A053602 -id:A053602

A068928

Number of incongruent ways to tile a 3 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

+10
7

2, 2, 2, 4, 5, 9, 12, 21, 30, 51, 76, 127, 195, 322, 504, 826, 1309, 2135, 3410, 5545, 8900, 14445, 23256, 37701, 60813, 98514, 159094, 257608, 416325, 673933, 1089648, 1763581, 2852242, 4615823, 7466468, 12082291, 19546175, 31628466

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..38.

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 10.

Index entries for linear recurrences with constant coefficients, signature (1,2,-1,0,-1,-1).

FORMULA

For n >= 8, a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6).

O.g.f.: x(2-4x^2-x^4+x^6)/((1-x-x^2)(1-x^2-x^4)). a(n) = (A000045(n+1)+A053602(n+1))/2, n>1. [From R. J. Mathar, Aug 30 2008]

CROSSREFS

Cf. A068922 for total number of tilings, A068926 for more info.

Essentially the same as A001224.

KEYWORD

easy,nonn

AUTHOR

Dean Hickerson, Mar 11 2002

STATUS

approved

A051792

a(n) = (-1)^(n-1)*(a(n-1) - a(n-2)), a(1)=1, a(2)=1.

+10
6

1, 1, 0, 1, 1, 0, -1, 1, 2, -1, -3, 2, 5, -3, -8, 5, 13, -8, -21, 13, 34, -21, -55, 34, 89, -55, -144, 89, 233, -144, -377, 233, 610, -377, -987, 610, 1597, -987, -2584, 1597, 4181, -2584, -6765, 4181, 10946, -6765, -17711, 10946, 28657, -17711, -46368

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,9

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (0,-1,0,1).

FORMULA

a(3-n) = A053602(n).

From Michael Somos: (Start)

G.f.: x*(1 + x + x^2 + 2*x^3)/(1 + x^2 - x^4).

a(n) = -a(n-2) + a(n-4). (End)

a(n) = b(n-1) + b(n-2) + b(n-3) + 2*b(n-4), where b(n) = i^n * A079977(n). - G. C. Greubel, Dec 06 2022

MATHEMATICA

LinearRecurrence[{0, -1, 0, 1}, {1, 1, 0, 1}, 60] (* Harvey P. Dale, May 08 2017 *)

PROG

(PARI) a(n)=fibonacci((3-n)\2+(3-n)%2*2)

(Sage)

def A051792():

x, y, b = 1, 1, true

while True:

yield x

x, y = y, x - y

y = -y if b else y

b = not b

a = A051792()

print([next(a) for _ in range(51)]) # Peter Luschny, Mar 19 2020

(Magma) [Fibonacci(1 -Floor((n-4)/2) -2*((n-4) mod 2)): n in [1..60]]; // G. C. Greubel, Dec 06 2022

CROSSREFS

Cf. A000045, A053602.

KEYWORD

easy,sign

AUTHOR

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 10 1999

STATUS

approved

A123231

Row sums of A123230.

+10
4

1, 2, 1, 3, 2, 5, 3, 8, 5, 13, 8, 21, 13, 34, 21, 55, 34, 89, 55, 144, 89, 233, 144, 377, 233, 610, 377, 987, 610, 1597, 987, 2584, 1597, 4181, 2584, 6765, 4181, 10946, 6765, 17711, 10946, 28657, 17711, 46368, 28657, 75025, 46368, 121393, 75025, 196418

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

All terms are Fibonacci numbers A000045: a(2n-1) = Fibonacci(n), a(2n) = Fibonacci(n+2), a(2n-1) = a(2n+2). - Alexander Adamchuk, Oct 08 2006

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

FORMULA

From Alexander Adamchuk, Oct 08 2006: (Start)

a(n) = Fibonacci(A028242(n+2)).

a(n) = Fibonacci(A030451(n+1)).

a(n) = Fibonacci(3/4 -(-1)^(n+1)*3/4 +(n+1)/2). (End)

a(n) = A053602(n+1) = A097594(n-5). - R. J. Mathar, Mar 08 2011

G.f. -x*(1+2*x+x^3) / ( -1+x^2+x^4 ). - R. J. Mathar, Mar 08 2011

a(n) = a(n-2) + a(n-4). - Muniru A Asiru, Oct 12 2018

MAPLE

seq(coeff(series(-x*(1+2*x+x^3)/(x^4+x^2-1), x, n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 12 2018

