A006130 -id:A006130 - cp's OEIS frontend

A228661 Number of 2 X n binary arrays with top left value 1 and no two ones adjacent horizontally, diagonally or antidiagonally.

2, 2, 8, 14, 38, 80, 194, 434, 1016, 2318, 5366, 12320, 28418, 65378, 150632, 346766, 798662, 1838960, 4234946, 9751826, 22456664, 51712142, 119082134, 274218560, 631464962, 1454120642, 3348515528, 7710877454, 17756424038, 40889056400

Offset: 1

R. H. Hardin, Aug 29 2013

nonn

Row 2 of A228660.

The recurrence is demonstrated as follows: For every 2X(n-1) array, we can add the column (0,0) to get an appropriate array of size 2Xn, and for every 2X(n-2) array, we can add the column (0,0) and either (1,0), (0,1) or (1,1) to get an appropriate sized array. Every admissible array is of one of these two forms, and these two forms do not overlap (since their last columns are different). - Tom Edgar, Aug 29 2013

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (1,3).

a(n) = a(n-1) +3*a(n-2).
G.f.: -2*x / ( -1+x+3*x^2 ). a(n) = 2*A006130(n-1). - R. J. Mathar, Aug 29 2013
a(n) = -2/13*sqrt(13)*(-1/2*sqrt(13)+1/2)^n + 2/13*sqrt(13)*(1/2*sqrt(13)+1/2)^n. - Tom Edgar, Aug 31 2013
G.f.: Q(0)/x -1/x, where Q(k) = 1 + 3*x^2 + (2*k+3)*x - x*(2*k+1 + 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013

A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 7, 7, 8, 9, 9, 12, 12, 15, 16, 17, 21, 21, 27, 28, 32, 37, 38, 48, 49, 59, 65, 70, 85, 87, 107, 114, 129, 150, 157, 192, 201, 236, 264, 286, 342, 358, 428, 465, 522, 606, 644, 770, 823, 950, 1071, 1166, 1376

Offset: 1

Vicente Jesús Maniega Granado, Oct 03 2016

Generalized Fibonacci growth sequence using i = 2 as maturity period, j = 5 as conception period, and k = 2 as growth factor.

Maturity period is the number of periods that a Fibonacci tree node needs for being able to start developing branches. Conception period is the number of periods in a Fibonacci tree node needed to develop new branches since its maturity. Growth factor is the number of additional branches developed by a Fibonacci tree node, plus 1, and equals the base of the exponential series related to the given tree if maturity factor would be zero. Standard Fibonacci would use 1 as maturity period, 1 as conception period, and 2 as growth factor as the series becomes equal to 2^n with a maturity period of 0. Related to Lucas sequences.

			For n = 13 the a(13) = a(8) + a(6) = 2 + 1 = 3.

Colin Barker, Table of n, a(n) for n = 1..1000
Julia Collins, Fibonacci Tree
Fractal Foundation, Fibonacci Fractals
D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,1).

Cf. A000079 (i = 0, j = 1, k = 2), A000244 (i = 0, j = 1, k = 3), A000302 (i = 0, j = 1, k = 4), A000351 (i = 0, j = 1, k = 5), A000400 (i = 0, j = 1, k = 6), A000420 (i = 0, j = 1, k = 7), A001018 (i = 0, j = 1, k = 8), A001019 (i = 0, j = 1, k = 9), A011557 (i = 0, j = 1, k = 10), A001020 (i = 0, j = 1, k = 11), A001021 (i = 0, j = 1, k = 12), A016116 (i = 0, j = 2, k = 2), A108411 (i = 0, j = 2, k = 3), A213173 (i = 0, j = 2, k = 4), A074872 (i = 0, j = 2, k = 5), A173862 (i = 0, j = 3, k = 2), A127975 (i = 0, j = 3, k = 3), A200675 (i = 0, j = 4, k = 2), A111575 (i = 0, j = 4, k = 3), A000045 (i = 1, j = 1, k = 2), A001045 (i = 1, j = 1, k = 3), A006130 (i = 1, j = 1, k = 4), A006131 (i = 1, j = 1, k = 5), A015440 (i = 1, j = 1, k = 6), A015441 (i = 1, j = 1, k = 7), A015442 (i = 1, j = 1, k = 8), A015443 (i = 1, j = 1, k = 9), A015445 (i = 1, j = 1, k = 10), A015446 (i = 1, j = 1, k = 11), A015447 (i = 1, j = 1, k = 12), A000931 (i = 1, j = 2, k = 2), A159284 (i = 1, j = 2, k = 3), A238389 (i = 1, j = 2, k = 4), A097041 (i = 1, j = 2, k = 10), A079398 (i = 1, j = 3, k = 2), A103372 (i = 1, j = 4, k = 2), A103373 (i = 1, j = 5, k = 2), A103374 (i = 1, j = 6, k = 2), A000930 (i = 2, j = 1, k = 2), A077949 (i = 2, j = 1, k = 3), A084386 (i = 2, j = 1, k = 4), A089977 (i = 2, j = 1, k = 5), A178205 (i = 2, j = 1, k = 11), A103609 (i = 2, j = 2, k = 2), A077953 (i = 2, j = 2, k = 3), A226503 (i = 2, j = 3, k = 2), A122521 (i = 2, j = 6, k = 2), A003269 (i = 3, j = 1, k = 2), A052942 (i = 3, j = 1, k = 3), A005686 (i = 3, j = 2, k = 2), A237714 (i = 3, j = 2, k = 3), A238391 (i = 3, j = 2, k = 4), A247049 (i = 3, j = 3, k = 2), A077886 (i = 3, j = 3, k = 3), A003520 (i = 4, j = 1, k = 2), A108104 (i = 4, j = 2, k = 2), A005708 (i = 5, j = 1, k = 2), A237716 (i = 5, j = 2, k = 3), A005709 (i = 6, j = 1, k = 2), A122522 (i = 6, j = 2, k = 2), A005710 (i = 7, j = 1, k = 2), A237718 (i = 7, j = 2, k = 3), A017903 (i = 8, j = 1, k = 2).

