A195978 -id:A195978 - cp's OEIS frontend

A195971 Number of n X 1 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

0, 1, 3, 4, 5, 9, 16, 25, 39, 64, 105, 169, 272, 441, 715, 1156, 1869, 3025, 4896, 7921, 12815, 20736, 33553, 54289, 87840, 142129, 229971, 372100, 602069, 974169, 1576240, 2550409, 4126647, 6677056, 10803705, 17480761, 28284464, 45765225

Offset: 0

R. H. Hardin, Sep 25 2011

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's.
Column 1 of A195978.
a(n) is the number of total dominating sets in the (n+1)-path graph. - Eric W. Weisstein, Apr 10 2018
Equivalently, a(n) is the number of 0-1 sequences (every term is "0" or "1") of length n+1 whose every term is adjacent to a term "1". - Yuda Chen, Apr 06 2022
From Wajdi Maaloul, Jun 23 2022: (Start)
For n > 1, a(n) is the number of ways to tile the figure below using squares and dominoes: a horizontal strip of length n-1 that contains a central vertical strip of length 3. Below are figures for a(2) through a(5):
                                     
      ||      |_|     ||_        |_|_
      ||     ||_|    |||_|     |||_|_|
      ||       ||      ||           ||
(End)
a(n) is the number of compositions of n+2 with 1's, 3's and 4's, with the restriction that you cannot begin with two 1's. - Greg Dresden and Yuan Shen, Aug 10 2024

			All solutions for n=4:
  0   0   1   1   0
  0   0   0   0   1
  0   0   0   0   1
  1   0   1   0   0

R. H. Hardin, Table of n, a(n) for n = 0..200 (corrected by _R. H. Hardin_, Jan 19 2019)
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Total Dominating Set
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. A195978.

a:=[1,3,4,5];; for n in [5..40] do a[n]:=a[n-1]+a[n-3]+a[n-4]; od; Concatenation([0], a); # G. C. Greubel, Apr 03 2019

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x)^2/(1-x-x^3-x^4) )); // G. C. Greubel, Apr 03 2019

Table[(LucasL[n + 3] - 2 Sin[n Pi/2] - 4 Cos[n Pi/2])/5, {n, 0, 40}] (* Eric W. Weisstein, Apr 10 2018 *)
LinearRecurrence[{1, 0, 1, 1}, {0, 1, 3, 4, 5}, 40] (* Eric W. Weisstein, Apr 10 2018; amended for a(0) by Georg Fischer, Apr 03 2019 *)
CoefficientList[Series[x*(1+x)^2/(1-x-x^3-x^4), {x, 0, 40}], x] (* Eric W. Weisstein, Apr 10 2018 *)

my(x='x+O('x^40)); concat([0], Vec(x*(1+x)^2/(1-x-x^3-x^4))) \\ G. C. Greubel, Apr 03 2019

(x*(1+x)^2/(1-x-x^3-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

a(n) = a(n-1) + a(n-3) + a(n-4).
G.f.: x*(1 + x)^2 / ((1 + x^2)*(1 - x - x^2)). - Colin Barker, Feb 17 2018
a(n) = (A000032(n + 3) - 2*sin(n*Pi/2) - 4*cos(n*Pi/2))/5. - Eric W. Weisstein, Apr 10 2018
a(n) = (Lucas(n+3) - (-1)^(floor(n/2))*(3+(-1)^n))/5. - G. C. Greubel, Apr 03 2019
From Wajdi Maaloul, Jun 23 2022: (Start)
a(2n) = A226205(n+1) = - A121646(n+1) = Fibonacci(n+1)^2 - Fibonacci(n)^2 = Fibonacci(n+2)*Fibonacci(n-1);
a(2n+1) = Fibonacci(n+2)^2 = A007598(n+2).
(End)

A195972 Number of n X 2 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

3, 9, 14, 23, 43, 78, 139, 249, 450, 815, 1475, 2674, 4859, 8841, 16102, 29359, 53587, 97894, 178971, 327425, 599394, 1097871, 2011875, 3688402, 6764595, 12410585, 22775742, 41808791, 76765147, 140977582, 258949451, 475718009, 874068802

Offset: 1

R. H. Hardin, Sep 25 2011

nonn

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's.

Column 2 of A195978.

			Some solutions for n=4:
..0..1....2..1....1..1....1..0....0..1....1..0....1..2....0..1....0..0....0..1
..1..1....1..0....0..0....1..0....1..1....1..0....0..1....0..1....0..0....0..1
..1..0....1..0....0..0....1..0....1..0....1..1....0..1....1..1....0..0....0..1
..1..0....2..1....1..1....1..0....2..1....0..1....1..2....1..0....0..0....0..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A195978.

