A006357 -id:A006357 - cp's OEIS frontend

A120747 Sequence relating to the 11-gon (or hendecagon).

0, 1, 4, 14, 50, 175, 616, 2163, 7601, 26703, 93819, 329615, 1158052, 4068623, 14294449, 50221212, 176444054, 619907431, 2177943781, 7651850657, 26883530748, 94450905714, 331837870408, 1165858298498, 4096053203771, 14390815650209, 50559786403254

Offset: 1

Gary W. Adamson, Jul 01 2006

The hendecagon is an 11-sided polygon. The preferred word in the OEIS is 11-gon.
The lengths of the diagonals of the regular 11-gon are r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5, where r[1] = 1 is the length of the edge.
The value of limit(a(n)/a(n-1),n=infinity) equals the longest diagonal r[5].
The a(n) equal the matrix elements M^n[1,2], where M = Matrix([[1,1,1,1,1], [1,1,1,1,0], [1,1,1,0,0], [1,1,0,0,0], [1,0,0,0,0]]). The characteristic polynomial of M is (x^5 - 3x^4 - 3x^3 + 4x^2 + x - 1) with roots x1 = -r[4]/r[3], x2 = -r[2]/r[4], x3 = r[1]/r[2], x4 = r[3]/r[5] and x5 = r[5]/r[1].
Note that M^4*[1,0,0,0,0] = [55, 50, 41, 29, 15] which are all terms of the 5-wave sequence A038201. This is also the case for the terms of M^n*[1,0,0,0,0], n>=1.

			From _Johannes W. Meijer_, Aug 03 2011: (Start)
The lengths of the regular hendecagon edge and diagonals are:
  r[1] = 1.000000000, r[2] = 1.918985948, r[3] = 2.682507066,
  r[4] = 3.228707416, r[5] = 3.513337092.
The first few rows of the T(n,k) array are, n>=1, 1 <= k <=5:
    0,   0,   0,   0,   1, ...
    1,   1,   1,   1,   1, ...
    1,   2,   3,   4,   5, ...
    5,   9,  12,  14,  15, ...
   15,  29,  41,  50,  55, ...
   55, 105, 146, 175, 190, ...
  190, 365, 511, 616, 671, ... (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jay Kappraff, Slavik Jablan, Gary W. Adamson and Radmila Sazdanovich, Golden Fields, Generalized Fibonacci Sequences and Chaotic Matrices, Forma, Vol. 19 No. 4, pp. 367-387, 2004.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165
Eric Weisstein's World of Mathematics, Hendecagon.
Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1).

Cf. A006358, A033304, A038201, A052963, A065941, A066170.
From Johannes W. Meijer, Aug 03 2011: (Start)
Cf. A006358 (T(n+2,1) and T(n+1,5)), A069006 (T(n+1,2)), A038342 (T(n+1,3)), this sequence (T(n,4)) (m=5: hendecagon or 11-gon).
Cf. A000045 (m=2; pentagon or 5-gon); A006356, A006054 and A038196 (m=3: heptagon or 7-gon); A006357, A076264, A091024 and A038197 (m=4: enneagon or 9-gon); A006359, A069007, A069008, A069009, A070778 (m=6; tridecagon or 13-gon); A025030 (m=7: pentadecagon or 15-gon); A030112 (m=8: heptadecagon or 17-gon). (End)

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) )); // G. C. Greubel, Nov 13 2022

nmax:=27: m:=5: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/3 do seq(T(n,k), k=1..m) od; for n from 1 to nmax do a(n):=T(n,4) od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{3, 3, -4, -1, 1}, {0, 1, 4, 14, 50}, 41] (* G. C. Greubel, Nov 13 2022 *)

def A120747_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) ).list()
A120747_list(40) # G. C. Greubel, Nov 13 2022

a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
From Johannes W. Meijer, Aug 03 2011: (Start)
a(n) = T(n,4) with T(n,k) = Sum_{k1 = 6-k..6} T(n-1, k1), T(1,1) = T(1,2) = T(1,3) = T(1,4) = 0 and T(1,5) = 1, n>=1 and 1 <= k <= 5. [Steinbach]
Sum_{k=1..5} T(n,k)*r[k] = r[5]^n, n>=1. [Steinbach]
r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5. [Kappraff]
Sum_{k=1..5} T(n,k) = A006358(n-1).
Limit_{n -> 00} T(n,k)/T(n-1,k) = r[5], 1 <= k <= 5.
sequence(sequence( T(n,k), k=2..5), n>=1) = A038201(n-4).
G.f.: (x^2*(x - x1)*(x - x2))/((x - x3)*(x - x4)*(x - x5)*(x - x6)*(x - x7)) with x1 = phi, x2 = (1-phi), x3 = r[1] - r[3], x4 = r[3] - r[5], x5 = r[5] - r[4], x6 = r[4] - r[2], x7 = r[2], where phi = (1 + sqrt(5))/2 is the golden ratio A001622. (End)

Edited and information added by Johannes W. Meijer, Aug 03 2011

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.

Original entry on oeis.org

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))

A060823 4-wave sequence beginning with 2's with middles dropped.

