A000126 -id:A000126 - cp's OEIS frontend

A107909 Numbers having no consecutive zeros or no consecutive ones in binary representation.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111

Offset: 0

Reinhard Zumkeller, May 28 2005

Union of A003754 and A003714, complement of A107911;
a(A023548(n+2)) = A052940(n+1) for n>0;
a(A001924(n)) = A000225(n) = 2^n - 1;
a(A000126(n)) = A000079(n) = 2^n for n>0;
A107910(n) = a(n+1) - a(n).

Ivan Neretin, Table of n, a(n) for n = 0..10000
J. M. Dusel, Balanced parabolic quotients and branching rules for Demazure crystals, 2014.
Index entries for 2-automatic sequences.

Cf. A007088, A107907.

foreach $n(1..100){$_=sprintf("%b",$n); print "$n\n" if !m/11/||!m/00/}
# Ivan Neretin, May 01 2016

A026646 a(n) = Sum_{i=0..n} Sum_{j=0..n} A026637(i,j).

Original entry on oeis.org

1, 3, 7, 17, 37, 79, 163, 333, 673, 1355, 2719, 5449, 10909, 21831, 43675, 87365, 174745, 349507, 699031, 1398081, 2796181, 5592383, 11184787, 22369597, 44739217, 89478459, 178956943, 357913913, 715827853, 1431655735

Offset: 0

Clark Kimberling

nonn

a(n) indexes the corner blocks on the Jacobsthal spiral built from blocks of unit area (using J(1) and J(2) as the sides of the first block). - Paul Barry, Mar 06 2008

Partial sums of A026644, which are the row sums of A026637. - Paul Barry, Mar 06 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A000126, A001045, A026637, A026644, A084639, A109613.
Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026643.
Cf. A026644, A026966, A026967, A026968, A026969, A026970.

[(2^(n+4) -(6*n+9+(-1)^n))/6: n in [0..40]]; // G. C. Greubel, Jul 01 2024

CoefficientList[Series[(1-x^2+2x^3)/((1-x)(1-2x)(1-x^2)), {x, 0, 29}], x] (* Metin Sariyar, Sep 22 2019 *)
LinearRecurrence[{3,-1,-3,2}, {1,3,7,17}, 41] (* G. C. Greubel, Jul 01 2024 *)

[(2^(n+4) - (-1)^n -9 - 6*n)/6 for n in range(41)] # G. C. Greubel, Jul 01 2024

G.f.: (1 -x^2 +2*x^3)/((1-x)*(1-2*x)*(1-x^2)). - Ralf Stephan, Apr 30 2004
From Paul Barry, Mar 06 2008: (Start)
a(n) = A001045(n+3) - 2*floor((n+2)/2).
a(n) = -n + Sum_{k=0..n} A001045(k+2) = A084639(n+1) - n. (End)
a(n+1) = 2*a(n) + A109613(n), a(0)=1. - Paul Curtz, Sep 22 2019

A145111 Square array A(n,k) of numbers of length n binary words with fewer than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 7, 5, 1, 2, 4, 8, 11, 6, 1, 2, 4, 8, 15, 16, 7, 1, 2, 4, 8, 16, 27, 22, 8, 1, 2, 4, 8, 16, 31, 47, 29, 9, 1, 2, 4, 8, 16, 32, 59, 80, 37, 10, 1, 2, 4, 8, 16, 32, 63, 111, 134, 46, 11, 1, 2, 4, 8, 16, 32, 64, 123, 207, 222, 56, 12, 1, 2, 4, 8, 16, 32, 64, 127, 239, 384, 365, 67, 13

Offset: 0

Alois P. Heinz, Oct 02 2008

			A(4,1) = 11, because 11 binary words of length 4 have fewer than 1 0-digit between any pair of consecutive 1-digits: 0000, 0001, 0010, 0100, 1000, 0011, 0110, 1100, 0111, 1110, 1111.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2, ...
  3,  4,  4,  4,  4,  4, ...
  4,  7,  8,  8,  8,  8, ...
  5, 11, 15, 16, 16, 16, ...
  6, 16, 27, 31, 32, 32, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns 0-9 give: A000027(n+1), A000124, A000126(n+1), A007800(n+1), A145112, A145113, A145114, A145115, A145116, A145117.
Main diagonal gives A000079.
Cf. A141539.

f:= proc(n,k) option remember; local j; if n=0 then 1 elif n<=k then 2^(n-1) else add(f(n-j, k), j=1..k) fi end: g:= proc(n,k) option remember; if n<0 then 0 else g(n-1,k) +f(n,k) fi end: A:= (n,k)-> `if`(n=0, g(0,k), A(n-1,k) +g(n-1,k)): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[n_, k_] := SeriesCoefficient[(1 - x + x^(k+1))/(1 - 3*x + 2*x^2 + x^(k+1) - x^(k+2)), {x, 0, n}]; a[0, ] = 1; Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover, Jan 15 2014 *)

G.f. of column k: (1-x+x^(k+1))/(1-3*x+2*x^2+x^(k+1)-x^(k+2)).

