A106803 -id:A106803 - cp's OEIS frontend

A006054 a(n) = 2*a(n-1) + a(n-2) - a(n-3), with a(0) = a(1) = 0, a(2) = 1.

0, 0, 1, 2, 5, 11, 25, 56, 126, 283, 636, 1429, 3211, 7215, 16212, 36428, 81853, 183922, 413269, 928607, 2086561, 4688460, 10534874, 23671647, 53189708, 119516189, 268550439, 603427359, 1355888968, 3046654856, 6845771321, 15382308530, 34563733525

Offset: 0

N. J. A. Sloane

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,... (A006356 with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (this sequence 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(2k). - Gary W. Adamson, Dec 23 2003
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006054 counts walks of length n between the vertex of degree 1 and the vertex of degree 3. - Paul Barry, Oct 02 2004
Form the digraph with matrix [1,1,0; 1,0,1; 1,1,1]. A006054(n) counts walks of length n between the vertices with loops. - Paul Barry, Oct 15 2004
Nonzero terms = INVERT transform of (1, 1, 2, 2, 3, 3, ...). Example: 56 = (1, 1, 2, 5, 11, 25) dot (3, 3, 2, 2, 1, 1) = (3 + 3 + 4 + 10 + 11 + 25). - Gary W. Adamson, Apr 20 2009
-a(n+1) appears in the formula for the nonpositive powers of rho:= 2*cos(Pi/7), the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with sigma:=rho^2-1, the ratio of the larger heptagon diagonal to the side length, as follows. rho^(-n) = C(n)*1 + C(n-1)*rho - a(n+1)*sigma, n >= 0, with C(n)=A077998(n), C(-1):=0. See the Steinbach reference, and a comment under A052547.
If, with the above notations, the power basis of the field Q(rho) is taken one has for nonpositive powers of rho, rho^(-n) = a(n+2)*1 + A077998(n-1)*rho - a(n+1)*rho^2. For nonnegative powers see A006053. See also the Steinbach reference. - Wolfdieter Lang, May 06 2011
a(n) appears also in the nonnegative powers of sigma,(defined in the above comment, where also the basis is given). See a comment in A106803.
The sequence b(n):=(-1)^(n+1)*a(n) forms the negative part (i.e., with nonpositive indices) of the sequence (-1)^n*A006053(n+1). In this way we obtain what we shall call the Ramanujan-type sequence number 2a for the argument 2*Pi/7 (see the comment to Witula's formula in A006053). We have b(n) = -2*b(n-1) + b(n-2) + b(n-3) and b(n) * 49^(1/3) = (c(1)/c(4))^(1/3) * (c(1))^(-n) + (c(2)/c(1))^(1/3) * (c(2))^(-n) + (c(4)/c(2))^(1/3) * (c(4))^(-n) = (c(2)/c(1))^(1/3) * (c(1))^(-n+1) + (c(4)/c(2))^(1/3) * (c(2))^(-n+1) + (c(1)/c(4))^(1/3) * (c(4))^(-n+1), where c(j) := 2*cos(2*Pi*j/7) (for the proof, see the comments to A215112). - Roman Witula, Aug 06 2012
(1, 1, 2, 5, 11, 25, 56, ...) * (1, 0, 1, 0, 1, ...) = the variant of A006356: (1, 1, 3, 6, 14, 31, ...). - Gary W. Adamson, May 15 2013
The limit of a(n+1)/a(n) for n -> infinity is, for all generic sequences with this recurrence of signature (2,1,-1), sigma = rho^2-1, approximately 2.246979603, the length ratio (largest diagonal)/side in the regular heptagon (7-gon). For rho = 2*cos(Pi/7) and sigma see a comment above, and the P. Steinbach reference. Proof: a(n+1)/a(n) = 2 + 1/(a(n)/a(n-1)) - 1/((a(n)/a(n-1))*(a(n-1)/a(n-2))), leading in the limit to sigma^3 -2*sigma^2 - sigma + 1, which is solved by sigma = rho^2-1, due to C(7, rho) = 0 , with the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of rho (see A187360). - Wolfdieter Lang, Nov 07 2013
Numbers of straight-chain aliphatic amino acids involving single, double or triple bonds (allowing adjacent double bonds) when cis/trans isomerism is neglected. - Stefan Schuster, Apr 19 2018
Let A(r,n) be the total number of ordered arrangements of an n+r tiling of r red squares and white tiles of total length n, where the individual tile lengths can range from 1 to n. A(r,0) corresponds to a tiling of r red squares only, and so A(r,0) = 1. Also, A(r,n)=0 for n<0. Let A_1(r,n) = Sum_{j=0..n} A(r,j). Then the expansion of 1/(1 - 2*x - x^2 + x^3) is A_1(0,n) + A_1(1,n-2) + A_1(n-4) + ... = a(n) without the initial two 0's. In general, the expansion of 1/(1 - 2*x -x^k + x^(k+1)) is equal to Sum_{j>=0} A_1(j, n-j*k). - Gregory L. Simay, May 25 2018
For n>1, a(n) is the number of ways to tile a strip of length n-1 with one color of squares and dominos, two colors of trominos and quadrominos, 3 colors of 5-minos and 6-minos, and so on. - Greg Dresden and Zhiyu Zhang, Jun 26 2025

