A289768
Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood.
Original entry on oeis.org
1, 1, 3, 3, 5, 5, 13, 13, 17, 17, 59, 59, 81, 81, 219, 219, 257, 257, 899, 899, 1349, 1349, 3437, 3437, 4353, 4353, 15235, 15235, 20805, 20805, 56173, 56173, 65537, 65537, 229379, 229379, 344069, 344069, 876557, 876557, 1118225, 1118225, 3913787, 3913787
Offset: 0
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
-
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 598; 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
Original entry on oeis.org
1, 2, 4, 9, 22, 57, 153, 420, 1170, 3293, 9339, 26642, 76363, 219728, 634312, 1836229, 5328346, 15494125, 45137995, 131712826, 384900937, 1126265986, 3299509114, 9676690939, 28407473191, 83470059532, 245465090758, 722406781935, 2127562036990, 6270020029353
Offset: 0
Cf.
A000069,
A001969,
A002144,
A002145,
A005773,
A005774,
A005775,
A010059,
A097893,
A122896,
A167630,
A210736,
A211278,
A257520.
-
gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)):
a := n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n = 0 .. 29);
# Recurrence:
a:= proc(n) option remember; `if`(n<=2, 2^n, 3*a(n-1) - (6/n-1)*a(n-2) + (6/n-3)*a(n-3)) end:
seq(a(n), n = 0 .. 29);
-
Module[{a, n}, RecurrenceTable[{a[n] == 3*a[n-1] - (6-n)*a[n-2]/n + 3*(2-n)*a[n-3]/n, a[0] == 1, a[1] == 2, a[2] == 4}, a, {n, 0, 30}]] (* Paolo Xausa, May 05 2025 *)
-
from math import comb as C
def a(n):
return sum(C(n, k)*abs(sum((-1)**j*C(k, j) for j in range(k//2 + 1))) for k in range(n + 1))
print([a(n) for n in range(30)])
A383609
Triangle read by rows: T(n,k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k) for 0 < k < n, T(n,0) = T(n,n) = 1.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 8, 1, 1, 5, 13, 20, 17, 1, 1, 6, 19, 38, 50, 38, 1, 1, 7, 26, 63, 107, 126, 89, 1, 1, 8, 34, 96, 196, 296, 322, 216, 1, 1, 9, 43, 138, 326, 588, 814, 834, 539, 1, 1, 10, 53, 190, 507, 1052, 1728, 2236, 2187, 1374, 1
Offset: 0
Triangle T(n, k) starts:
n\k : 0 1 2 3 4 5 6 7
====================================================================
0 : 1
1 : 1 1
2 : 1 2 1
3 : 1 3 4 1
4 : 1 4 8 8 1
5 : 1 5 13 20 17 1
6 : 1 6 19 38 50 38 1
7 : 1 7 26 63 107 126 89 1
...
-
T := proc (n, k) option remember; if k = n or k = 0 then 1 elif k < 0 then 0 else T(n-1, k-2)+T(n-1, k-1)+T(n-1, k) end if end proc:
seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 8);
A167655
Riordan array (1-u,u) where u=x/(1+x+x^2).
Original entry on oeis.org
1, -1, 1, 1, -2, 1, 0, 2, -3, 1, -1, 0, 4, -4, 1, 1, -3, -1, 7, -5, 1, 0, 4, -6, -4, 11, -6, 1, -1, -1, 11, -9, -10, 16, -7, 1, 1, -4, -6, 24, -10, -20, 22, -8, 1, 0, 6, -9, -21, 44, -6, -35, 29, -9, 1, -1, -2, 21, -12, -55, 70, 7, -56, 37, -10, 1
Offset: 0
Triangle begins:
1;
-1, 1;
1, -2, 1;
0, 2, -3, 1;
-1, 0, 4, -4, 1;
1, -3, -1, 7, -5, 1;
0, 4, -6, -4, 11, -6, 1;
-1, -1, 11, -9, -10, 16, -7, 1;
1, -4, -6, 24, -10, -20, 22, -8, 1;
0, 6, -9, -21, 44, -6, -35, 29, -9, 1;
-1, -2, 21, -12, -55, 70, 7, -56, 37, -10, 1;
... _Philippe Deléham_, Jan 22 2014
Showing 1-4 of 4 results.
Comments