[n le 7 select 1 else Self(n-5)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Nov 30 2016

LinearRecurrence[{0, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1}, 70] (*  or *)
CoefficientList[ Series[(1+x+x^2+x^3+x^4)/(1-x^5-x^7), {x, 0, 70}], x] (* Robert G. Wilson v, Nov 25 2016 *)
nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,a+c}; NestList[nxt,{1,1,1,1,1,1,1},70][[;;,1]] (* Harvey P. Dale, Oct 22 2024 *)

Vec(x*(1+x+x^2+x^3+x^4)/((1-x+x^2)*(1+x-x^3-x^4-x^5)) + O(x^100)) \\ Colin Barker, Oct 27 2016

@CachedFunction # a = A242763
def a(n): return 1 if n<8 else a(n-5) +a(n-7)
[a(n) for n in range(1,76)] # G. C. Greubel, Oct 23 2024

Generic a(n) = 1 for n <= i+j; a(n) = a(n-j) + (k-1)*a(n-(i+j)) for n>i+j where i = maturity period, j = conception period, k = growth factor.
G.f.: x*(1+x+x^2+x^3+x^4) / ((1-x+x^2)*(1+x-x^3-x^4-x^5)). - Colin Barker, Oct 09 2016
Generic g.f.: x*(Sum_{l=0..j-1} x^l) / (1-x^j-(k-1)*x^(i+j)), with i > 0, j > 0 and k > 1.

A249139 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 3, 1, 5, 2, 11, 7, 1, 21, 16, 3, 43, 41, 12, 1, 85, 94, 34, 4, 171, 219, 99, 18, 1, 341, 492, 261, 60, 5, 683, 1101, 678, 195, 25, 1, 1365, 2426, 1692, 576, 95, 6, 2731, 5311, 4149, 1644, 340, 33, 1, 5461, 11528, 9959, 4488, 1106, 140, 7, 10923, 24881

Offset: 0

Clark Kimberling, Oct 23 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = 1 + (x + 2)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A006130(n+1) for n >= 0.
(Column 1) is essentially A001045.

			f(0,x) = 1/1, so that p(0,x) = 1
f(1,x) = (3 + x)/1, so that p(1,x) = 3 + x;
f(2,x) = (5 + 2 x)/(3 + x), so that p(2,x) = 5 + 2 x.
First 6 rows of the triangle of coefficients:
1
3    1
5    2
11   7    1
21   16   3
43   41   12  1

Clark Kimberling, Rows 0..100, flattened

Cf. A006130, A001045.

z = 15; f[x_, n_] := 1 + (x + 2)/f[x, n - 1]; f[x_, 1] = 1;
t = Table[Factor[f[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249139 array *)
Flatten[CoefficientList[u, x]] (* A249139 sequence *)

A052666 E.g.f. 1/(1-x-3x^2).

Original entry on oeis.org

1, 1, 8, 42, 456, 4800, 69840, 1093680, 20482560, 420577920, 9736070400, 245887488000, 6806133734400, 203555082931200, 6565920180019200, 226728504946944000, 8355118608764928000, 327047476385710080000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 613

spec := [S,{S=Sequence(Union(Z,Prod(Z,Union(Z,Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

E.g.f.: -1/(-1+x+3*x^2)
Recurrence: {a(1)=1, a(0)=1, (-3*n^2-9*n-6)*a(n)+(-2-n)*a(n+1)+a(n+2)=0}
Sum(1/13*(1+6*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+3*_Z^2))*n!
a(n) = n!*A006130(n). - R. J. Mathar, Nov 27 2011

A347636 Number of ways to tile an n X n square with 1 X 1 squares and (n-2) X 2 vertical or horizontal rectangles.

Original entry on oeis.org

193, 399, 783, 1601, 3283, 6947, 14897, 32607, 72175, 161649, 364611, 827555, 1885729, 4310639, 9874319, 22654881, 52032883, 119601123, 275058321, 632823743, 1456319215, 3352072913, 7716633443, 17765737443, 40904125825, 94182711375

Offset: 5

Greg Dresden and Osondu Ugochukwu, Sep 09 2021

			Here are two of the 193 possible tilings for a 5 X 5 square (using 1 X 1 squares and 3 X 2 rectangles):
._________   ._________
|_|     |_|  |_|_|     |
|_|_ _ _|_|  |   |_ _ _|
|   |_|   |  |   |_|   |
|   |_|   |  |___|_|   |
|___|_|___|  |_|_|_|___|

Index entries for linear recurrences with constant coefficients, signature (3,1,-7,1,3).

Cf. A000045, A006130.

Cf. A335560 which is the same problem but with 1 X 1 squares and (n-1) X 1 rectangles, and A337024 which uses 1 X 1 squares and 2 X 2 squares.

a(n) = 2*A006130(n) + 12*F(n + 1) + 16*F(n - 1) - 31 for F(n) = A000045(n) the Fibonacci sequence.

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

cp's OEIS Frontend

A228661 Number of 2 X n binary arrays with top left value 1 and no two ones adjacent horizontally, diagonally or antidiagonally.

Views

Author

Keywords

Comments

Examples

Links

Formula

A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A249139 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A052666 E.g.f. 1/(1-x-3x^2).

Views

Author

Keywords

Links

Programs

Maple

Formula

A347636 Number of ways to tile an n X n square with 1 X 1 squares and (n-2) X 2 vertical or horizontal rectangles.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula