A038196 -id:A038196 - cp's OEIS frontend

A006356 a(n) = 2*a(n-1) + a(n-2) - a(n-3) for n >= 3, starting with a(0) = 1, a(1) = 3, and a(2) = 6.

1, 3, 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

Offset: 0

N. J. A. Sloane

Number of distributive lattices; also number of paths with n turns when light is reflected from 3 glass plates.
Let u(k), v(k), w(k) be defined by u(1) = 1, v(1) = 0, w(1) = 0 and u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k) + v(k), w(k+1) = u(k); then {u(n)} = 1, 1, 3, 6, 14, 31, ... (this sequence with an extra initial 1), {v(n)} = 0, 1, 2, 5, 11, 25, ... (A006054 with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = A077998 with an extra initial 0. - Benoit Cloitre, Apr 05 2002
Also u(k)^2 + v(k)^2 + w(k)^2 = u(2*k). - Gary W. Adamson, Dec 23 2003
The n-th term of the series is the number of paths for a ray of light that enters two layers of glass and then is reflected exactly n times before leaving the layers of glass.
One such path (with 2 plates of glass and 3 reflections) might be:
...\........./..................
--------------------------------
....\/\..../....................
--------------------------------
........\/......................
--------------------------------
For a k-glass sequence, say a(n,k), a(n,k) is always asymptotic to z(k)*w(k)^n where w(k) = (1/2)/cos(k*Pi/(2*k+1)) and it is conjectured that z(k) is the root 1 < x < 2 of a polynomial of degree Phi(2k+1)/2.
Number of ternary sequences of length n-1 such that every pair of consecutive digits has a sum less than 3. That is to say, the pairs (1,2), (2,1) and (2,2) do not appear. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 07 2004
Number of weakly up-down sequences of length n using the digits {1,2,3}. When n=2 the sequences are 11, 12, 13, 22, 23, 33.
Form the graph with matrix A = [1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006356 counts walks of length n that start at the degree 4 vertex. - Paul Barry, Oct 02 2004
In general, the g.f. for p glass plates is: A(x) = F_{p-1}(-x)/F_p(x) where F_p(x) = Sum_{k=0..p} (-1)^[(k+1)/2]*C([(p+k)/2],k)*x^k. - Paul D. Hanna, Feb 06 2006
Equals the INVERT transform of (1, 2, 1, 1, 1, ...) equivalent to a(n) = a(n-1) + 2*a(n-2) + a(n-3) + a(n-4) + ... + 1. a(6) = 70 = (31 + 2*14 + 6 + 3 + 1 + 1). - Gary W. Adamson, Apr 27 2009
a(n) = the number of terms in the n-th iterate of sequence A179542 generated from the rules a(0) = 1, then (1->1,2,3), (2->1,2), (3->1).
Example: 3rd iterate = (1,2,3,1,2,1,1,2,3,1,2,1,2,3) = 14 terms composed of a frequency of (6, 5, 3): (1's, 2's, and 3's), where a(3) = 14, and the [6, 5, 3] = top row and left column of the 3rd power of M, the matrix generator [1,1,1; 1,1,0; 1,0,0] or a(2) = 6, A006054(4) = 5, and a(1) = 3.
Given the heptagon diagonal lengths with edge = 1: (a = 1, b = 1.80193773..., c = 2.24697...) = (1, 2*cos(Pi/7), (1 + 2*cos(2*Pi/7))), and using the diagonal product formulas in [Steinbach], we obtain: c^n = c*a(n-2) + b*A006054(n) + a(n-3) corresponding to the top row of M^(n-1), in the case M^3 = [6, 5, 3]. Example: c^4 = 25.491566... = 6*c + 5*b + 3 = 13.481... + 9.00968... + 3. - Gary W. Adamson, Jul 18 2010
Equals row sums of triangle A180262. - Gary W. Adamson, Aug 21 2010
The number of the one-sided n-step prudent walks, avoiding 2 or more consecutive east steps. - Shanzhen Gao, Apr 27 2011
a(n) = [A_{7,2}^(n+2)](1,1), where A{7,2} is the 3 X 3 unit-primitive matrix (see [Jeffery]) A_{7,2} = [0,0,1; 0,1,1; 1,1,1]. The denominator of the generating function for this sequence is also the characteristic polynomial of A_{7,2}. - L. Edson Jeffery, Dec 06 2011 [See the comments for sequence A306334. - Petros Hadjicostas, Nov 17 2019]
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 0, 0; 1, 0, 1] or of the 3 X 3 matrix [1, 1, 1; 1, 1, 0; 1, 0, 0]. - R. J. Mathar, Feb 03 2014
Successive sequences in this set (A006356, A006357, A006358, etc.) can be generated as follows: Begin with (1, 1, 1, 1, 1, 1, ...); and perform an operation with three steps to get the next sequence in the series. First, put alternate signs in the current series: With (1, 1, 1, ...) this equals (1, -1, 1, -1, ...); then take the inverse, getting (1, 1, 0, 0, 0, ...). Take the INVERT transform of the last step, getting (1, 2, 3, 5, 8, ...). Repeat the three steps using (1, 2, 3, 5, ...) --> (1, -2, 3, -5) --> (1, 2, 1, 1, 1, ...) --> (1, 3, 6, 14, 31, ...). Repeat the three steps using (1, 3, 6, 14, 31, ...), getting (1, 4, 10, 30, 85, ...) = A006357; and so on. - Gary W. Adamson, Aug 08 2019
Let W_n be the fence poset (a.k.a. zig-zag poset) of size n.  Let [2] be a chain of size 2.  Then a(n) is the number of antichains in the product poset W_n X [2]. See Berman- Koehler link. - Geoffrey Critzer, Jun 13 2023
a(n) is the number of double-dimer covers of the 2 X (n+1) square grid graph. See Musiker et al. link. - Nicholas Ovenhouse, Jan 07 2024
In general, the number of weakly up-down words of length n over an alphabet of size k is given by 4/(2*k+1)*|Sum_{j = 1..k} sin^2(2*j*Pi/(2*k+1))/(2*cos^2(2*j*Pi/(2*k+1)))^(n+1)| and the corresponding g. f. is given by V_(k-1)(-x/2)/W_k(x/2) if k is even and -W_(k-1)(-x/2) / V_k(x/2) if k is odd, where V_m(x) and W_m(x) are the Chebyshev polynomials of the third and fourth kind, respectively (see Paul D. Hanna's comment above and the Fried link). - Sela Fried, Apr 01 2025

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd edition, p. 291 (very briefly without generalizations).
J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.
Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
J. Berman and P. Köhler, On Dedekind Numbers and Two Sequences of Knuth, J. Int. Seq., Vol. 24 (2021), Article 21.10.7.
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.
Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
V. E. Hoggatt Jr. and M. Bicknell-Johnson, Reflections across two and three glass plates, Fibonacci Quarterly, volume 17 (1979), 118-142.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 451
L. E. Jeffery, Unit-primitive matrices
B. Junge and V. E. Hoggatt, Jr., Polynomials arising from reflections across multiple plates, Fib. Quart., 11 (1973), 285-291.
Peter Köhler, The Central Decomposition of FD_01(n), Order (2021).
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
Julien Leroy, Michel Rigo, and Manon Stipulanti, Behavior of Digital Sequences Through Exotic Numeration Systems, Electronic Journal of Combinatorics 24(1) (2017), #P1.44.
Leo Moser, Problem B-6: some reflections, Fib. Quart. Vol. 1, No. 4 (1963), 75-76.
Leo Moser and Max Wyman, Multiple reflections, Fib. Quart., 11 (1973).
Gregg Musiker, Ralf Schiffler, Nicholas Ovenhouse, and Sylvester Zhang, Higher Dimer Covers on Snake Graphs, arXiv:2306.14389 [math.CO], 2023.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Lingjuan Shi and Kai Deng, The Numbers of Perfect and Maximal Matchings in Double Hexagonal Chains, Match Comm. Math. Comp. Chem. (2025) 659-685. See p. 8.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
R. Witula, D. Slota and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Cf. A000217, A000330, A050446, A050447, A006054, A077998, A052534, A052994, A052949.
See also A006357-A006359, A025030, A030112-A030116.
Cf. A038196 (3-wave sequence).
Cf. A179542. - Gary W. Adamson, Jul 18 2010
Cf. A180262. - Gary W. Adamson, Aug 21 2010
Cf. A033303, A190360, A306334.

