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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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

References

  • 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).

Links

Crossrefs

Cf. A005578, A006053, A006356, A007583, A080937, A094790, A214683, A214699, A214779, A215112, A306334.
Row sums of A144159 and A180264.

Programs

  • Haskell
    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
  • Maple
    A006054:=z**2/(1-2*z-z**2+z**3); # Simon Plouffe in his 1992 dissertation
  • Mathematica
    LinearRecurrence[{2, 1, -1}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
  • Maxima
    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 */
  • PARI
    x='x+O('x^66);
concat([0, 0], Vec(x^2/(1-2*x-x^2+x^3))) \\ Joerg Arndt, May 05 2011

Formula

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

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.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Haskell
    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
  • Magma
    [ 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
  • Maple
    A006356:=-(-1-z+z**2)/(1-2*z-z**2+z**3); # conjectured by Simon Plouffe in his 1992 dissertation
  • Mathematica
    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 *)
  • Maxima
    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 */
  • PARI
    {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
  • PARI
    Vec((1+x-x^2)/(1-2*x-x^2+x^3)+O(x^66)) \\ Joerg Arndt, Apr 30 2013
  • Python
    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

Formula

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

Extensions

Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)
Alternative definition added by Andrew Niedermaier, Nov 11 2008

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

Views

Author

Tomasz Dziekanski, Feb 18 2024

Keywords

Examples

			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

Links

Crossrefs

Cf. A006356, A006054, A306334.

Programs

  • Mathematica
    LinearRecurrence[{0, 2, 0, 1, 0, -1}, {1, 3, 2, 6, 5, 14}, 50] (* Paolo Xausa, Feb 22 2024 *)
  • PARI
    Vec(O(x^55)+(1+3*x-x^5)/(1-2*x^2-x^4+x^6)) \\ Joerg Arndt, Feb 18 2024
  • Python
    a = [1, 3, 2, 6, 5, 14]
for i in range(30):
    a.append(2*a[-2]+a[-4]-a[-6])
print(a)

Formula

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
Showing 1-3 of 3 results.

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