			G.f. = x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 56*x^7 + 126*x^8 + 283*x^9 + ... - _Michael Somos_, Jun 25 2018

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..150
C. P. de Andrade, J. P. de Oliveira Santos, E. V. P. da Silva and K. C. P. Silva, Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers, Open Journal of Discrete Mathematics, 2013, 3, 25-32 doi:10.4236/ojdm.2013.31006. - From _N. J. A. Sloane_, Feb 20 2013
Maximilian Fichtner, K. Voigt, and S. Schuster, The tip and hidden part of the iceberg: Proteinogenic and non-proteinogenic aliphatic amino acids, Biochimica et Biophysica Acta (BBA)-General, 2016, Volume 1861, Issue 1, Part A, January 2017, Pages 3258-3269.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 434
S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z. 281 (2015) 1061.
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
R. Sachdeva and A. K. Agarwal, Combinatorics of certain restricted n-color composition functions, Discrete Mathematics, 340, (2017), 361-372.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
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 (2,1,-1).

Cf. A005578, A006053, A006356, A007583, A080937, A094790, A214683, A214699, A214779, A215112, A306334.

Row sums of A144159 and A180264.

a006054 n = a006053_list !! n
a006054_list = 0 : 0 : 1 : zipWith (+) (map (2 *) $ drop 2 a006054_list)
   (zipWith (-) (tail a006054_list) a006054_list)
-- Reinhard Zumkeller, Oct 14 2011

A006054:=z**2/(1-2*z-z**2+z**3); # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{2, 1, -1}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

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

x='x+O('x^66);
concat([0, 0], Vec(x^2/(1-2*x-x^2+x^3))) \\ Joerg Arndt, May 05 2011

G.f.: x^2/(1-2*x-x^2+x^3).
Sum_{k=0..n+2} a(k) = A077850(n). - Philippe Deléham, Sep 07 2006
Let M = the 3 X 3 matrix [1,1,0; 1,2,1; 0,1,2], then M^n*[1,0,0] = [A080937(n-1), A094790(n), A006054(n-1)]. E.g., M^3*[1,0,0] = [5,9,5] = [A080937(2), A094790(3), A006054(2)]. - Gary W. Adamson, Feb 15 2006
a(n) = round(k*A006356(n-1)), for n>1, where k = 0.3568958678... = 1/(1+2*cos(Pi/7)). - Gary W. Adamson, Jun 06 2008
a(n+1) = A187070(2n+1) = A187068(2n+3). - L. Edson Jeffery, Mar 10 2011
a(n+3) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-1)^(j-k)*binomial(k,j)*2^(-n+3*k-j); a(0)=0, a(1)=0, a(2)=1. - Vladimir Kruchinin, May 05 2011
7*a(n) = (c(2)-c(4))*(1+c(1))^n + (c(4)-c(1))*(1+c(2))^n + (c(1)-c(2))*(1+c(4))^n, where c(j):=2*cos(2*Pi*j/7) - for the proof see Witula et al. papers. - Roman Witula, Aug 07 2012
a(n) = -A006053(1-n) for all n in Z. - Michael Somos, Jun 25 2018

A077998 Expansion of (1-x)/(1-2*x-x^2+x^3).

Original entry on oeis.org

1, 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, 43100270734, 96845429254

Offset: 0

N. J. A. Sloane, Nov 17 2002

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,... (A006356 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 = this sequence with an extra initial 0. - Benoit Cloitre, Apr 05 2002 [Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary W. Adamson, Dec 23 2003]
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A077998 counts closed walks of length n at the vertex of degree 4. - Paul Barry, Oct 02 2004
a(n) is the number of Motzkin (n+2)-sequences with no flatsteps at ground level and whose height is <=2. For example, a(3)=6 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD, UUFDD. - David Callan, Dec 09 2004
Number of compositions of n if there are two kinds of part 2. Example: a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1). Row sums of A105477. - Emeric Deutsch, Apr 09 2005
Diagonal sums of A056242. - Paul Barry, Dec 26 2007
Diagonal sums of triangle in A105306. - Philippe Deléham, Nov 16 2008
a(n) appears in the formula for the nonpositive powers of rho:= 2*cos(Pi/7), the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with sigma:=rho^2-1, the ratio of the larger heptagon diagonal to the side length, as follows. rho^(-n) = a(n)*1 + a(n-1)*rho - C(n)*sigma, n>=0, with C(n)=A006054(n+1). Put a(-1):=0. See the Steinbach reference, and a comment under A052547.
The limit a(n+1)/a(n) for n -> infinity is sigma = rho^2-1, approximately 2.246979603. See a Nov 07 2013 comment on A006054 for the proof, and the preceding comment for rho and sigma and the P. Steinbach reference. - Wolfdieter Lang, Nov 07 2013
From Greg Dresden and Aaron Zhou, Jun 15 2023: (Start)
a(n) is the number of ways to tile a skew double-strip of 3*n cells using all possible "trominos". Here is the skew double-strip corresponding to n=4, with 12 cells:
   _ ___ _ ___ _ ___
  |   |   |   |   |   |   |
 |__|___|_|___| |___|
|   |   |   |   |   |   |
|_|___|_|___|_|___|,
and here are the three possible "tromino" tiles, which can be rotated or reflected as needed:
   _              _
  |   |            |   |
 |__|_      ___|___|     _________
|   |   |    |   |   |      |   |   |   |
|_|___|,   |_|___|  ,   |_|___|_|.
As an example, here is one of the a(4) = 14 ways to tile the skew double-strip of 12 cells:
   _ ___ _____ _______
  |   |   |       |       |
 |   |  |___  |     |
|       |       |   |   |
|_____|_______|_|___|. (End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 14*x^4 + 31*x^5 + 70*x^6 + 157*x^7 + 353*x^8 + ... - _Michael Somos_, Dec 12 2023

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z. 281 (2015) 1061.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015.
Floor van Lamoen, Wave sequences
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).

Apart from initial term, same as A006356, which is the main entry for this sequence. A106803 is yet another version.

Cf. A096976, A105477.

a:=[1,1,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jun 27 2019

I:=[1,1,3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 01 2017

CoefficientList[Series[(1-x)/(1-2*x-x^2+x^3), {x, 0, 40}], x] (* Stefan Steinerberger, Sep 11 2006 *)
LinearRecurrence[{2,1,-1},{1,1,3},40] (* Roman Witula, Aug 07 2012 *)
a[ n_] := {1, 0, 0} . MatrixPower[{{0, 1, 0}, {0, 0, 1}, {-1, 1, 2}}, n] . {1, 1, 3}; (* Michael Somos, Dec 12 2023 *)

a(n)=([0,1,0; 0,0,1; -1,1,2]^n*[1;1;3])[1,1] \\ Charles R Greathouse IV, May 10 2016

((1-x)/(1-2*x-x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

a(0)=a(1)=1, a(2)=3, a(n+1) = 2*a(n) + a(n-1) - a(n-2) for n>=2. - Philippe Deléham, Sep 07 2006
7*a(n) = (s(2))^2*(1+c(1))^n + (s(4))^2*(1+c(2))^n + (s(1))^2(1+c(4))^n, where c(j) = 2*Cos(2Pi*j/7) and s(j) = 2*Sin(2Pi*j/7) - for the proof of this one and many other relations for the sequences u(k), v(k) and w(k) defined on the top of the comments by Benoit Cloitre - see Witula et al.'s paper. - Roman Witula, Aug 07 2012
a(n) = b(n+2)- b(n+1), first differences of b(n) = A006054(n). - Wolfdieter Lang, Nov 07 2013; corrected by Kai Wang, May 31 2017
a(n) = A096976(-n) for all n in Z. - Michael Somos, Dec 12 2023

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar

A052547 Expansion of (1-x)/(1-x-2*x^2+x^3).