a006056 n = a006056_list !! n
a006056_list = 1 : 3 : 6 : zipWith (+) (map (2 *) $ drop 2 a006056_list)
   (zipWith (-) (tail a006056_list) a006056_list)
-- Reinhard Zumkeller, Oct 14 2011

[ n eq 1 select 1 else n eq 2 select 3 else n eq 3 select 6 else 2*Self(n-1)+Self(n-2)- Self(n-3): n in [1..40] ] ; // Vincenzo Librandi, Aug 20 2011

A006356:=-(-1-z+z**2)/(1-2*z-z**2+z**3); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{2,1,-1},{1,3,6},30] (* or *) CoefficientList[ Series[ (1+x-x^2)/(1-2x-x^2+x^3),{x,0,30}],x] (* Harvey P. Dale, Jul 06 2011 *)
Table[If[n==0, a2=0; a1=1; a0=1, a3=a2; a2=a1; a1=a0; a0=2*a1+a2-a3], {n, 0, 29}] (* Jean-François Alcover, Apr 30 2013 *)

a(n):=sum(sum((sum(binomial(j,-3*k+2*j+i)*(-1)^(j-k)*binomial(k,j),j,0,k))*binomial(n+k-i-1,k-1),i,k,n),k,1,n); /* Vladimir Kruchinin, May 05 2011 */

{a(n)=local(p=3);polcoeff(sum(k=0,p-1,(-1)^((k+1)\2)*binomial((p+k-1)\2,k)* (-x)^k)/sum(k=0,p,(-1)^((k+1)\2)*binomial((p+k)\2,k)*x^k+x*O(x^n)),n)} \\ Paul D. Hanna, Feb 06 2006

Vec((1+x-x^2)/(1-2*x-x^2+x^3)+O(x^66)) \\ Joerg Arndt, Apr 30 2013

from math import comb
def A006356(n): return sum(comb(j,a)*comb(k,j)*comb(n+k-i,k-1)*(-1 if j-k&1 else 1) for k in range(1,n+2) for i in range(k,n+2) for j in range(k+1) if (a:=-3*k+2*j+i)>=0) # Chai Wah Wu, Feb 19 2024

a(n) is asymptotic to z(3)*w(3)^n where w(3) = (1/2)/cos(3*Pi/7) and z(3) is the root 1 < X < 2 of P(3, X) = 1 - 14*X - 49*X^2 + 49*X^3. w(3) = 2.2469796.... z(3) = 1.220410935...
G.f.: (1 + x - x^2)/(1 - 2*x - x^2 + x^3). - Paul D. Hanna, Feb 06 2006
a(n) = a(n-1) + a(n-2) + A006054(n+1). - Gary W. Adamson, Jun 05 2008
a(n) = A006054(n+2) + A006054(n+1) - A006054(n). - R. J. Mathar, Apr 07 2011
a(n-1) = Sum_{k = 1..n} Sum_{i = k..n} Sum_{j = 0..k} binomial(j, -3*k+2*j+i) * (-1)^(j-k) * binomial(k, j) * binomial(n+k-i-1, k-1). - Vladimir Kruchinin, May 05 2011
Sum_{k=0..n} a(k) = a(n+1) - a(n-1) - 1. - Greg Dresden and Mina BH Arsanious, Aug 23 2023

Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)

Alternative definition added by Andrew Niedermaier, Nov 11 2008

A038197 4-wave sequence.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 7, 9, 10, 19, 26, 30, 56, 75, 85, 160, 216, 246, 462, 622, 707, 1329, 1791, 2037, 3828, 5157, 5864, 11021, 14849, 16886, 31735, 42756, 48620, 91376, 123111, 139997, 263108, 354484, 403104, 757588, 1020696, 1160693, 2181389

Offset: 0

Floor van Lamoen

This sequence is related to the nonagon or 9-gon.

