A176971 -id:A176971 - cp's OEIS frontend

A005251 a(0) = 0, a(1) = a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

0, 1, 1, 1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786, 26931732, 47261895, 82938844, 145547525, 255418101, 448227521

Offset: 0

N. J. A. Sloane and R. K. Guy

a(n+3) is the number of n-bit sequences that avoid 010. Example: For n=4 the 12 sequences are all 4-bit sequences except 0100, 0101, 0010, 1010. - David Callan, Mar 25 2004
a(n+2) is the number of compositions (ordered partitions) of n where no two adjacent parts are != 1, see example. - Joerg Arndt, Jan 26 2013
a(n+1) is the number of compositions of n avoiding the part 2. - Joerg Arndt, Jul 13 2014
Number of different positive braids with n crossings of 3 strands.
This is a_2(n) in the Doroslovacki reference. Note that there is a typo in the paper in the formula for a_2(n): the upper bound in the inner sum should be "n-i" not "i-1". - Max Alekseyev, Jun 26 2007
a(n) is the number of peakless Motzkin paths of length n-1 with no UHH...HD's starting at level > 0 (here n > 0 and U=(1,1), H=(1,0), D=(1,-1)). Example: a(5)=7 because from all 8 peakless Motzkin paths of length 5 (see A004148) only UUHDD does not qualify. - Emeric Deutsch, Sep 13 2004
Equals the INVERT transform of (1, 0, 1, 1, 1, ...); equivalent to a(n) = a(n-1) + a(n-3) + a(n-4) + ... - Gary W. Adamson, Apr 27 2009
a(n) is the number of length n-1 words on {0,1} such that each string of 1's is followed by a string of at least two 0's. For example, a(5) = 4 because we have: 0000, 0100, 1000, and 1100. - Geoffrey Critzer, Aug 09 2013
a(n+1) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 0; 0, 1, 1; 1, 0, 0] or [1, 0, 1; 1, 1, 0; 0, 1, 0] or [1, 1, 0; 0, 0, 1; 1, 0, 1] or [1, 0, 1; 1, 0, 0; 0, 1, 1]. - R. J. Mathar, Feb 03 2014
For n >= 2, a(n) is the number of (n-2)-length binary words with no isolated zeros. - Milan Janjic, Mar 07 2015
Apart from the first three terms, the total number of bargraphs of semiperimeter n of height at most two for n >= 2 starts 1, 2, 4, 7, 12, ... - Arnold Knopfmacher, Nov 02 2016
Number of DD-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are DD-equivalent iff the positions of pattern DD are identical in these paths. - Sergey Kirgizov, Apr 08 2018
From Gus Wiseman, Nov 25 2019: (Start)
For n > 0, also the number of subsets of {1, ..., n - 3} such that if x and x + 2 are both in the subset, then so is x + 1. For example, the a(3) = 1 through a(7) = 12 subsets are:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {2,3}    {1,2}
                  {1,2,3}  {1,4}
                           {2,3}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}
(End)
The two-dimensional version, which counts sets of pairs where, if two pairs are separated by graph-distance 2, then the intermediate pair or pairs are also in the set, is A329871. - Gus Wiseman, Nov 30 2019
a(n+1) is the number of ways to tile a strip of length n with squares, dominoes, and tetrominoes, where the first tile cannot be a domino. - Greg Dresden and Myanna Nash, Aug 18 2020
For n>=3, a(n) is the number of binary strings of length n-2 without any maximal runs of ones of length 1. - Félix Balado, Aug 25 2025

			From _Joerg Arndt_, Jan 26 2013: (Start)
The a(5+2) = 12 compositions of 5 where no two adjacent parts are != 1 are
  [ 1]  [ 1 1 1 1 1 ]
  [ 2]  [ 1 1 1 2 ]
  [ 3]  [ 1 1 2 1 ]
  [ 4]  [ 1 1 3 ]
  [ 5]  [ 1 2 1 1 ]
  [ 6]  [ 1 3 1 ]
  [ 7]  [ 1 4 ]
  [ 8]  [ 2 1 1 1 ]
  [ 9]  [ 2 1 2 ]
  [10]  [ 3 1 1 ]
  [11]  [ 4 1 ]
  [12]  [ 5 ]
(End)
G.f. = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 12*x^7 + 21*x^8 + 37*x^9 + ...

S. Burckel, Efficient methods for three strand braids (submitted). [Apparently unpublished]
P. Chinn and S. Heubach, "Compositions of n with no occurrence of k", Congressus Numeratium, 2002, v. 162, pp. 33-51.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 205.
R. K. Guy, "Anyone for Twopins?" in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
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..500
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). See Table 4.
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Shifted quasisymmetric functions and the Hopf algebra of peak functions, arXiv:math/9904105 [math.CO], 1999.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 11.
A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 112.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Properties of a Ternary Infinite Word, arXiv:2206.01776 [cs.DM], 2022.
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Complement Avoidance in Binary Words, arXiv:2209.09598 [math.CO], 2022.
J. Demetrovics et al., On the number of unions in a family of sets, in Combinatorial Math., Proc. 3rd Internat. Conf., Annals NY Acad. Sci., 555 (1989), 150-158.
R. Doroslovacki, Binary sequences without 011...110 (k-1 1's) for fixed k, Mat. Vesnik 46 (1994), no. 3-4, 93-98.
Nazim Fatès, Biswanath Sethi, and Sukanta Das, On the Reversibility of ECAs with Fully Asynchronous Updating: The Recurrence Point of View, in Reversibility and Universality, Andrew Adamatzky, editor, Emergence, Complexity and Computation Vol. 30. Springer, 2018.
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.
R. K. Guy, Letter to N. J. A. Sloane, Feb 1986
R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,2).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 98
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Vedran Krcadinac, A new generalization of the golden ratio, Fibonacci Quart. 44 (2006), no. 4, 335-340.
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras II. Satisfaction of bracketing identities, spectrum dichotomy, arXiv:2011.08522 [math.CO], 2020.
J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=2, k=0.
T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Lemma 2.1, k=2, 0 peaks.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Nicolas Ollinger and Jeffrey Shallit, The Repetition Threshold for Rote Sequences, arXiv:2406.17867 [math.CO], 2024.
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.
A. G. Shannon, Some recurrence relations for binary sequence matrices, NNTDM 17 (2011), 4, 913.
Bojan Vučković and Miodrag Živković, Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3), ResearchGate, January 2017, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Cf. A001608, A004148, A005314, A006498, A011973, A049864, A049853, A078065, A118891, A173022, A176971, A178470, A261041, A303696, A329871, A384153.
Bisection of Padovan sequence A000931.
Partial sums of A005314 shifted 3 times to the right, if we assume A005314(0) = 1.
Compositions without adjacent equal parts are A003242.
Compositions without isolated parts are A114901.
Row sums of A097230(n-2) for n>1.

a005251 n = a005251_list !! n
a005251_list = 0 : 1 : 1 : 1 : zipWith (+) a005251_list
   (drop 2 $ zipWith (+) a005251_list (tail a005251_list))
-- Reinhard Zumkeller, Dec 28 2011

I:=[0,1,1,1]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-4): n in [1..45]]; // Vincenzo Librandi, Nov 30 2018

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-x)/(1-2*x + x^2 - x^3) )); // Marius A. Burtea, Oct 24 2019

A005251 := proc(n) option remember; if n <= 2 then n elif n = 3 then 4 else 2*A005251(n - 1) - A005251(n - 2) + A005251(n - 3); fi; end;
A005251:=(-1+z)/(-1+2*z-z**2+z**3); # Simon Plouffe in his 1992 dissertation
a := n -> `if`(n<=1, n, hypergeom([(2-n)/3, 1-n/3, (1-n)/3], [1/2, -n+1], 27/4)):
seq(simplify(a(n)), n=0..36); # Peter Luschny, Apr 08 2018

