A032085 -id:A032085 - cp's OEIS frontend

A167993 Expansion of x^2/((3x-1)(3*x^2-1)).

0, 0, 1, 3, 12, 36, 117, 351, 1080, 3240, 9801, 29403, 88452, 265356, 796797, 2390391, 7173360, 21520080, 64566801, 193700403, 581120892, 1743362676, 5230147077, 15690441231, 47071500840, 141214502520, 423644039001, 1270932117003, 3812797945332, 11438393835996

Offset: 0

Paul Curtz, Nov 16 2009

The terms satisfy a(n) = 3*a(n-1) +3*a(n-2) -9*a(n-3), so they follow the pattern a(n) = p*a(n-1) +q*a(n-2) -p*q*a(n-3) with p=q=3. This could be called the principal sequence for that recurrence because we have set all but one of the initial terms to zero. [p=q=1 leads to the principal sequence A004526. p=q=2 leads essentially to A032085. The common feature is that the denominator of the generating function does not have a root at x=1, so the sequences of higher order successive differences have the same recurrence as the original sequence. See A135094, A010036, A006516.]

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

Cf. A138587, A107767 (partial sums).

CoefficientList[Series[x^2/((3*x - 1)*(3*x^2 - 1)), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016 *)
LinearRecurrence[{3,3,-9},{0,0,1},30] (* Harvey P. Dale, Nov 05 2017 *)

Vec(x^2/((3*x-1)*(3*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 29 2011

a(2*n+1) = 3*a(2*n).
a(2*n) = A122006(2*n)/2.
a(n) = 3*a(n-1) + 3*a(n-4) - 9*a(n-3).
a(n+1) - a(n) = A122006(n).
a(n) = (3^n - A108411(n+1))/6.
G.f.: x^2/((3*x-1)*(3*x^2-1)).
From Colin Barker, Sep 23 2016: (Start)
a(n) = 3^(n-1)/2-3^(n/2-1)/2 for n even.
a(n) = 3^(n-1)/2-3^(n/2-1/2)/2 for n odd.
(End)

Formulae corrected by Johannes W. Meijer, Jun 28 2011

A289405 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 110, 1100, 11100, 111000, 1111000, 11110000, 111110000, 1111100000, 11111100000, 111111000000, 1111111000000, 11111110000000, 111111110000000, 1111111100000000, 11111111100000000, 111111111000000000, 1111111111000000000, 11111111110000000000

Offset: 0

Robert Price, Jul 05 2017

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

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. A289404, A052551, A032085.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 566; 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}]

Conjectures from Colin Barker, Jul 05 2017: (Start)
G.f.: 1 / ((1 - 10*x)*(1 - 10*x^2)).
a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>2.
(End)

A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 4, 4, 2, 0, 0, 3, 6, 10, 6, 3, 0, 0, 3, 9, 16, 16, 9, 3, 0, 0, 4, 12, 28, 32, 28, 12, 4, 0, 0, 4, 16, 40, 60, 60, 40, 16, 4, 0, 0, 5, 20, 60, 100, 126, 100, 60, 20, 5, 0, 0, 5, 25, 80, 160, 226, 226, 160, 80, 25, 5, 0, 0, 6, 30, 110, 240

Offset: 0

N. J. A. Sloane

Also number of linear unbranched n-4-catafusenes of C_{2v} symmetry.
Number of n-bead black-white reversible strings with k black beads; also binary grids; string is not palindromic. - Yosu Yurramendi, Aug 08 2008
The first seven columns are A004526, A002620, A006584, A032091, A032092, A032093, A032094. Row sums give essentially A032085. - Yosu Yurramendi, Aug 08 2008
From Álvar Ibeas, Jun 01 2020: (Start)
T(n, k) is the sum of odd-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for A034851(n, k) paths and odd for T(n, k) of them.
For a (non-reversible) string of k black and n-k white beads, consider the minimum number of bead transpositions needed to place the black ones to the left and the white ones to the right (in other words, the number of inversions of the permutation obtained by labeling the black beads by integers 1,...,k and the white ones by k+1,...,n, in the same order they take on the string). It is even for A034851(n, k) strings and odd for T(n, k) cases.
(End)

			Triangle begins:
  0;
  0 0;
  0 1 0;
  0 1 1 0;
  0 2 2 2 0;
  0 2 4 4 2 0;
  ...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences

Cf. A007318, A034851, A051159.

Essentially the same as A034877.

a034852 n k = a034852_tabl !! n !! k
a034852_row n = a034852_tabl !! n
a034852_tabl = zipWith (zipWith (-)) a007318_tabl a034851_tabl
-- Reinhard Zumkeller, Mar 24 2012

nmax = 12; t[n_?EvenQ, k_?EvenQ] := (Binomial[n, k] - Binomial[n/2, k/2])/ 2; t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_?OddQ, k_?EvenQ] := (Binomial[n, k] - Binomial[(n-1)/2, k/2])/2; t[n_?OddQ, k_?OddQ] := (Binomial[n, k] - Binomial[(n-1)/2, (k-1)/2])/2; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 15 2011, after Yosu Yurramendi *)

Equals (A007318-A051159)/2. - Yosu Yurramendi, Aug 08 2008

T(n, k) = T(n - 1, k - 1) + T(n - 1, k); except when n is even and k odd, in which case T(n, k) = A034851(n, k) = T(n - 1, k - 1) + A034841(n - 1, k) = A034841(n - 1, k - 1) + T(n - 1, k) = C(n, k) / 2. - Álvar Ibeas, Jun 01 2020

More terms from James Sellers, May 04 2000

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A167993 Expansion of x^2/((3*x-1)*(3*x^2-1)).

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A289405 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood.

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A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

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A032069 Number of reversible strings with n labeled beads of 2 colors, no palindromes of more than 1 bead.

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A167993 Expansion of x^2/((3x-1)(3*x^2-1)).