			The first few rows of the T(n,k) array are, n>=1, 1 <= k <=4:
  0,  0,   0,   1
  1,  1,   1,   1
  1,  2,   3,   4
  4,  7,   9,   10
  10, 19,  26,  30
  30, 56,  75,  85
  85, 160, 216, 246

F. v. Lamoen, Wave sequences
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Nonagon.
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-3,3,0,0,0,-1,1,-1).

The a(3*n) lead to A006357; The T(n,k) lead to A076264 and A091024.
Cf. A038196, A038201.
Cf. A120747 (m = 5: hendecagon or 11-gon)

m:=4: nmax:=15: 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/2 do seq(T(n,k), k=1..m) od; a(0):=1: Tx:=1: for n from 2 to nmax do for k from 2 to m do a(Tx):= T(n,k): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{1,-1,3,-3,3,0,0,0,-1,1,-1},{1,1,1,1,2,3,4,7,9,10,19},50] (* Harvey P. Dale, Oct 02 2015 *)

a(n) = a(n-1)+a(n-2) if n=3*m+1, a(n) = a(n-1)+a(n-4) if n=3*m+2, a(n) = a(n-1)+a(n-6) if n=3*m. Also: a(n) = 2*a(n-3)+3*a(n-6)-a(n-9)-a(n-12).
G.f.: -(-1-x-x^2+x^3-x^5+x^6)/(1-2*x^3-3*x^6+x^9+x^12)
a(n-1) = sequence(sequence(T(n,k), k=2..4), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 5-k..4) with T(1,1) = T(1,2) = T(1,3) = 0 and T(1,4) = 1; n>=1 and 1 <= k <= 4. [Steinbach]

Edited by Floor van Lamoen, Feb 05 2002

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

A187070 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 5, 6, 11, 14, 25, 31, 56, 70, 126, 157, 283, 353, 636, 793, 1429, 1782, 3211, 4004, 7215, 8997, 16212, 20216, 36428, 45425, 81853, 102069, 183922, 229347, 413269, 515338, 928607, 1157954, 2086561, 2601899

Offset: 0

L. Edson Jeffery, Mar 05 2011

Theory. (Start)
1. Definitions. Let T_(7,j,0) denote the rhombus with sides of unit length (=1), interior angles given by the pair (j*Pi/7,(7-j)*Pi/7) and Area(T_(7,j,0))=sin(j*Pi/7), j in {1,2,3}. Associated with T_(7,j,0) are its angle coefficients (j, 7-j) in which one coefficient is even while the other is odd. A half-tile is created by cutting T_(7,j,0) along a line extending between its two corners with even angle coefficient; let H_(7,j,0) denote this half-tile. Similarly, a T_(7,j,r) tile is a linearly scaled version of T_(7,j,0) with sides of length x^r and Area(T_(7,j,r))=x^(2*r)*sin(j*Pi/7), r>=0 an integer, where x is the positive, constant square root x=sqrt[(2*cos(j*Pi/7))^2 - 1]; likewise let H_(7,j,r) denote the corresponding half-tile. Often H_(7,i,r) (i in {1,2,3}) can be subdivided into an integral number of each equivalence class H_(7,j,0). But regardless of whether or not H_(7,j,r) subdivides, in theory such a proposed subdivision for each j can be represented by the matrix M=(m_(i,j)), i,j=1,2,3, in which the entry m_(i,j) gives the quantity of H_(7,j,0) tiles that should be present in a subdivided H_(7,i,r) tile. The number x^(2*r) (the square of the scaling factor) is an eigenvalue of M=(U_2)^r, where
  U_2= (0 0 1)
       (0 1 1)
       (1 1 1).
2. The sequence. Let r>=0, and let C_r be the r-th "block" defined by C_r={a(2*r),a(2*r+1),a(2*r+2)}. Note that C_r-2*C_(r-1)-C_(r-2)+C_(r-3)={0,0,0}. Let n=2*r+i-1. Then a(n)=a(2*r+i-1)=m_(i,3), where M=(m_(i,j))=(U_2)^r was defined above. Hence the block C_r corresponds component-wise to the third column of M, and a(n)=m_(i,3) gives the quantity of H_(7,3,0) tiles that should appear in a subdivided H_(7,i,r) tile. (End)
Combining blocks A_r, B_r and C_r, from A187068, A187069 and this sequence, respectively, as matrix columns [A_r,B_r,C_r] generates the matrix (U_2)^r, and a negative index (-1)*r yields the corresponding inverse [A_(-r),B_(-r),C_(-r)]=(U_2)^(-r) of (U_2)^r. Therefore, the three sequences need not be causal.
Since a(2*r+2)=a(2*(r+1)) for all r, this sequence arises by concatenation of third-column entries m_(1,3) and m_(2,3) from successive matrices M=(U_2)^r.
This sequence is a trivial extension of A038196.