Empirical: a(n) = 2*a(n-1) +a(n-3) -a(n-4) -2*a(n-5) -2*a(n-6) -a(n-7).

Empirical g.f.: x*(1 + x)^2*(3 - 3*x - x^2 - 3*x^3 - 2*x^4) / ((1 + x^2)*(1 - x - x^2)*(1 - x - x^2 - x^3)). - Colin Barker, May 08 2018

A195973 Number of n X 3 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

4, 14, 24, 36, 67, 134, 240, 432, 803, 1501, 2764, 5118, 9519, 17718, 32927, 61310, 114257, 213023, 397223, 741197, 1383497, 2583168, 4824204, 9012010, 16838364, 31466993, 58813148, 109939804, 205534006, 384287357, 718564103, 1343717638

Offset: 1

R. H. Hardin, Sep 25 2011

nonn

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's.

Column 3 of A195978.

			Some solutions for n=4:
..0..1..2....1..0..0....0..0..1....0..0..1....1..1..0....1..0..1....1..0..0
..1..1..1....2..1..1....0..0..1....0..0..1....0..1..1....1..0..1....1..1..1
..1..0..0....2..1..1....0..0..1....1..1..1....1..1..1....1..0..1....0..1..1
..2..1..1....1..0..0....0..0..1....2..1..0....1..0..0....1..0..1....1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A195978.

Empirical: a(n) = 2*a(n-1) +a(n-3) +a(n-4) -6*a(n-5) -a(n-6) -4*a(n-7) +2*a(n-8) +2*a(n-9) +5*a(n-10) +2*a(n-11) -2*a(n-14) -a(n-15).

Empirical g.f.: x*(4 + 6*x - 4*x^2 - 16*x^3 - 23*x^4 - 14*x^5 + 23*x^7 + 26*x^8 + 19*x^9 + 9*x^10 - x^11 - 7*x^12 - 5*x^13 - x^14) / ((1 - x)*(1 + x^2)*(1 - x - x^2)*(1 - x^2 - 2*x^3 - 4*x^4 - 2*x^5 - x^6 + x^7 + x^8 + 2*x^9 + x^10)). - Colin Barker, May 08 2018

A195974 Number of n X 4 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

5, 23, 36, 67, 133, 256, 455, 931, 1830, 3565, 7001, 13872, 27433, 54401, 108164, 215285, 428869, 855426, 1707759, 3411577, 6819860, 13640635, 27294311, 54635604, 109402661, 219127235, 438997334, 879659285, 1762952379, 3533677212

Offset: 1

R. H. Hardin, Sep 25 2011

nonn

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's.

Column 4 of A195978.

			Some solutions for n=5:
..2..1..0..1....2..1..1..2....2..1..1..0....2..1..0..1....0..1..1..1
..1..1..0..1....1..0..0..1....1..0..1..1....1..1..1..2....1..1..0..0
..0..1..1..1....2..1..1..2....1..0..1..1....0..0..1..2....1..1..1..1
..0..1..1..0....2..1..1..2....1..1..1..0....1..1..1..1....0..0..1..1
..0..1..1..0....1..0..0..1....0..1..1..0....2..1..0..0....1..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 4*a(n-1) -7*a(n-2) +13*a(n-3) -17*a(n-4) +9*a(n-5) -7*a(n-6) -11*a(n-7) +28*a(n-8) -20*a(n-9) +37*a(n-10) -7*a(n-11) +11*a(n-13) -41*a(n-14) +14*a(n-15) -32*a(n-16) +19*a(n-18) -14*a(n-19) +30*a(n-20) -10*a(n-21) +9*a(n-22) +7*a(n-23) -9*a(n-24) +10*a(n-25) -11*a(n-26) +3*a(n-28) -4*a(n-29) +a(n-30) -2*a(n-31) -2*a(n-32).

A195975 Number of nX5 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

9, 43, 67, 133, 244, 519, 1072, 2225, 4456, 9098, 19066, 39689, 82372, 171498, 358146, 748911, 1567225, 3279792, 6866722, 14390098, 30170057, 63261378, 132676528, 278340227, 584041659, 1225661652, 2572438981, 5399619168, 11334969352