MATHEMATICA

p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = x*p[k - 1, x] + (-1)^(n + 1)p[k - 2, x]; Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, n + 1}], {n, 0, 20}]

Rest[Flatten[Reverse/@Partition[Fibonacci[Range[30]], 2, 1]]] (* Harvey P. Dale, Mar 19 2013 *)

PROG

(PARI) vector(50, n, fibonacci(3/4 -(-1)^(n+1)*3/4 +(n+1)/2)) \\ G. C. Greubel, Oct 12 2018

(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 + 2*x + x^3)/(1 - x^2 - x^4))); // G. C. Greubel, Oct 12 2018

(GAP) a:=[1, 2, 1, 3];; for n in [5..50] do a[n]:=a[n-2]+a[n-4]; od; a; # Muniru A Asiru, Oct 12 2018

CROSSREFS

Cf. A051792, A000045, A028242, A030451.

KEYWORD

nonn

AUTHOR

Roger L. Bagula, Oct 06 2006

EXTENSIONS

More terms from Alexander Adamchuk, Oct 08 2006

STATUS

approved

A233323

Triangle read by rows: T(n,k) = number of palindromic compositions of n in which the largest part is equal to k, 1 <= k <= n.

+10
4

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 1, 0, 1, 1, 2, 3, 0, 1, 0, 1, 1, 7, 3, 3, 0, 1, 0, 1, 1, 4, 6, 1, 2, 0, 1, 0, 1, 1, 12, 7, 7, 1, 2, 0, 1, 0, 1, 1, 7, 12, 3, 5, 0, 2, 0, 1, 0, 1, 1, 20, 16, 15, 3, 5, 0, 2, 0, 1, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

A palindromic composition of a natural number m is an ordered partition of m into N+1 natural numbers (or parts), p_0, p_1, ..., p_N, of the form m = p_0 + p_1 + ... + p_N such that p_j = p_{N-j}, for each j in {0,...,N}. Two palindromic compositions, sum_{j=0..N} p_j and sum_{j=0..N} q_j (say), are identical if and only if p_j = q_j, j = 0,...,N; otherwise they are taken to be distinct.

LINKS

Alois P. Heinz, Rows n = 1..141, flattened (rows n = 1..50 from Charles R Greathouse IV)

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

EXAMPLE

There are eight palindromic compositions of n=7, namely, {7}, {3,1,3}, {2,3,2}, {2,1,1,1,2}, {1,5,1}, {1,2,1,2,1}, {1,1,3,1,1}, {1,1,1,1,1,1,1}, and three of them have the largest part equal to 3, so T(7,3) = 3.

Triangle T(n,k) begins:

1;

1, 1;

1, 0, 1,

1, 2, 0, 1;

1, 1, 1, 0, 1;

1, 4, 1, 1, 0, 1;

1, 2, 3, 0, 1, 0, 1;

1, 7, 3, 3, 0, 1, 0, 1;

1, 4, 6, 1, 2, 0, 1, 0, 1;

1, 12, 7, 7, 1, 2, 0, 1, 0, 1;

...

MAPLE

b:= proc(n, k) option remember; `if`(n<=k, 1, 0)+

add(b(n-2*j, k), j=1..min(k, iquo(n, 2)))

end:

T:= (n, k)-> b(n, k) -b(n, k-1):

seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Dec 11 2013

MATHEMATICA

b[n_, k_] := b[n, k] = If[n <= k, 1, 0] + Sum[b[n-2*j, k], { j, 1, Min[k, Quotient[n, 2]]}]; t[n_, k_] := b[n, k] - b[n, k-1]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Alois P. Heinz's Maple code *)

PROG

(PARI) T(n, k, ok=0)={

if(n<1, return(n==0 && ok));

if(k>n/2 && !ok,

n-=k;

if(n<0||n%2, return(0));

return(2^max(n/2-1, 0))

);

sum(i=1, k,

T(n-2*i, k, ok||i==k)

)+(n==k || (ok && n<k))

}; \\ Charles R Greathouse IV, Dec 11 2013

CROSSREFS

Cf. A016116 (row sums), A233324 (row partial sums).

