A083579 -id:A083579 - cp's OEIS frontend

A083580 Binomial transform of A083579.

0, 1, 3, 10, 34, 114, 374, 1202, 3798, 11842, 36550, 111954, 340982, 1034210, 3127206, 9434866, 28419286, 85503618, 257035142, 772219538, 2319017910, 6962034466, 20896589158, 62711787570, 188181500054, 564640969154

Offset: 0

Paul Barry, May 01 2003

			a(0) = 2/3-5/12-1/4 = 0 (use 0^0=1).

Index entries for linear recurrences with constant coefficients, signature (7,-16,12).

Cf. A083579.

CoefficientList[Series[x*(1 - 4*x + 5*x^2)/((1 - 2*x)^2*(1 - 3*x)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 28 2023 *)

a(n) = 2*3^n/3-5*0^n/12-(n+1)*2^(n-2).

G.f.: x*(1-4*x+5*x^2)/((1-2*x)^2*(1-3*x)). - Colin Barker, Apr 16 2012

A084172 a(n) = 3a(n-1) - a(n-2) - 3a(n-3) + 2*a(n-4).

Original entry on oeis.org

1, 2, 4, 9, 19, 40, 82, 167, 337, 678, 1360, 2725, 5455, 10916, 21838, 43683, 87373, 174754, 349516, 699041, 1398091, 2796192, 5592394, 11184799, 22369609, 44739230, 89478472, 178956957, 357913927, 715827868, 1431655750, 2863311515

Offset: 0

Paul Barry, May 18 2003

Original name was: Generalized Jacobsthal numbers.

Sums of rows of the triangle in A109225. - Reinhard Zumkeller, Jun 23 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

[2^(n+2)/3-(-1)^n/12-(2*n+1)/4: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

LinearRecurrence[{3,-1,-3,2},{1,2,4,9},40] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = 2^(n+2)/3 - (-1)^n/12 - (2*n+1)/4.
G.f: (2*x^3 - x^2 - x + 1)/( (x+1)*(1-2*x)*(1-x)^2).
a(n+2) = a(n+1) + 2*a(n) + n, a(0)=0, a(1)=2.
a(n) = A001045(n+1) + A083579(n).
a(n+1) = 2*a(n) + floor(n/2). Franklin T. Adams-Watters, Oct 17 2013
a(n)+a(n+1) = A095768(n+1). - R. J. Mathar, Apr 15 2024

A284353 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 7, 13, 31, 61, 127, 253, 511, 1021, 2047, 4093, 8191, 16381, 32767, 65533, 131071, 262141, 524287, 1048573, 2097151, 4194301, 8388607, 16777213, 33554431, 67108861, 134217727, 268435453, 536870911, 1073741821, 2147483647, 4294967293, 8589934591

Offset: 0

Robert Price, Mar 25 2017

Initialized with a single black (ON) cell at stage zero.

If one begins the Generalized Jacobsthal numbers (A083579) with a(0)=1, instead of a(0)=0, the same sequence will be obtained. - Henrik Lipskoch, Jan 28 2021

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A000975, A284351, A284352, A284354.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 899; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]

Conjectures from Colin Barker, Mar 26 2017: (Start)
G.f.: (1 - x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
a(n) = 2^(n+1) - 1 for n even.
a(n) = 2^(n+1) - 3 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
Conjecture: For n > 0, a(n) = Sum_{k=0..n-1} C(n,k) * (2-(-1)^k). - Wesley Ivan Hurt, Sep 23 2017
Apparently, a(n) = 6*A000975(n-1) + 1 for n >= 1. - Hugo Pfoertner, Jan 28 2021

A083578 a(n) = (6^n + (-4)^n)/2.

Original entry on oeis.org

1, 1, 26, 76, 776, 3376, 25376, 131776, 872576, 4907776, 30757376, 179301376, 1096779776, 6496792576, 39316299776, 234555621376, 1412702437376, 8454739787776, 50814338072576, 304542431051776, 1828628975845376

Offset: 0

Paul Barry, Apr 30 2003

Index entries for linear recurrences with constant coefficients, signature (2,24).

Cf. A083579. First differences of A051958.

LinearRecurrence[{2,24},{1,1},30] (* Harvey P. Dale, Nov 29 2017 *)

a(n) = (6^n + (-4)^n)/2.
G.f.: (1-x)/((1+4x)(1-6x)).
E.g.f.: (exp(6x) + exp(-4x))/2.

A084173 a(n) = 3a(n-1) - a(n-2) - 3a(n-3) + 2*a(n-4).

Original entry on oeis.org

1, 3, 5, 13, 27, 59, 121, 249, 503, 1015, 2037, 4085, 8179, 16371, 32753, 65521, 131055, 262127, 524269, 1048557, 2097131, 4194283, 8388585, 16777193, 33554407, 67108839, 134217701, 268435429, 536870883, 1073741795, 2147483617

Offset: 0

Paul Barry, May 18 2003

Original name was: A sum of generalized Jacobsthal numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

[2^(n+1)-(2*n+1)/2-(-1)^n/2: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

a[n_]:=2^(n+1) - (2*n+1)/2 - (-1)^n/2; Array[a, 50, 0] (* or *)
CoefficientList[Series[(4*x^3 - 3*x^2 + 1)/(-2*x^4 + 3*x^3 + x^2 - 3*x + 1), {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -1, -3, 2}, {1, 3, 5, 13}, 50] (* Stefano Spezia, Oct 08 2018 *)

x='x+O('x^33); Vec((4*x^3-3*x^2+1)/(-2*x^4+3*x^3+x^2-3*x+1)) \\ Altug Alkan, Oct 08 2018

a(n) = 2^(n+1) - (2*n+1)/2 - (-1)^n/2.
G.f.: (4*x^3 - 3*x^2 + 1)/(-2*x^4 + 3*x^3 + x^2 - 3*x + 1).
a(n) = A084172(n) + A083579(n).

A338198 Triangle read by rows, T(n,k) = ((k+1)2^(n-k)-(k-2)(-1)^(n-k))/3 for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 2, 3, 2, 1, 6, 5, 4, 3, 1, 10, 11, 8, 5, 4, 1, 22, 21, 16, 11, 6, 5, 1, 42, 43, 32, 21, 14, 7, 6, 1, 86, 85, 64, 43, 26, 17, 8, 7, 1, 170, 171, 128, 85, 54, 31, 20, 9, 8, 1, 342, 341, 256, 171, 106, 65, 36, 23, 10, 9, 1, 682, 683, 512, 341, 214, 127, 76, 41, 26, 11, 10, 1

Offset: 0

Werner Schulte, Oct 15 2020

This triangle is related to the Jacobsthal numbers (A001045).

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :    0     1     2    3    4    5    6   7   8   9
======================================================
  0 :    1
  1 :    0     1
  2 :    2     1     1
  3 :    2     3     2    1
  4 :    6     5     4    3    1
  5 :   10    11     8    5    4    1
  6 :   22    21    16   11    6    5    1
  7 :   42    43    32   21   14    7    6   1
  8 :   86    85    64   43   26   17    8   7   1
  9 :  170   171   128   85   54   31   20   9   8   1
etc.

For columns k = 0 to 8 see A078008, A001045, A000079, A001045, A084214, A014551, A083595, A083582, A259713 respectively.

Table[((k + 1)*2^(n - k) - (k - 2)*(-1)^(n - k))/3, {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 15 2020 *)

T(n,n) = 1 for n >= 0; T(n,n-1) = n-1 for n > 0.
T(n,k) = T(n-1,k) + 2 * T(n-2,k) for 0 <= k <= n-2.
T(n,k) = 2 * T(n-1,k) - (k-2) * (-1)^(n-k) for 0 <= k < n.
T(n,k) = T(n+1-k,1) + (k-1) * T(n-k,1) for 0 <= k < n.
T(n+1,k) * T(n-1,k) - T(n,k+1) * T(n,k-1) = T(n-k,1)^2 for 0 < k < n.
Row sums are A083579(n+1) for n >= 0.
G.f. of column k >= 0: (1+(k-1)*t) * t^k / (1-t-2*t^2).
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = (1 - (1+x)*t + 2*x*t^2) / ((1 - x*t)^2 * (1 - t - 2*t^2)).
Conjecture: Let M(n,k) be the matrix inverse of T(n,k), seen as a matrix. Then M(i,j) = 0 if j < 0 or j > i, M(n,n) = 1 for n >= 0, M(n,n-1) = 1-n for n > 0, and M(n,k) = (-1)^(n-k) * (k^2-2) * (n-2)! / k! for 0 <= k <= n-2.

cp's OEIS Frontend

A083580 Binomial transform of A083579.

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A084172 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

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A284353 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood.

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Mathematica

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A083578 a(n) = (6^n + (-4)^n)/2.

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A084173 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

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Magma

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A338198 Triangle read by rows, T(n,k) = ((k+1)*2^(n-k)-(k-2)*(-1)^(n-k))/3 for 0 <= k <= n.

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Mathematica

Formula

A084172 a(n) = 3a(n-1) - a(n-2) - 3a(n-3) + 2*a(n-4).

A084173 a(n) = 3a(n-1) - a(n-2) - 3a(n-3) + 2*a(n-4).

A338198 Triangle read by rows, T(n,k) = ((k+1)2^(n-k)-(k-2)(-1)^(n-k))/3 for 0 <= k <= n.