LinearRecurrence[{2,-1,1},{0,1,1},40]  (* Harvey P. Dale, May 05 2011 *)
a[ n_]:= If[n<0, SeriesCoefficient[ -x(1-x)/(1 -x + 2x^2 -x^3), {x, 0, -n}], SeriesCoefficient[ x(1-x)/(1 -2x +x^2 -x^3), {x, 0, n}]] (* Michael Somos, Dec 13 2013 *)
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n-2] + a[n-3]; Table[a[2 n-1], {n, 1, 20}] (* Rigoberto Florez, Oct 15 2019 *)
Table[If[n==0,0,Length[DeleteCases[Subsets[Range[n-3]],{_,x_,y_,_}/;x+2==y]]],{n,0,10}] (* Gus Wiseman, Nov 25 2019 *)

Vec((1-x)/(1-2*x+x^2-x^3)+O(x^99)) /* Charles R Greathouse IV, Nov 20 2012 */

{a(n) = if( n<0, polcoeff( -x*(1-x)/(1 -x +2*x^2 -x^3) + x*O(x^-n), -n), polcoeff( x*(1-x)/(1 -2*x +x^2 -x^3) + x*O(x^n), n))} /* Michael Somos, Dec 13 2013 */

[sum( binomial(n-j-1, 2*j) for j in (0..floor((n-1)/3)) ) for n in (0..50)] # G. C. Greubel, Apr 13 2022

a(n) = 2*a(n-1) - a(n-2) + a(n-3).
G.f.: z*(1-z)/(1 - 2*z + z^2 - z^3). - Emeric Deutsch, Sep 13 2004
23*a_n = 3*P_{2n+1} + 7*P_{2n} - 2*P_{2n-1}, where P_n are the Perrin numbers, A001608. - Don Knuth, Dec 09 2008
a(n+1) = Sum_{k=0..n} binomial(n-k, 2k). - Richard L. Ollerton, May 12 2004
From Henry Bottomley, Feb 21 2001: (Start)
a(n) = (Sum_{ja(n) = A005314(n) - A005314(n-1).
a(n) = A049853(n-1) - a(n-1).
a(n) = A005314(n) - a(n-2). (End)
Conjecture: a(n+1) + |A078065(n)| = 2*A005314(n+1). - Creighton Dement, Dec 21 2004
a(n+2) has g.f. (F_3(-x) + F_2(-x))/(F_4(-x) + F_3(-x)) = 1/(-x+1/(-x+1/(-x+1))) where F_n(x) is the n-th Fibonacci polynomial; see A011973. - Qiaochu Yuan (qchu(AT)mit.edu), Feb 19 2009
a(n) = A173022(2^(n-2) - 1) for n > 1. - Reinhard Zumkeller, Feb 07 2010
BINOMIAL transform of A176971 is a(n+1). - Michael Somos, Dec 13 2013
a(n) = hypergeom([(2-n)/3, 1-n/3, (1-n)/3], [1/2, -n+1], 27/4) for n > 1. - Peter Luschny, Apr 08 2018
G.f.: z/(1-z-z^3-z^4-z^5-...) for the compositions of n-1 avoiding 2. The g.f. for the number of compositions of n avoiding the part k is 1/(1-z-...-z^(k-1) - z^(k+1)-...). - Gregory L. Simay, Sep 09 2018
If p,q,r are the three solutions to x^3 = 2x^2 - x + 1, then a(n) = (p-1)*p^n/((p-q)*(p-r)) + (q-1)*q^n/((q-p)*(q-r)) + (r-1)*r^n/((r-p)*(r-q)). - Greg Dresden and AnXing Yang, Aug 12 2025

A182097 Expansion of 1/(1-x^2-x^3).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456, 696081, 922111, 1221537, 1618192, 2143648, 2839729, 3761840, 4983377, 6601569, 8745217

Offset: 0

N. J. A. Sloane, Apr 11 2012

Number of compositions (ordered partitions) into parts 2 and 3. - Joerg Arndt, Aug 21 2013
a(n) is the top left entry of the n-th power of any of the 3X3 matrices [0, 1, 1; 0, 0, 1; 1, 0, 0], [0, 1, 0; 1, 0, 1; 1, 0, 0], [0, 1, 1; 1, 0, 0; 0, 1, 0] or [0, 0, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
Conjectured values of d(n), the dimension of a Z-module in MZV(conv). See the Waldschmidt link. - Michael Somos, Mar 14 2014
Shannon et al. (2006) call these the Van der Laan numbers. - N. J. A. Sloane, Jan 11 2022

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

A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, Volume 37:7 (2006), 825-831. See R_n.
Michel Waldschmidt, "Multiple Zeta values and Euler-Zagier numbers", in Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Hoffman, The algebra of multiharmonic series, Journ. of Alg., Vol. 192, Issue 2 (Aug 1997), 477-495.
I. E. Leonard and A. C. F. Liu, A familiar recurrence occurs again, Amer. Math. Monthly, 119 (2012), 333-336.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (32).
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Michel Waldschmidt, Multiple Zeta values and Euler-Zagier numbers, Slides, Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A000931, A001608.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3))); // G. C. Greubel, Aug 11 2018

a[ n_] := If[n < 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, -n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, n}]]; (* Michael Somos, Dec 13 2013 *)
CoefficientList[Series[1/(1-x^2-x^3),{x,0,60}],x] (* or *) LinearRecurrence[ {0,1,1},{1,0,1},70] (* Harvey P. Dale, Dec 04 2014 *)

{a(n) = if( n<0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n))}; /* Michael Somos, Dec 13 2013 */

Vec(1/(1-x^2-x^3) + O(x^99)) \\ Altug Alkan, Sep 02 2016

G.f.: 1 / (1 - x^2 - x^3).
a(n) = A000931(n+3).
From Michael Somos, Dec 13 2013: (Start)
a(n) = A176971(-n).
a(n) = a(n-2) + a(n-3) for all n in Z.
a(n-7) = A133034(n).
a(n-5) = A078027(n).
a(n-3) = A000931(n).
a(n+2) = A134816(n).
a(n+4) = A164001(n) if n > 1. - (End)
a(n) = (A001608(n) - A000931(n))/2. - Elmo R. Oliveira, Dec 31 2022

A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595, -350, -174, 769, -1119, 945, -176, -943, 1888, -2064, 1121

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 - x + x^2 - x^4 + 2*x^5 - 2*x^6 + x^7 + x^8 - 3*x^9 + 4*x^10 + ...

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

Cf. A000931, A050935, A078013, A104769, A176971.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + x - x^3)));  // G. C. Greubel, Aug 04 2018

CoefficientList[Series[1/(1 + x - x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)
LinearRecurrence[{-1,0,1},{1,-1,1},60] (* Harvey P. Dale, Apr 10 2025 *)

{a(n) = if( n<0, n = -3-n; polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n), polcoeff( 1 / (1 + x - x^3) + x * O(x^n), n))};

G.f.: 1 / (1 + x - x^3).
0 = a(n) - a(n+2) - a(n+3) for all n in Z.
a(-n) = A000931(n) for all n in Z.
a(n) = A176971(n+3) for all n in Z.
-a(n) = A104769(n+1) for all n in Z.
(-1)^n * a(n) = A050935(n+3) for all n in Z.
-(-1)^n * a(n) = A078013(n+3) for all n in Z.

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42

Offset: 0

Philippe Deléham, Nov 12 2011

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A247918 Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 2, 0, -1, 2, -1, 2, 1, -2, 4, -3, 1, 4, -5, 7, -4, -2, 10, -12, 11, -1, -11, 22, -23, 13, 11, -33, 45, -35, 3, 44, -78, 81, -37, -41, 122, -158, 119, 4, -163, 281, -276, 115, 167, -443, 558, -391, -52, 611, -1000, 949

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 + x + x^5 + x^6 + x^8 + x^11 + 2*x^13 - x^15 + 2*x^16 - x^17 + ...

