A006356 -id:A006356 - cp's OEIS frontend

A373424 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(n) is the continued fraction (-1)^n/(~x - 1/(~x - ... 1/(~x - 1)))...) and where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 5, 1, 0, 1, 5, 10, 14, 8, 1, 0, 1, 6, 15, 30, 31, 13, 1, 0, 1, 7, 21, 55, 85, 70, 21, 1, 0, 1, 8, 28, 91, 190, 246, 157, 34, 1, 0, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 0, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 0

Offset: 0

Peter Luschny, Jun 09 2024

A variant of both A050446 and A050447 which are the main entries. Differs in indexing and adds a first row to the array resp. a diagonal to the triangle.

			Generating functions of the rows:
   gf0 =  1;
   gf1 = -1/( x-1);
   gf2 =  1/(-x-1/(-x-1));
   gf3 = -1/( x-1/( x-1/( x-1)));
   gf4 =  1/(-x-1/(-x-1/(-x-1/(-x-1))));
   gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));
   gf6 =  1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));
   ...
Array A(n, k) starts:
  [0] 1, 0,  0,  0,   0,    0,    0,     0,      0,      0, ...  A000007
  [1] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ...  A000012
  [2] 1, 2,  3,  5,   8,   13,   21,    34,     55,     89, ...  A000045
  [3] 1, 3,  6, 14,  31,   70,  157,   353,    793,   1782, ...  A006356
  [4] 1, 4, 10, 30,  85,  246,  707,  2037,   5864,  16886, ...  A006357
  [5] 1, 5, 15, 55, 190,  671, 2353,  8272,  29056, 102091, ...  A006358
  [6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ...  A006359
   A000027,A000330,   A085461,     A244881, ...
       A000217, A006322,    A108675, ...
.
Triangle T(n, k) = A(n - k, k) starts:
  [0] 1;
  [1] 1,  0;
  [2] 1,  1,  0;
  [3] 1,  2,  1,  0;
  [4] 1,  3,  3,  1,  0;
  [5] 1,  4,  6,  5,  1,  0;
  [6] 1,  5, 10, 14,  8,  1, 0;

T. Kyle Petersen and Yan Zhuang, Zig-zag Eulerian polynomials, arXiv:2403.07181 [math.CO], 2024. (Table 3)

Cf. A050446, A050447, A276313 (main diagonal), A373353 (row sums of triangle).

Cf. A373423.

row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then
a :=  x - 1; for j from 1 to n do a :=  x - 1 / a od: a :=  a - x; else
a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;
fi; ser := series(a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]:      # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form

def Arow(n, len):
    R. = PowerSeriesRing(ZZ, len)
    if n == 0: return [1] + [0]*(len - 1)
    x = -x if n % 2 else x
    a = x + 1
    for _ in range(n):
        a = x - 1 / a
    a = x - a if n % 2 else a - x
    return a.list()
for n in range(7): print(Arow(n, 10))

A095310 a(n+3) = 2a(n+2) + 3(n+1) - a(n).

Original entry on oeis.org

1, 5, 12, 38, 107, 316, 915, 2671, 7771, 22640, 65922, 191993, 559112, 1628281, 4741905, 13809541, 40216516, 117119750, 341079507, 993301748, 2892722267, 8424270271, 24533405595, 71446899736, 208069745986, 605946785585

Offset: 1

Gary W. Adamson, Jun 02 2004

Let M = the 3 X 3 matrix [1 1 1 / 3 1 0 / 1 0 0], then M^n * [1 0 0] = [a(n) q a(n-1)] where q is another sequence with the same recursion rule.

			a(6) = 316 = 2*107 + 3*38 - 12.
a(5) = 107 since M^5 * [1 0 0] = [107 q 38].

Index entries for linear recurrences with constant coefficients, signature (2, 3, -1).

Cf. A001263, A006356, A006054, A061646, A061647, A095125, A095126.