A111639 Expansion of (3+8x-3x^2-2x^3)/((x^2+4x+1)(x^2-2x-1)).

Original entry on oeis.org

-3, 10, -33, 114, -403, 1450, -5281, 19394, -71619, 265450, -986241, 3670002, -13670803, 50957770, -190026433, 708824834, -2644492803, 9867263050, -36820012641, 137401810674, -512760729619, 1913577130090, -7141393334881, 26651623320002, -99464199710403

Offset: 0

Creighton Dement, Aug 10 2005

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-8,2,1).

Cf. A111640, A111641, A111642, A111643, A000126.

LinearRecurrence[{-6,-8,2,1},{-3,10,-33,114},30] (* Harvey P. Dale, Jul 04 2019 *)

Vec(-(3 + 8*x - 3*x^2 - 2*x^3) / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^25)) \\ Colin Barker, May 11 2019

From Colin Barker, May 11 2019: (Start)
a(n) = ((-1-sqrt(2))^(1+n) + (-1+sqrt(2))^(1+n) - 2*(-2-sqrt(3))^n - sqrt(3)*(-2-sqrt(3))^n - 2*(-2+sqrt(3))^n + sqrt(3)*(-2+sqrt(3))^n) / 2.
a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3. (End)

A111640 Expansion of (-1+3x+x^2-x^3)/((x^2+4x+1)(x^2-2x-1)).

Original entry on oeis.org

1, -9, 45, -195, 793, -3117, 12013, -45751, 172961, -650849, 2441917, -9144539, 34203161, -127829669, 477505565, -1783134255, 6657304833, -24851573497, 92762239373, -346229372851, 1292232479961, -4822886991709, 17999765604237, -67177262104679, 250711906290721

Offset: 0

Creighton Dement, Aug 10 2005

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-8,2,1).

Cf. A111639, A111641, A111642, A111643, A000126.

Vec((1 - 3*x - x^2 + x^3) / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^25)) \\ Colin Barker, Apr 29 2019

a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3. - Colin Barker, Apr 29 2019

A111641 Expansion of -(1+x+3x^2+x^3)/((x^2+4x+1)(x^2-2x-1)).

Original entry on oeis.org

1, -5, 25, -107, 433, -1697, 6529, -24839, 93841, -352973, 1323961, -4957139, 18539041, -69282185, 258790465, -966364367, 3607837153, -13467809237, 50270219929, -187629535739, 700287673681, -2613617125553, 9754412512321, -36404592257879, 135865306871281

Offset: 0

Creighton Dement, Aug 10 2005

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].

Floretion Algebra Multiplication Program, FAMP Code: 1lestesseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-8,2,1).

Cf. A111639, A111640, A111642, A111643, A111644, A000126.

CoefficientList[Series[-(1+x+3x^2+x^3)/((x^2+4x+1)(x^2-2x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{-6,-8,2,1},{1,-5,25,-107},30] (* Harvey P. Dale, Oct 12 2017 *)

Vec((1 + x + 3*x^2 + x^3) / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^25)) \\ Colin Barker, Apr 29 2019

a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3. - Colin Barker, Apr 29 2019

A111642 Expansion of 2(x-1)(x+1)/((x^2+4x+1)(x^2-2*x-1)).

Original entry on oeis.org

2, -12, 54, -224, 890, -3452, 13198, -50016, 188498, -707916, 2652678, -9925760, 37105802, -138631292, 517742494, -1933118784, 7216615970, -26937891852, 100545928278, -375272321696, 1400607336218, -5227311479036, 19509011469358, -72809634633120, 271731700422002

Offset: 0

Creighton Dement, Aug 10 2005

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].

Floretion Algebra Multiplication Program, FAMP Code: 1ibaseseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-8,2,1).

Cf. A111639, A111640, A111641, A111643, A111644, A000126.

Vec(2*(1 - x)*(1 + x) / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^25)) \\ Colin Barker, Apr 29 2019

a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3. - Colin Barker, Apr 29 2019

A111643 Expansion of 2(x+1)^2/((x^2+4x+1)(x^2-2x-1)).