			Suppose r=3. Then
C_r = C_3 = {a(2*r),a(2*r+1),a(2*r+2)} = {a(6),a(7),a(8)} = {3,5,6},
corresponding to the entries in the third column of
M = (U_2)^3 = (1 2 3)
              (2 4 5)
              (3 5 6).
Choose i=2 and set n=2*r+i-1. Then a(n) = a(2*r+i-1) = a(6+2-1) = a(7) = 5, which equals the entry in row 2 and column 3 of M. Hence a subdivided H_(7,2,3) tile should contain a(7) = m_(2,3) = 5 H_(7,3,0) tiles.

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. Edson Jeffery, Unit-primitive matrices
Roman Witula, D. Slota and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-1).

Cf. A038196, A187068, A187069.

a[0] = a[1] = 0; a[2] = a[3] = a[4] = 1; a[?Negative] = 0; a[n] := a[n] = 2*a[n-2] + a[n-4] - a[n-6]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 02 2013 *)

x='x+O('x^50); concat([0,0], Vec(x^2*(1+x-x^2)/(1-2*x^2-x^4+x^6))) \\ G. C. Greubel, Jul 05 2017

Recurrence: a(n) = 2*a(n-2) + a(n-4) - a(n-6).
G.f.: x^2*(1+x-x^2)/(1-2*x^2-x^4+x^6).
a(2*n)=A106803(n); a(2*n+1)=A006054(n+1); a(2*n+2)=A077998(n).
Closed-form: a(n) = (1/14)*[[X_1+Y_1*(-1)^(n-1)]*[(w_2)^2-(w_3)^2]*(w_1)^(n-1)+[X_2+Y_2*(-1)^(n-1)]*[(w_3)^2-(w_1)^2]*(w_2)^(n-1)+[X_3+Y_3*(-1)^(n-1)]*[(w_1)^2-(w_2)^2]*(w_3)^(n-1)], where w_k = sqrt[(2cos(k*Pi/7))^2-1], X_k = (w_k)^3+(w_k)^2-w_k and Y_k = -(w_k)^3+(w_k)^2+w_k, k=1,2,3.

A038201 5-wave sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 9, 12, 14, 15, 29, 41, 50, 55, 105, 146, 175, 190, 365, 511, 616, 671, 1287, 1798, 2163, 2353, 4516, 6314, 7601, 8272, 15873, 22187, 26703, 29056, 55759, 77946, 93819, 102091, 195910, 273856, 329615, 358671, 688286, 962142

Offset: 0

Floor van Lamoen

This sequence is related to the hendecagon or 11-gon, see A120747.

Sequence of perfect distributions for a cascade merge with six tapes according to Knuth. - Michael Somos, Feb 07 2004

			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
G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 9*x^8 + 12*x^9 + ...

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.3, Eq. (1).

F. v. Lamoen, Wave sequences
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Hendecagon.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,3,0,0,0,-4,0,0,0,-1,0,0,0,1).

Cf. A038196, A038197.

The a(4*n) values (column 0) lead to A006358; the T(n,k) lead to A069006, A038342 and A120747.

m:=5: nmax:=12: 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/2 do seq(T(n,k), k=1..m) od; a(0):=1: Tx:=1: for n from 2 to nmax do for k from 2 to m do a(Tx):= T(n,k): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{0,0,0,3,0,0,0,3,0,0,0,-4,0,0,0,-1,0,0,0,1},{1,1,1,1,1,2,3,4,5,9,12,14,15,29,41,50,55,105,146,175},50] (* Harvey P. Dale, Dec 13 2012 *)

{a(n) = local(m); if( n<=0, n==0, m = (n-1)\4 * 4; sum(k=2*m - n, m, a(k)))};

a(n) = a(n-1)+a(n-2) if n=4*m+1, a(n) = a(n-1)+a(n-4) if n=4*m+2, a(n) = a(n-1)+a(n-6) if n=4*m+3 and a(n) = a(n-1)+a(n-8) if n=4*m.
G.f.: -(1+x+x^2+x^3-2*x^4-x^5+x^7-x^8-x^11+x^12)/(-1+3*x^4+3*x^8-4*x^12-x^16+x^20).
a(n) =  3*a(n-4)+3*a(n-8)-4*a(n-12)-a(n-16)+a(n-20).
a(n-1) = sequence(sequence(T(n,k), k=2..5), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 6-k..5) with 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]

Edited by Floor van Lamoen, Feb 05 2002

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

A187068 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,1,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 2, 3, 5, 6, 11, 14, 25, 31, 56, 70, 126, 157, 283, 353, 636, 793, 1429, 1782, 3211, 4004, 7215, 8997, 16212, 20216, 36428, 45425, 81853, 102069, 183922, 229347, 413269, 515338, 928607, 1157954, 2086561, 2601899

Offset: 0

L. Edson Jeffery, Mar 06 2011

(Start) See A187070 for supporting theory. Define the matrix
U_2=
(0 0 1)
(0 1 1)
(1 1 1).
Let r>=0, and let A_r be the r-th "block" defined by A_r={a(2*r),a(2*r+1),a(2*r+2)}. Note that A_r-2*A_(r-1)-A_(r-2)+A_(r-3)={0,0,0}. Let n=2*r+i-1 and M=(m_(i,j))=(U_2)^r. Then A_r corresponds component-wise to the first column of M, and a(n)=a(2*r+i-1)=m_(i,1) gives the quantity of H_(7,1,0) tiles that should appear in a subdivided H_(7,i,r) tile.  (End)
Since a(2*r+2)=a(2*(r+1)) for all r, this sequence arises by concatenation of first-column entries m_(1,1) and m_(2,1) from successive matrices M=(U_2)^r.
This sequence is a nontrivial extension of both A038196 and A187070.

			(Start) Suppose r=3. Then
A_r = A_3 = {a(2*r),a(2*r+1),a(2*r+2)} = {a(6),a(7),a(8)} = {1,2,3},
corresponding to the entries in the first column of
M = m_(i,j) = (U_2)^3 =
(1 2 3)
(2 4 5)
(3 5 6).
Suppose i=2. Setting n=2*r+i-1, then a(n) = a(2*r+i-1) = a(6+2-1) = a(7) = m_(2,1) = 2. Hence a subdivided H_(7,2,3) tile should contain a(7) = m_(2,1) = 2 H_(7,1,0) tiles. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unit-primitive matrices
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-1).

Cf. A038196, A187069, A187070.

a[0] = 1; a[1] = a[2] = a[3] = 0; a[4] = a[5] = 1; a[?Negative] = 0; a[n] := a[n] = 2*a[n-2] + a[n-4] - a[n-6]; Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jan 02 2013 *)
CoefficientList[Series[(1 - 2*x^2 + x^5)/(1 - 2*x^2 - x^4 + x^6), {x, 0, 50}], x] (* G. C. Greubel, Jul 06 2017 *)

x='x+O('x^50); Vec((1-2*x^2+x^5)/(1-2*x^2-x^4+x^6)) \\ G. C. Greubel, Jul 06 2017

{a(n+2)} = A187070.
a(n) = 2*a(n-2) + a(n-4) - a(n-6).
G.f.: (1-2*x^2+x^5)/(1-2*x^2-x^4+x^6).
Closed-form: a(n) = (1/14)*[[X_1+Y_1*(-1)^(n-1)]*[(w_2)^2-(w_3)^2]*(w_1)^(n-1)+[X_2+Y_2*(-1)^(n-1)]*[(w_3)^2-(w_1)^2]*(w_2)^(n-1)+[X_3+Y_3*(-1)^(n-1)]*[(w_1)^2-(w_2)^2]*(w_3)^(n-1)], where w_k = sqrt[(2cos(k*Pi/7))^2-1], X_k = (w_k)^5-2*(w_k)^3+1 and Y_k = -(w_k)^5+2*(w_k)^3+1, k=1,2,3.

A187069 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 2, 4, 5, 9, 11, 20, 25, 45, 56, 101, 126, 227, 283, 510, 636, 1146, 1429, 2575, 3211, 5786, 7215, 13001, 16212, 29213, 36428, 65641, 81853, 147494, 183922, 331416, 413269, 744685, 928607, 1673292, 2086561, 3759853, 4688460, 8448313, 10534874

Offset: 0

L. Edson Jeffery, Mar 06 2011

See A187070 for supporting theory. Define the matrix
U_2 = (0 0 1)
      (0 1 1)
      (1 1 1).
Let r>=0, and let B_r be the r-th "block" defined by B_r={a(2*r),a(2*r+1),a(2*r+2)}. Note that B_r-2*B_(r-1)-B_(r-2)+B_(r-3)={0,0,0}. Let n=2*r+i-1 and M=(m_(i,j))=(U_2)^r. Then B_r corresponds component-wise to the second column of M, and a(n)=a(2*r+i-1)=m_(i,2) gives the quantity of H_(7,2,0) tiles that should appear in a subdivided H_(7,i,r) tile.
Since a(2*r+2)=a(2*(r+1)) for all r, this sequence arises by concatenation of second-column entries m_(1,2) and m_(2,2) from successive matrices M=(U_2)^r.

			Suppose r=3.
Then B_r = B_3 = {a(2*r),a(2*r+1),a(2*r+2)} = {a(6),a(7),a(8)} = {2,4,5}, corresponding to the entries in the second column of
  M = (U_2)^3 = (1 2 3)
                (2 4 5)
                (3 5 6).
Suppose i=2. Setting n=2*r+i-1, then a(n) = a(2*r+i-1) = a(6+2-1) = a(7) = m_(2,2) = 4. Hence a subdivided H_(7,2,3) tile should contain a(7) = m_(2,2) = 4 H_(7,2,0) tiles.

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. Edson Jeffery, Unit-primitive matrices
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-1).

Cf. A038196, A187068, A187070, A006054.

CoefficientList[Series[x*(1 - x^2 + x^3 - x^4)/(1 - 2*x^2 - x^4 + x^6), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)
LinearRecurrence[{0,2,0,1,0,-1},{0,1,0,1,1,2},50] (* Harvey P. Dale, Dec 16 2017 *)

my(x='x+O('x^50)); concat([0], Vec(x*(1-x^2+x^3-x^4)/(1-2*x^2-x^4+x^6))) \\ G. C. Greubel, Oct 20 2017

Recurrence: a(n) = 2*a(n-2) + a(n-4) - a(n-6).
G.f.: x*(1-x^2+x^3-x^4)/(1-2*x^2-x^4+x^6).
Closed-form: a(n) = (1/14)*[[X_1+Y_1*(-1)^(n-1)]*[(w_2)^2-(w_3)^2]*(w_1)^(n-1)+[X_2+Y_2*(-1)^(n-1)]*[(w_3)^2-(w_1)^2]*(w_2)^(n-1)+[X_3+Y_3*(-1)^(n-1)]*[(w_1)^2-(w_2)^2]*(w_3)^(n-1)], where w_k = sqrt[(2cos(k*Pi/7))^2-1], X_k = (w_k)^4-(w_k)^2+w_k-1 and Y_k = (w_k)^4+(w_k)^2-w_k-1, k=1,2,3.
a(2*n) = A006054(n), a(2*n+3) = A052534(n).

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

Original entry on oeis.org

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

cp's OEIS Frontend

A038223 Bottom line of 3-wave sequence A038196, also bisection of A006356.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

Extensions

A038213 Top line of 3-wave sequence A038196, also bisection of A006356.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A006356 a(n) = 2*a(n-1) + a(n-2) - a(n-3) for n >= 3, starting with a(0) = 1, a(1) = 3, and a(2) = 6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Formula

Extensions

A038197 4-wave sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A187070 Let i be in {1,2,3}, let r >= 0 be an integer and n=2*r+i-1. Then a(n)=a(2*r+i-1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A038201 5-wave sequence.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

A187070 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

A187068 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,1,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

A187069 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).