Original entry on oeis.org

2, 2, 8, 20, 60, 170, 492, 1414, 4074, 11728, 33772, 97240, 279994, 806208, 2321386, 6684162, 19246280, 55417452, 159568196, 459458306, 1322957468, 3809304206, 10968454314, 31582405472, 90937912212, 261845282320, 753953441490

Offset: 2

Jason Earls, May 01 2001

Harry J. Smith, Table of n, a(n) for n = 2..200
F. v. Lamoen, Wave sequences
Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Cf. A038197.

LinearRecurrence[{2,3,-1,-1},{2,2,8,20},30] (* Harvey P. Dale, Aug 31 2016 *)

{ for (n=2, 200, if (n>5, a=2*a1 + 3*a2 - a3 - a4; a4=a3; a3=a2; a2=a1; a1=a, if (n==2, a=a4=2, if (n==3, a=a3=2, if (n==4, a=a2=8, a=a1=20)))); write("b060823.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 12 2009

a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4). - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 22 2003
From R. J. Mathar, Jul 15 2009: (Start)
G.f.: -2*x^2*(-1+x+x^2)/((1+x)*(x^3-3*x+1)).
a(n) = 2*A006357(n-3). (End)

A123609 Quasiperiodic 9-gonal (nonagonal) sequence as a 1-dimensional tiling.

Original entry on oeis.org

4, 1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 03 2006

The frequency of each distinct term (1, 2, 3, or 4) tends to converge to the ratio of each diagonal (a, b, c, or d) to the sum of the 4 diagonal lengths. The four 9-gon (nonagon) diagonals are a=1, b=1.87938524..., c=2.53208888... and d=2.87938524..., with the sum a+b+c+d = d^2 = 8.29085936.... Converting these terms to fractions of the sum, a=0.12061475..., b=0.22668159..., c=0.30540728..., and d=0.34729635.... Through n = 45, we can thus expect sixteen 4's (correct), since round(45*.34729635...) = 16. The numbers of terms in each subset strung together is found in A006357: (1, 4, 10, 30, 85...), thus: (4), (1,2,3,4), (4,3,4,2,3,4,1,2,3,4), ..., while the distributive breakdown of numbers of 1's, 2's, 3's, and 4's may be found in the 4-termed set of vectors in A076264: 1 1 1 1 4 3 2 1 10 9 7 4 30 26 19 10 ... where the sum of 4 terms in a row = the left term in the next row. For example, the frequency distribution of 30 includes ten 4's, nine 3's, seven 2's, and four 1's. Check: the subset of 30 terms generated from the previous subset of 10: (1,2,3,4,2,3,4,1,2,3,4,3,4,2,3,4,1,2,3,4,4,3,4,2,3,4,1,2,3,4).

A fractal structure is suggested by parsing each subset into groups: (1,2,3,4), (2,3,4), (1,2,3,4), (3,4), (2,3,4), (1,2,3,4), (4), (3,4), (2,3,4), (1,2,3,4). That is, 10 groups: four with four terms, three with three terms, two with two terms, and one with one term. Replacing the terms (4,3,2,1) with the diagonal lengths (d,c,b,a) and referring to the set of vectors: (1,1,1,1; 4,3,2,1; 10,9,7,4; ...), label these rows 2,3,4,... and consider (2,3,4,...) exponents to diagonal d=2.87938524..., such that, for example, "4" corresponds to (10,9,7,4), and (Cf. Steinbach) d^4 = 68.738349... = (10*d + 9*c + 7*b + 4*a). Such relationships are a consequence of the "Diagonal Product Formulas" mentioned on p. 23.

			1=>4, then 4=>1,2,3,4, which, in turn, generates 4,3,4,2,3,4,1,2,3,4 (append next result to right of previous result, getting an infinite aperiodic sequence).

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, p. 29.

Cf. A123508, A006357, A076264, A091024.

Drop[SubstitutionSystem[{1->{4},2->{3,4},3->{2,3,4},4->{1,2,3,4}},{1},{5}][[1]],5] (* Harvey P. Dale, Mar 02 2022 *)

Using the seed "1", we use the recurrence rules 1=>4; 2=>3,4; 3=>2,3,4; 4=>1,2,3,4; to form iterative subsets which are appended in succession to form a continuous string.

Partially edited by Jon E. Schoenfield, Sep 15 2013

A373568 Expansion of x - 1/(x - 1/(x - 1/(x - 1/(x + 1)))).

Original entry on oeis.org

1, 5, 10, 30, 85, 246, 707, 2037, 5864, 16886, 48620, 139997, 403104, 1160693, 3342081, 9623140, 27708726, 79784098, 229729153, 661478734, 1904652103, 5484227157, 15791202736, 45468956106, 130922641160, 376976720745, 1085461206128, 3125460977225, 8999406210929

Offset: 0

Peter Luschny, Jun 10 2024

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

Essentially the same as A006357.

CoefficientList[Series[x - 1/(x - 1/(x - 1/(x - 1/(x + 1)))), {x, 0, 28}], x] (* Michael De Vlieger, Jun 10 2024 *)

a(n) = [x^n] (x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1)/(x^4 + x^3 - 3*x^2 - 2*x + 1).

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A120747 Sequence relating to the 11-gon (or hendecagon).

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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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A060823 4-wave sequence beginning with 2's with middles dropped.

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A123609 Quasiperiodic 9-gonal (nonagonal) sequence as a 1-dimensional tiling.

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A373568 Expansion of x - 1/(x - 1/(x - 1/(x - 1/(x + 1)))).

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