Original entry on oeis.org

1, 0, 2, 1, 5, 5, 14, 19, 42, 66, 131, 221, 417, 728, 1341, 2380, 4334, 7753, 14041, 25213, 45542, 81927, 147798, 266110, 479779, 864201, 1557649, 2806272, 5057369, 9112264, 16420730, 29587889, 53317085, 96072133, 173118414, 311945595, 562110290, 1012883066

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Form the graph with matrix A=[0,1,1;1,0,0;1,0,1] (P_3 with a loop at an extremity). Then A052547 counts closed walks of length n at the degree 2 vertex. - Paul Barry, Oct 02 2004
The characteristic polynomial x^3 - x^2 - 2*x + 1 generates a 3 step recursion: a(0)=1,a(1)=0,a(2)=2, for n>2 a(n)=a(n-1)+2*a(n-2)-a(n-3) so we can also prepend the term 1,0 to a(n) and get the same sequence, i.e. start with a(0)=1,a(1)=0,a(2)=1. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005
The length of the diagonals (including the side) of a regular 7-gon (heptagon) inscribed in a circle of radius r=1 are d_1=2*sin(Pi/7) (the side length), d_2=2*cos(Pi/7)*d_1, and d_3=2*sin(3*Pi/7). The two ratios are rho := R_2 = d_2/d_1 = 2*cos(Pi/7) approximately 1.801937736, and sigma:= R_3 = d_3/d_1 = S(2,rho) = rho^2-1, approximately 2.246979604. See A049310 for Chebyshev S-polynomials. See the Steinbach reference  where the basis <1,rho,sigma> has been considered for an extension of the rational field Q, which is there called Q(rho). This rho is the largest zero of S(6,x). For nonnegative powers of rho one has rho^n = C(n)*1 + B(n)*rho + A(n)*sigma, with B(n)=a(n-1), a(-1):=0, a(-2):=1, A(n)=B(n+1)-B(n-1)= A006053(n), and C(n)=B(n-1)=a(n-2), n>=0. For negative powers see A106803 and -A006054. For nonnegative and negative powers of sigma see A006054, A106803 and a(n), -A006053, respectively.
a(n) appears also in the formula for the nonpositive powers of sigma (see the comment above for the definition and the Steinbach basis) as sigma^(-n) = a(n)*1 - A006053(n+1)*rho - a(n-1)*sigma, n>=0. Put a(-1):=0. 1/sigma=sigma-rho, the smallest positive zero of S(6,x) (see A049310 for Chebyshev S-polynomials). - Wolfdieter Lang, Dec 01 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.
Man Gao, Colin Tan, and Jie Wu, Loop homological invariants associated to real projective spaces, Topology Appl. 209, 275-288 (2016).
M. Gao and J. Wu, Homology Decompositions of the Loops on 1-Stunted Borel Constructions of C_2-Actions, arXiv:1301.0864 [math.AT], 2013.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 483
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A096976; second differences of A028495 and first differences of A006053 (up to an offset).

a:=[1,0,2];; for n in [4..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 08 2019

I:=[1,0,2]; [n le 3 select I[n] else Self(n-1) + 2*Self(n-2) - Self(n-3): n in [1..40]]; // G. C. Greubel, May 08 2019

spec := [S,{S=Sequence(Prod(Z,Union(Z,Prod(Z, Sequence(Z)))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..40);

LinearRecurrence[{1, 2, -1}, {1, 0, 2}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{a(n) = if(n==0,1,if(n==1,0,if(n==2,2,a(n-1)+2*a(n-2)-a(n-3))))};
for(i=0,40,print1(a(i),",")) \\ Lambert Klasen, Jan 30 2005

((1-x)/(1-x-2*x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 08 2019

a(n) = a(n-1) + 2*a(n-2) - a(n-3), with a(0)=1, a(1)=0, a(2)=2.
a(n) = Sum(-1/7*_alpha*(-3+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^3-2*_Z^2-_Z+1)).
a(n) = 5*a(n-2) - 6*a(n-4) + a(n-6). - Floor van Lamoen, Nov 02 2005

More terms from James Sellers, Jun 05 2000

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.

cp's OEIS Frontend

A006054 a(n) = 2*a(n-1) + a(n-2) - a(n-3), with a(0) = a(1) = 0, a(2) = 1.

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A077998 Expansion of (1-x)/(1-2*x-x^2+x^3).

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A052547 Expansion of (1-x)/(1-x-2*x^2+x^3).

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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).

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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).