Offset: 1

R. H. Hardin Sep 25 2011

nonn

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's

Column 5 of A195978

			Some solutions for n=4
..1..2..2..1..2....0..0..1..1..0....0..1..1..1..2....1..1..1..0..0
..0..1..1..0..1....0..0..1..1..0....1..1..0..0..1....0..0..1..1..1
..1..1..1..0..1....1..1..1..1..0....1..1..1..1..1....1..1..1..1..1
..1..0..1..1..2....2..1..0..1..1....0..0..1..1..0....2..1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 5*a(n-1) -9*a(n-2) +11*a(n-3) -6*a(n-4) -36*a(n-5) +90*a(n-6) -142*a(n-7) +197*a(n-8) -46*a(n-9) -183*a(n-10) +478*a(n-11) -942*a(n-12) +792*a(n-13) -498*a(n-14) -210*a(n-15) +1415*a(n-16) -1639*a(n-17) +1834*a(n-18) -709*a(n-19) -862*a(n-20) +1990*a(n-21) -3015*a(n-22) +1957*a(n-23) -374*a(n-24) -1742*a(n-25) +3816*a(n-26) -3839*a(n-27) +2182*a(n-28) -480*a(n-29) -3921*a(n-30) +3721*a(n-31) -4788*a(n-32) +2840*a(n-33) +2416*a(n-34) -577*a(n-35) +7322*a(n-36) -1770*a(n-37) +1322*a(n-38) -1236*a(n-39) -6838*a(n-40) -789*a(n-41) -4538*a(n-42) -89*a(n-43) +3459*a(n-44) +1446*a(n-45) +4611*a(n-46) +1551*a(n-47) -859*a(n-48) -465*a(n-49) -2777*a(n-50) -1070*a(n-51) +363*a(n-52) -79*a(n-53) +1222*a(n-54) -43*a(n-55) -809*a(n-56) -357*a(n-57) -616*a(n-58) +368*a(n-59) +965*a(n-60) +619*a(n-61) +408*a(n-62) -203*a(n-63) -570*a(n-64) -385*a(n-65) -205*a(n-66) +35*a(n-67) +181*a(n-68) +122*a(n-69) +57*a(n-70) +5*a(n-71) -33*a(n-72) -26*a(n-73) -14*a(n-74) -8*a(n-75) -2*a(n-76)

A195976 Number of nX6 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

16, 78, 134, 256, 519, 1280, 2718, 5830, 12799, 28969, 64536, 144480, 323887, 730358, 1647281, 3720888, 8406355, 19026721, 43091379, 97668401, 221445855, 502417344, 1140305840, 2589100946, 5880051614, 13357939515, 30351874554, 68979138844

Offset: 1

R. H. Hardin Sep 25 2011

nonn

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's

Column 6 of A195978

			Some solutions for n=4
..1..2..2..1..1..2....0..1..1..0..0..1....1..0..1..2..1..0....0..1..2..1..0..0
..0..1..1..0..0..1....0..1..1..0..0..1....1..1..2..2..1..1....0..1..2..1..0..0
..0..1..1..1..1..1....1..1..1..0..0..1....0..1..2..1..0..1....0..1..2..1..0..0
..1..1..0..1..1..0....1..0..1..1..1..2....0..1..2..1..0..1....0..1..2..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..200

A195977 Number of nX7 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

25, 139, 240, 455, 1072, 2718, 5637, 12823, 30486, 72042, 166884, 390514, 917562, 2167323, 5114140, 12071815, 28523614, 67538866, 159964100, 378996636, 898240293, 2130429190, 5054033816, 11993264778, 28464315995, 67574829560

Offset: 1

R. H. Hardin Sep 25 2011

nonn

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's

Column 7 of A195978

			Some solutions for n=4
..1..0..1..1..0..0..1....0..0..1..1..0..0..1....1..0..0..0..0..1..2
..2..1..2..2..1..1..1....0..0..1..1..0..0..1....2..1..1..1..1..1..1
..2..1..2..2..1..1..0....1..1..1..1..1..1..1....2..1..1..1..1..0..0
..1..0..1..1..0..1..1....2..1..0..0..1..1..0....1..0..0..0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..200

A195970 Number of n X n 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

1, 9, 24, 67, 244, 1280, 5637, 32909, 212185, 1609756, 13476199, 134461394

Offset: 1

R. H. Hardin Sep 25 2011

nonn

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's

Diagonal of A195978

			Some solutions for n=4
..1..0..0..1....1..2..2..1....1..0..1..1....1..0..1..1....1..0..0..0
..1..0..0..1....0..1..1..0....1..1..1..0....2..1..1..0....2..1..1..1
..1..0..0..1....0..1..1..1....0..1..1..0....2..1..1..0....2..1..1..1
..1..0..0..1....1..1..0..1....0..1..1..0....1..0..1..1....1..0..0..0

cp's OEIS Frontend

A195971 Number of n X 1 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A195972 Number of n X 2 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A195973 Number of n X 3 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A195974 Number of n X 4 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

Formula

A195975 Number of nX5 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

Formula

A195976 Number of nX6 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

A195977 Number of nX7 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

A195970 Number of n X n 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples