A077936 -id:A077936 - cp's OEIS frontend

A002478 Bisection of A000930.

1, 1, 3, 6, 13, 28, 60, 129, 277, 595, 1278, 2745, 5896, 12664, 27201, 58425, 125491, 269542, 578949, 1243524, 2670964, 5736961, 12322413, 26467299, 56849086, 122106097, 262271568, 563332848, 1209982081, 2598919345, 5582216355, 11990037126, 25753389181

Offset: 0

N. J. A. Sloane

Number of ways to tile a 3 X n region with 1 X 1, 2 X 2 and 3 X 3 tiles.
Number of ternary words with subwords (0,0), (0,1) and (1,1) not allowed. - Olivier Gérard, Aug 28 2012
Diagonal sums of A063967. - Paul Barry, Nov 09 2005
Row sums of number triangle A116088. - Paul Barry, Feb 04 2006
Sequence is identical to its second differences negated, minus the first 3 terms. - Paul Curtz, Feb 10 2008
a(n) = term (3,3) in the 3 X 3 matrix [0,1,0; 0,0,1; 1,2,1]^n. - Gary W. Adamson, May 30 2008
a(n)/a(n-1) tends to 2.147899035..., an eigenvalue of the matrix and a root to x^3 - x^2 - 2x - 1 = 0. - Gary W. Adamson, May 30 2008
INVERT transform of (1, 2, 1, 0, 0, 0, ...) = (1, 3, 6, 13, 28, ...); such that (1, 2, 1, 0, 0, 0, ...) convolved with (1, 1, 3, 6, 13, 28, 0, 0, 0, ...) shifts to the left. - Gary W. Adamson, Apr 18 2010
a(n) is the top left entry in the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 0, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014

			a(3)=6 as there is one tiling of a 3 X 3 region with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 1 tiling consisting of the 3 X 3 tile.

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 322.
S. Heubach, Tiling an m X n Area with Squares of Size up to k X k (m<=5), Congressus Numerantium 140 (1999), pp. 43-64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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..300
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Emeric Deutsch, Counting tilings with L-tiles and squares, Problem 10877, Amer. Math. Monthly, 110 (March 2003), 245-246.
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, arXiv:1907.06517 [math.CO], 2019.
Leonhard Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 412
Milan Janjić, Pascal Triangle and Restricted Words, arXiv:1705.02497 [math.CO], 2017.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Richard J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 19 (halved...).
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 5.
Sam Northshield, Some generalizations of a formula of Reznick, SUNY Plattsburgh (2022).
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
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Cf. A000930, A054856, A054857, A025234, A078007, A078039, A226546, A077936 (INVERT transform), A008346 (inverse INVERT transform).

I:=[1,1,3]; [n le 3 select I[n] else Self(n-1) +2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 14 2023

f[A_]:= Module[{til = A}, AppendTo[til, A[[-1]] + 2A[[-2]] + A[[-3]]]]; NumOfTilings[ n_ ]:= Nest[ f, {1,1,3}, n-2]; NumOfTilings[30]
LinearRecurrence[{1,2,1},{1,1,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
CoefficientList[Series[1/(1-x-2x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Oct 17 2024 *)

a(n)=([0,1,0; 0,0,1; 1,2,1]^n*[1;1;3])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

@CachedFunction
def a(n): # A002478
    if (n<3): return (1,1,3)[n]
    else: return sum(binomial(2,j)*a(n-j) for j in range(1,4))
[a(n) for n in (0..40)] # G. C. Greubel, Apr 14 2023

G.f.: 1 / (1-x-2*x^2-x^3). [Simon Plouffe in his 1992 dissertation.]
a(n) = a(n-1) + 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} binomial(2*n-2*k, k). - Paul Barry, Nov 13 2004
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j, n-k-j)*C(j, k). - Paul Barry, Nov 09 2005
a(n) = Sum_{k=0..n} C(2*k,n-k) = Sum_{k=0..n} C(n,k)*C(3*k,n)/C(3*k,k). - Paul Barry, Feb 04 2006
a(n) = A000930(n) + 2*Sum_{i=0..n-2} a(i)*A000930(n-2-i). - Michael Tulskikh, Jun 07 2020

Additional comments from Silvia Heubach (silvi(AT)cine.net), Apr 21 2000

A158687 Riordan array (1/(1-x),x(1+x)^2/(1-x)).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 7, 1, 1, 12, 24, 10, 1, 1, 16, 56, 49, 13, 1, 1, 20, 104, 160, 83, 16, 1, 1, 24, 168, 400, 351, 126, 19, 1, 1, 28, 248, 832, 1120, 656, 178, 22, 1, 1, 32, 344, 1520, 2912, 2561, 1102, 239, 25, 1

Offset: 0

Paul Barry, Mar 24 2009

Row sums are A077936. Diagonal sums are A129847. Central terms are A059304.

Inverse of alternating signed version is A100326.

			Number triangle begins
1,
1, 1,
1, 4, 1,
1, 8, 7, 1,
1, 12, 24, 10, 1,
1, 16, 56, 49, 13, 1,
1, 20, 104, 160, 83, 16, 1

Cf. A059304, A077936, A100326, A129847.

Number triangle T(n,k) = Sum_{j=0..n-k} C(n-j,k)*C(2k,j).
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 11 2013
G.f.: 1/(1-y-x*(1+y)^2). - Vladimir Kruchinin, Apr 21 2015

A077983 Expansion of 1/(1 + 2x - 2x^2 + x^3).

Original entry on oeis.org

1, -2, 6, -17, 48, -136, 385, -1090, 3086, -8737, 24736, -70032, 198273, -561346, 1589270, -4499505, 12738896, -36066072, 102109441, -289089922, 818464798, -2317218881, 6560457280, -18573817120, 52585767681, -148879626882, 421504606246, -1193354233937, 3378597307248

Offset: 0

N. J. A. Sloane, Nov 17 2002

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

Cf. A078054 (first differences), A077936.

a:=[1,-2,6];; for n in [4..30] do a[n]:=-2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jun 25 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1+2*x-2*x^2+x^3) )); // G. C. Greubel, Jun 25 2019

LinearRecurrence[{-2,2,-1}, {1,-2,6}, 30] (* or *) CoefficientList[ Series[1/(1+2*x-2*x^2+x^3), {x,0,30}], x] (* G. C. Greubel, Jun 25 2019 *)

my(x='x+O('x^30)); Vec(1/(1+2*x-2*x^2+x^3)) \\ G. C. Greubel, Jun 25 2019

(1/(1+2*x-2*x^2+x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 25 2019

a(n) = (-1)^n * A077936(n). - Ivan Neretin, Jul 05 2015

a(n) = -2*a(n-1) + 2*a(n-2) - a(n-3) with a(0) = 1, a(1) = -2, a(2) = 6. - Taras Goy, Aug 04 2017

A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 7, 5, 1, 7, 17, 16, 7, 1, 13, 38, 46, 29, 9, 1, 24, 82, 122, 99, 46, 11, 1, 44, 174, 304, 303, 184, 67, 13, 1, 81, 362, 728, 857, 641, 309, 92, 15, 1, 149, 743, 1690, 2291, 2031, 1212, 482, 121, 17, 1, 274, 1509, 3827, 5869, 6004, 4260, 2108, 711, 154, 19, 1

Offset: 0

Emanuele Munarini, Apr 18 2011

Row sums are A077936, diagonal sums are A077946

			Triangle begins:
1
1,1
2,3,1
4,7,5,1
7,17,16,7,1
13,38,46,29,9,1
24,82,122,99,46,11,1
44,174,304,303,184,67,13,1
81,362,728,857,641,309,92,15,1

Cf. A104580, A187889, A077936, A077946.

Flatten[Table[Sum[Binomial[i+k,k]Sum[Binomial[i+k,j]Binomial[n-i-j,i+k],{j,0,n-k-2i}],{i,0,n}],{n,0,20},{k,0,n}]]

create_list(sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j,0,n-k-2*i),i,0,n),n,0,8,k,0,n);

T(n,k) = [x^n](x+x^2)^k/(1-x-x^2-x^3)^(k+1).
T(n,k) = sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j=0..n-k-2*i),i=0..n).
T(n,k) = sum(binomial(k,i)*(-1)^(k-i)*sum(binomial(j+k,k)*trinomial(i+j,n-3*k+2*i-j),j=0..n-k),i=0..k)
Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) + T(n+1,k) + T(n,k+1)

a(23) and a(40) corrected by Georg Fischer, Feb 20 2021 and Apr 29 2022

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A002478 Bisection of A000930.

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A158687 Riordan array (1/(1-x),x(1+x)^2/(1-x)).

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A077983 Expansion of 1/(1 + 2*x - 2*x^2 + x^3).

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A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).

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A077983 Expansion of 1/(1 + 2x - 2x^2 + x^3).