T(n,2)+1 = A053602(n+1) = A123231(n). T(4n-2,2n) = A011782(n-1). - Alois P. Heinz, Dec 11 2013

KEYWORD

nonn,tabl

AUTHOR

L. Edson Jeffery, Dec 10 2013

STATUS

approved

A233324

Triangle read by rows: T(n,k) = number of palindromic compositions of n in which no part exceeds k, 1 <= k <= n.

+10
4

1, 1, 2, 1, 1, 2, 1, 3, 3, 4, 1, 2, 3, 3, 4, 1, 5, 6, 7, 7, 8, 1, 3, 6, 6, 7, 7, 8, 1, 8, 11, 14, 14, 15, 15, 16, 1, 5, 11, 12, 14, 14, 15, 15, 16, 1, 13, 20, 27, 28, 30, 30, 31, 31, 32, 1, 8, 20, 23, 28, 28, 30, 30, 31, 31, 32, 1, 21, 37, 52, 55, 60, 60, 62, 62, 63, 63, 64

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

A palindromic composition of a natural number m is an ordered partition of m into N+1 natural numbers (or parts), p_0, p_1, ..., p_N, of the form m = p_0 + p_1 + ... + p_N such that p_j = p_{N-j}, for each j in {0,...,N}. Two palindromic compositions, sum_{j=0..N} p_j and sum_{j=0..N} q_j (say), are identical if and only if p_j = q_j, j = 0,...,N; otherwise they are taken to be distinct.

Partial sums of rows of A233323.

T(n,k) is defined for n,k >= 0. T(n,k) = T(n,n) = A016116(n) for k>= 0. - Alois P. Heinz, Dec 11 2013

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

EXAMPLE

Triangle T(n,k) begins:

1;

1, 2;

1, 1, 2;

1, 3, 3, 4;

1, 2, 3, 3, 4;

1, 5, 6, 7, 7, 8;

1, 3, 6, 6, 7, 7, 8;

1, 8, 11, 14, 14, 15, 15, 16;

1, 5, 11, 12, 14, 14, 15, 15, 16;

1, 13, 20, 27, 28, 30, 30, 31, 31, 32;

MAPLE

T:= proc(n, k) option remember; `if`(n<=k, 1, 0)+

add(T(n-2*j, k), j=1..min(k, iquo(n, 2)))

end:

seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Dec 11 2013

MATHEMATICA

T[n_, k_] := T[n, k] = If[n <= k, 1, 0] + Sum[T[n-2*j, k], {j, 1, Min[k, Quotient[ n, 2]]}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

PROG

(PARI) T(n, k)=if(n<1, return(n==0)); sum(i=1, k, T(n-2*i, k))+(n<=k) \\ Charles R Greathouse IV, Dec 11 2013

CROSSREFS

Cf. A233323.

T(n,2) = A053602(n+1) = A123231(n). T(2n,3) = A001590(n+3). T(2n,4) = A001631(n+4). - Alois P. Heinz, Dec 11 2013

KEYWORD

nonn,tabl

AUTHOR

L. Edson Jeffery, Dec 11 2013

STATUS

approved

A032089

"BHK" (reversible, identity, unlabeled) transform of 1,0,1,0...(odds).

+10
2

1, 0, 1, 1, 2, 3, 5, 9, 14, 25, 39, 68, 107, 182, 289, 483, 772, 1275, 2047, 3355, 5402, 8811, 14213, 23112, 37325, 60580, 97905, 158717, 256622, 415715, 672337, 1088661, 1760998, 2850645, 4611643, 7463884, 12075527, 19541994

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

C. G. Bower, Transforms (2)

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-2,0,-1,1,1).

FORMULA

G.f.: x*(1-x-2*x^2+2*x^3+x^6)/((1-x)*(1+x)*(1-x-x^2)*(1-x^2-x^4)).

a(n) = a(n-1) + 3*a(n-2) - 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) + a(n-8) for n > 8. - Andrew Howroyd, Aug 31 2018

2*a(n) = 2*A000035(n) + A000045(n) - A053602(n). - R. J. Mathar, Mar 02 2022

PROG

(PARI) Vec((1-x-2*x^2+2*x^3+x^6)/((1-x)*(1+x)*(1-x-x^2)*(1-x^2-x^4)) + O(x^40)) \\ Andrew Howroyd, Aug 31 2018

KEYWORD

nonn,easy

AUTHOR

Christian G. Bower

STATUS

approved

A097594

a(n) = (a(n-1) mod a(n-2)) + a(n-2), a(0) = 3, a(1) = 2.

+10
2

2, 5, 3, 8, 5, 13, 8, 21, 13, 34, 21, 55, 34, 89, 55, 144, 89, 233, 144, 377, 233, 610, 377, 987, 610, 1597, 987, 2584, 1597, 4181, 2584, 6765, 4181, 10946, 6765, 17711, 10946, 28657, 17711, 46368, 28657, 75025, 46368, 121393, 75025, 196418, 121393, 317811, 196418, 514229, 317811, 832040, 514229

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

FORMULA

a(2n) = Fibonacci(n+4), a(2n+1) = Fibonacci(n+3).

a(n) = A053602(n+6).

a(n) = abs( A051792(n+11) ).

G.f.: (2 + 5*x + x^2 + 3*x^3)/(1 - x^2 - x^4). - G. C. Greubel, Dec 06 2022

MATHEMATICA

LinearRecurrence[{0, 1, 0, 1}, {2, 5, 3, 8}, 60] (* G. C. Greubel, Dec 06 2022 *)

PROG

(Magma) [Fibonacci(3 +Floor(n/2) +2*(n mod 2)): n in [0..60]]; // G. C. Greubel, Dec 06 2022

(SageMath) [fibonacci(3 +(n//2) + 2*(n%2)) for n in range(61)] # G. C. Greubel, Dec 06 2022

CROSSREFS

Cf. A000045, A051792, A053602.

KEYWORD

nonn

AUTHOR

Gerald McGarvey, Aug 29 2004

STATUS

approved

A272912

Difference sequence of the sequence A116470 of all distinct Fibonacci numbers and Lucas numbers (A000032).

+10
2

1, 1, 1, 1, 2, 1, 3, 2, 5, 3, 8, 5, 13, 8, 21, 13, 34, 21, 55, 34, 89, 55, 144, 89, 233, 144, 377, 233, 610, 377, 987, 610, 1597, 987, 2584, 1597, 4181, 2584, 6765, 4181, 10946, 6765, 17711, 10946, 28657, 17711, 46368, 28657, 75025, 46368, 121393, 75025

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Every term is a Fibonacci number (A000045).

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

FORMULA

From Colin Barker, May 10 2016: (Start)

a(n) = a(n-2)+a(n-4) for n>4.

G.f.: x*(1+x-x^5) / (1-x^2-x^4).

(End)

a(n) = A053602(n-2), n>2. - R. J. Mathar, May 20 2016

a(n) = A123231(n-3), n>3. - Georg Fischer, Oct 23 2018

EXAMPLE

A116470 = (1, 2, 3, 4, 5, 7, 8, 11, 13, 18, 21, 29, 34, 47, 55, 76,...), so that (a(n)) = (1,1,1,1,2,1,3,2,5,3,8,5,13,8,12,...).

MATHEMATICA

u = Table[Fibonacci[n], {n, 1, 200}]; v = Table[LucasL[n], {n, 1, 200}];

Take[Differences[Union[u, v]], 100]

PROG

(PARI) Vec(x*(1+x-x^5)/(1-x^2-x^4) + O(x^50)) \\ Colin Barker, May 10 2016

CROSSREFS

Cf. A000045, A000032, A116470, A053602, A123231.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 10 2016

STATUS

approved

A239366

Triangular array read by rows: T(n,k) is the number of palindromic compositions of n having exactly k 1's, n>=0, 0<=k<=n.

+10
1

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 3, 0, 3, 0, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 1, 5, 0, 5, 0, 4, 0, 1, 0, 1, 3, 2, 3, 2, 1, 3, 1, 0, 0, 1, 8, 0, 10, 0, 7, 0, 5, 0, 1, 0, 1, 5, 3, 5, 5, 4, 3, 1, 4, 1, 0, 0, 1, 13, 0, 18, 0, 16, 0, 9, 0, 6, 0, 1, 0, 1, 8, 5, 10, 8, 7, 9, 5, 4, 1, 5, 1, 0, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,11

COMMENTS

Row sums = 2^floor(n/2).

T(n,0) = A053602(n-1) for n>0, T(n,1) = A079977(n-5) for n>4, T(2n+1,3) = A006367(n-1) for n>0, both bisections of column k=2 contain A010049. - Alois P. Heinz, Mar 21 2014

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

G.f.: G(x,y) = ((1 + x)*(1 - x + x^2 + x*y - x^2*y))/(1 - x^2 - x^4 - x^2*y^2 + x^4*y^2). Satisfies G(x,y) = 1/(1 - x) - x + y*x + (x^2/(1 - x^2) - x^2 +y^2*x^2)*G(x,y).

EXAMPLE

1,

0, 1,

1, 0, 1,

1, 0, 0, 1,

2, 0, 1, 0, 1,

1, 1, 1, 0, 0, 1,

3, 0, 3, 0, 1, 0, 1,

2, 1, 1, 2, 1, 0, 0, 1,

5, 0, 5, 0, 4, 0, 1, 0, 1,

3, 2, 3, 2, 1, 3, 1, 0, 0, 1

There are eight palindromic compositions of 6: T(6,0)=3 because we have: 6, 3+3, 2+2+2. T(6,2)=3 because we have: 1+4+1, 2+1+1+2, 1+2+2+1. T(6,4)=1 because we have: 1+1+2+1+1. T(6,6)=1 because we have: 1+1+1+1+1+1.

MAPLE

b:= proc(n) option remember; `if`(n=0, 1, expand(

add(b(n-j)*`if`(j=1, x^2, 1), j=1..n)))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))

(add(b(i)*`if`(n-2*i=1, x, 1), i=0..n/2)):

seq(T(n), n=0..30); # Alois P. Heinz, Mar 21 2014

MATHEMATICA

nn=15; Table[Take[CoefficientList[Series[((1+x)*(1-x+x^2+x*y-x^2*y))/(1-x^2-x^4-x^2*y^2+x^4*y^2), {x, 0, nn}], {x, y}][[n]], n], {n, 1, nn}]//Grid

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Mar 20 2014

STATUS

approved

A275435

Triangle read by rows: T(n,k) is the number of 00-avoiding binary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=floor(n/2)).

+10
1

1, 2, 1, 2, 3, 2, 2, 4, 2, 5, 6, 2, 3, 8, 8, 2, 8, 14, 10, 2, 5, 16, 20, 12, 2, 13, 30, 30, 14, 2, 8, 30, 48, 40, 16, 2, 21, 60, 78, 54, 18, 2, 13, 56, 106, 112, 68, 20, 2, 34, 116, 184, 166, 86, 22, 2, 21, 102, 224, 286, 224, 104, 24, 2, 55, 218, 408, 452, 310, 126, 26, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).

A sequence is palindromic if and only if its degree of asymmetry is 0.

Number of entries in row n is 1+floor(n/2).

Sum of entries in row n is A000045(n+2) (Fibonacci).

T(n,0) = A053602(n+2) (= number of palindromic 00-avoiding binary words of length n).

Sum(k*T(n,k), k>=0) = A275436(n).

LINKS

Table of n, a(n) for n=0..71.

FORMULA

G.f.: G(t,z) = (1 + 2z + tz^2 +z^3 -tz^5)/(1 - z^2 - tz^2 - z^4 - tz^4 + tz^6).

EXAMPLE

Row 3 is [3,2] because the 00-avoiding binary words of length 3 are 010, 011, 101, 110, 111, having asymmetry degrees 0, 1, 0, 1, 0, respectively.

Triangle starts:

1;

2;

1,2;

3,2;

2,4,2;

5,6,2.

MAPLE

G := (1+2*z+t*z^2+z^3-t*z^5)/(1-z^2-t*z^2-z^4-t*z^4+t*z^6): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. (1/2)*n) end do; # yields sequence in triangular form

MATHEMATICA

Table[BinCounts[#, {0, Floor[n/2] + 1, 1}] &@ Map[Total@ BitXor[Take[#, Ceiling[Length[#]/2]], Reverse@ Take[#, -Ceiling[Length[#]/2]]] &, Select[PadLeft[IntegerDigits[#, 2], n] & /@ Range[0, 2^n - 1], Length@ SequenceCases[#, {0, 0}] == 0 &]], {n, 0, 15}] // Flatten (* Michael De Vlieger, Aug 15 2016, Version 10.1 *)

CROSSREFS

Cf. A000045, A053602, A275436

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Aug 15 2016

STATUS

approved