2021
From Woosuk Kwak, Jan 25 2024: (Start)
a(n) = n + floor(2^n/3).
a(n) = n + A000975(n-1) for n >= 1. (End)

cp's OEIS Frontend

A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.

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A340660 A000079 is the first row. For the second row, subtract A001045. For the third row, subtract A001045 from the second one, etc. The resulting array is read by ascending antidiagonals.

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A340849 a(n) = A001045(n) + A052928(n).

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