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

Cf. A010892, A056594, A077905, A112553, A176971, A247907.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!((1 + x)/((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 04 2018

CoefficientList[Series[(1+x)/((1-x^4)(1+x^4-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)

{a(n) = if( n<0, n=-8-n; polcoeff( -1/((1-x)*(1-x+x^2)*(1+x^2)*(1 - x^2 - x^3)) + x * O(x^n), n), polcoeff( 1/((1-x)*(1-x+x^2)*(1+x^2)*(1+x-x^3)) + x * O(x^n), n))};

def A247918_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/((1-x^4)*(1+x^4-x^5)) ).list()
A247918_list(70) # G. C. Greubel, Aug 08 2022

G.f.: 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 + x - x^3)).
a(n) = a(n+1) + a(n+5) - mod(floor((n-1)/2), 2) for all n in Z.
a(n) = -A247907(-8-n) for all n in Z.
Convolution of A077905 and A112553.
a(n) = (35 + 7*(A056594(n) + 3*A056594(n-1)) + 10*(3*A010892(n) - A010892(n-1)) - 2*(A176971(n) + 4*A172971(n-1) + 12*A176971(n-2)))/70. - G. C. Greubel, Aug 08 2022

A124304 Riordan array (1, x*(1-x^2)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -2, 0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 1, 0, -4, 0, 1, 0, 0, 0, 3, 0, -5, 0, 1, 0, 0, 0, 0, 6, 0, -6, 0, 1, 0, 0, 0, -1, 0, 10, 0, -7, 0, 1, 0, 0, 0, 0, -4, 0, 15, 0, -8, 0, 1, 0, 0, 0, 0, 0, -10, 0, 21, 0, -9, 0, 1, 0, 0, 0, 0, 1, 0, -20, 0, 28, 0, -10, 0, 1

Offset: 0

Paul Barry, Oct 25 2006

T(2n,n) is a signed aerated version of C(2n,n).

Inverse is A124305.

			Triangle begins
  1;
  0,  1;
  0,  0,  1;
  0, -1,  0,  1;
  0,  0, -2,  0,  1;
  0,  0,  0, -3,  0,  1;
  0,  0,  1,  0, -4,  0,  1;
  0,  0,  0,  3,  0, -5,  0,  1;
  0,  0,  0,  0,  6,  0, -6,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

Cf. A014021 (diagonal sums), A050935 (row sums), A124305 (inverse).

Cf. A002054, A079977, A126869, A138364, A176971.

A124304:= func< n,k | (&+[(-1)^j*Binomial(k,k-j)*Binomial(k,n-k-j) : j in [0..n]]) >;
[A124304(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023

A124304[n_, k_]:= Binomial[k, (n-k)/2]*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2;
Table[A124304[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)

def A124304(n, k): return binomial(k, (n-k)//2)*(-1)^((n-k)//2)*(1+(-1)^(n-k))/2
flatten([[A124304(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023

T(n, k) = Sum_{j=0..n} C(k,k-j)*C(k,n-k-j)*(-1)^j.
T(n, k) = C(k,(n-k)/2)*(-1)^((n-k)/2)*(1 + (-1)^(n-k))/2.
Sum_{k=0..n} T(n, k) = A050935(n+2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A014021(n).
T(2*n, n) = (1 - 2*0^(n+2 mod 4))*A126869(n).
From G. C. Greubel, Aug 18 2023: (Start)
T(2*n-1, n-1) = (1 - 2*0^(n+1 mod 4))*A138364(n-1).
T(2*n-1, n+1) = (1 - 2*0^(n mod 4))*((1+(-1)^n)/2)*A002054(floor(n/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = A176971(n+3).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1 - 2*0^(n+2 mod 4))*A079977(n).
G.f.: 1/(1 - x*y*(1-x^2)). (End)

Showing 1-6 of 6 results.

cp's OEIS Frontend

A005251 a(0) = 0, a(1) = a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

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A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

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A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A247918 Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.

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A124304 Riordan array (1, x*(1-x^2)).

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