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {0}, {0}})[[1, 1]]; Table[ a[n], {n, 27}] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{2,3,-1},{1,5,12},30] (* Harvey P. Dale, Jan 25 2014 *)

G.f.: (-x^2+3*x+1)/(x^3-3*x^2-2*x+1). - Harvey P. Dale, Jan 25 2014

Corrected and extended by Robert G. Wilson v, Jun 05 2004

Edited by N. J. A. Sloane, Jun 07 2004

A370377 a(n) is the number of symmetrical linear hydrocarbon chains with n C-C bonds.

Original entry on oeis.org

1, 3, 2, 6, 5, 14, 11, 31, 25, 70, 56, 157, 126, 353, 283, 793, 636, 1782, 1429, 4004, 3211, 8997, 7215, 20216, 16212, 45425, 36428, 102069, 81853, 229347, 183922, 515338, 413269, 1157954, 928607, 2601899, 2086561, 5846414, 4688460, 13136773, 10534874

Offset: 0

Tomasz Dziekanski, Feb 18 2024

			For n = 1: a(1) = A006356(1) = 3
 CH3-CH3, CH2=CH2, CH≡CH
For n = 3: a(3) = A006356(2) = 6
 CH3-CH2-CH2-CH3, CH3-CH=CH-CH3, CH3-C≡C-CH3, CH2=CH-CH=CH2, CH≡C-C≡CH, CH2=C=C=CH2
For n = 4: a(4) = A006356(2) - A006356(0) = 6 - 1 = 5
 CH3-CH2-CH2-CH2-CH3, CH3-CH=C=CH-CH3, CH2=CH-CH2-CH=CH2, CH≡C-CH2-C≡CH, CH2=C=C=C=CH2

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

Cf. A006356, A006054, A306334.

LinearRecurrence[{0, 2, 0, 1, 0, -1}, {1, 3, 2, 6, 5, 14}, 50] (* Paolo Xausa, Feb 22 2024 *)

Vec(O(x^55)+(1+3*x-x^5)/(1-2*x^2-x^4+x^6)) \\ Joerg Arndt, Feb 18 2024

a = [1, 3, 2, 6, 5, 14]
for i in range(30):
    a.append(2*a[-2]+a[-4]-a[-6])
print(a)

a(n) = 2*A306334(n) - A006356(n).
Also:
a(0) = 1;
a(2) = 2;
a(n) = A006356((n+1)/2) if n is odd;
a(n) = A006356(n/2) - A006356((n-4)/2) if n is even.
G.f.: (1+3*x-x^5)/(1-2*x^2-x^4+x^6). - Joerg Arndt, Feb 18 2024

A373567 Expansion of x + 1/(-x - 1/(-x - 1/(-x + 1))).

Original entry on oeis.org

1, 4, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209, 19181424995, 43100270734, 96845429254

Offset: 0

Peter Luschny, Jun 10 2024

a(n) is the number of up-down words of length n over an alphabet of size 4. - Sela Fried, Apr 08 2025

L. Carlitz and R. Scoville, Up-down sequences, Duke Math. J. (39) (1972), 583-598.

Sela Fried, A formula for the number of up-down words, arXiv:2503.02005 [math.CO], 2025.
Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, Pattern-avoiding alternating words, arXiv:1505.04078 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Essentially the same as A006356.

Cf. A050446.

CoefficientList[Series[x + 1/(-x - 1/(-x - 1/(-x + 1))), {x, 0, 31}], x] (* Michael De Vlieger, Jun 10 2024 *)

a(n) = [x^n] (x^4 - x^3 - 3*x^2 + 2*x + 1) / (x^3 - x^2 - 2*x + 1).

A006363 Number of antichains (or order ideals) in the poset B_4 X [n]; or size of the distributive lattice J(B_4 X [n]).

Original entry on oeis.org

1, 168, 7581, 160948, 2068224, 18561984, 127234008, 706987164, 3320153661, 13583619496, 49530070161, 163806121656, 498180781144, 1408758106368, 3737505070344, 9372218674824, 22351423903953, 50960797533096, 111574385244253, 235475590500876, 480631725411720, 951504952784320, 1831615165328400, 3435931869872580

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of order preserving maps from B_4 into [n+1]. a(n) is also the number of length n+1 multichains from bottom to top in J(B_4). See Stanley reference for bijections with description in title. - Geoffrey Critzer, Jan 15 2021

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, page 256, Proposition 3.5.1.

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.

Cf. A056932, A002415.

p = Subsets[Range[4]];
f[list1_, list2_] := If[ContainsAll[list2, list1], 1, 0]; \[Zeta] = Table[Table[f[p[[i]], p[[j]]], {j, 1, 16}], {i, 1, 16}]; JB4 =
Complement[Subsets[Range[16]],Level[Table[Select[Subsets[Range[16]],MemberQ[#, i] && !ContainsAll[Level[Position[\[Zeta][[All, i]], 1], {2}]][#] &], {i, 2,16}], {2}] // DeleteDuplicates]; \[Zeta]JB4 =Table[Table[f[JB4[[i]], JB4[[j]]], {j, 1, 168}], {i, 1,168}]; \[CapitalOmega][n_] := Expand[InterpolatingPolynomial[
Table[{k, MatrixPower[\[Zeta]JB4, k][[1, 168]]}, {k, 1, 17}],n]]; Table[\[CapitalOmega][n], {n, 1, 30}] (* Geoffrey Critzer, Jan 15 2021 *)

Title corrected by Geoffrey Critzer, Jan 15 2021

a(11)-a(23) from Geoffrey Critzer, Jan 15 2021

A120771 Expansion of ( 1-x^3+x^4+x^5-x^8 ) / ( 1-2*x^3-x^6+x^9 ).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 2, 1, 6, 5, 3, 14, 11, 6, 31, 25, 14, 70, 56, 31, 157, 126, 70, 353, 283, 157, 793, 636, 353, 1782, 1429, 793, 4004, 3211, 1782, 8997, 7215, 4004, 20216, 16212, 8997, 45425, 36428, 20216, 102069, 81853, 45425, 229347, 183922, 102069, 515338, 413269, 229347, 1157954, 928607, 515338

Offset: 0

Gary W. Adamson, Jul 03 2006

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,1,0,0,-1).

Cf. A077998 (trisection), A006054 (trisection), A006356 (trisection), A038196.

CoefficientList[Series[(1-x^3+x^4+x^5-x^8)/(1-2*x^3-x^6+x^9),{x,0,60}],x] (* or *) LinearRecurrence[{0,0,2,0,0,1,0,0,-1},{1,0,0,1,1,1,3,2,1},60] (* Harvey P. Dale, Feb 19 2016 *)

Three consecutive coefficients are generated from the left row of the n-th power of the matrix [1,1,1; 1,1,0; 1,0,0].

A268534 Size of free Kleene algebra on n generators.

Original entry on oeis.org

2, 6, 84, 43918

Offset: 0

N. J. A. Sloane, Feb 22 2016

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]

cp's OEIS Frontend

A373424 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(n) is the continued fraction (-1)^n/(~x - 1/(~x - ... 1/(~x - 1)))...) and where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

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A095310 a(n+3) = 2*a(n+2) + 3*(n+1) - a(n).

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A370377 a(n) is the number of symmetrical linear hydrocarbon chains with n C-C bonds.

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A373567 Expansion of x + 1/(-x - 1/(-x - 1/(-x + 1))).

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A006363 Number of antichains (or order ideals) in the poset B_4 X [n]; or size of the distributive lattice J(B_4 X [n]).

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A120771 Expansion of ( 1-x^3+x^4+x^5-x^8 ) / ( 1-2*x^3-x^6+x^9 ).

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A268534 Size of free Kleene algebra on n generators.

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A095310 a(n+3) = 2a(n+2) + 3(n+1) - a(n).