Original entry on oeis.org

-2, 8, -34, 136, -530, 2032, -7714, 29104, -109378, 410040, -1534722, 5738360, -21441682, 80083808, -299027394, 1116348896, -4167148290, 15554127592, -58053908834, 216672484584, -808662529938, 3018041612880, -11263658377442, 42036964786320, -156885101002562

Offset: 0

Creighton Dement, Aug 10 2005

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].

Floretion Algebra Multiplication Program, FAMP Code: 1baseiseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-8,2,1).

Cf. A111639, A111640, A111641, A111642, A111644, A000126.

Vec(-2*(1 + x)^2 / ((1 + 2*x - x^2)*(1 + 4*x + x^2)) + O(x^40)) \\ Colin Barker, May 01 2019

From Colin Barker, May 01 2019: (Start)
a(n) = (-3*(-1-sqrt(2))^(1+n) - 3*(-1+sqrt(2))^(1+n) - 9*(-2-sqrt(3))^n - 5*sqrt(3)*(-2-sqrt(3))^n - 9*(-2+sqrt(3))^n + 5*sqrt(3)*(-2+sqrt(3))^n) / 6.
a(n) = -6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4) for n>3.
(End)

A306417 Number of self-conjugate set partitions of {1, ..., n}.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 7, 7, 46, 39, 321

Offset: 0

Gus Wiseman, Feb 14 2019

This sequence counts set partitions fixed under Callan's conjugation operation.

			The  a(3) = 1 through a(7) = 7 self-conjugate set partitions:
  {{12}{3}}  {{13}{24}}  {{123}{4}{5}}  {{135}{246}}    {{13}{246}{57}}
                         {{13}{2}{45}}  {{124}{35}{6}}  {{15}{246}{37}}
                                        {{13}{25}{46}}  {{1234}{5}{6}{7}}
                                        {{14}{2}{356}}  {{124}{3}{56}{7}}
                                        {{14}{236}{5}}  {{134}{2}{5}{67}}
                                        {{14}{25}{36}}  {{14}{2}{3}{567}}
                                        {{145}{26}{3}}  {{14}{23}{57}{6}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A001610, A032032, A052841, A080107, A169985, A306416, A324011, A324012.

A323952 Regular triangle read by rows where if k > 1 then T(n, k) is the number of connected subgraphs of an n-cycle with any number of vertices other than 2 through k - 1, n >= 1, 1 <= k <= n - 1. Otherwise T(n, 1) = n.

Original entry on oeis.org

1, 2, 3, 3, 7, 4, 4, 13, 9, 5, 5, 21, 16, 11, 6, 6, 31, 25, 19, 13, 7, 7, 43, 36, 29, 22, 15, 8, 8, 57, 49, 41, 33, 25, 17, 9, 9, 73, 64, 55, 46, 37, 28, 19, 10, 10, 91, 81, 71, 61, 51, 41, 31, 21, 11, 11, 111, 100, 89, 78, 67, 56, 45, 34, 23, 12, 12, 133, 121

Offset: 1

Gus Wiseman, Feb 10 2019

			Triangle begins:
   1
   2   3
   3   7   4
   4  13   9   5
   5  21  16  11   6
   6  31  25  19  13   7
   7  43  36  29  22  15   8
   8  57  49  41  33  25  17   9
   9  73  64  55  46  37  28  19  10
  10  91  81  71  61  51  41  31  21  11
  11 111 100  89  78  67  56  45  34  23  12
  12 133 121 109  97  85  73  61  49  37  25  13
Row 4 counts the following connected sets:
  {1}  {1}     {1}     {1}
  {2}  {2}     {2}     {2}
  {3}  {3}     {3}     {3}
  {4}  {4}     {4}     {4}
       {12}    {123}   {1234}
       {14}    {124}
       {23}    {134}
       {34}    {234}
       {123}   {1234}
       {124}
       {134}
       {234}
       {1234}

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

First column is A000027. Second column is A002061. Third column is A000290. Fourth column is A028387.

Cf. A000126, A000325, A306351, A323950, A323951, A323953, A323954, A323956.

anesw[n_,k_]:=Length[If[k==1,List/@Range[n],Union[Sort/@Select[Union[List/@Range[n],Join@@Table[Partition[Range[n],i,1,1],{i,k,n}]],UnsameQ@@#&&#!={}&]]]];
Table[anesw[n,k],{n,0,16},{k,n}]

T(n,k) = if(k==1, n, 1 + n * (n - k + 1)) \\ Andrew Howroyd, Jan 18 2023

T(n, 1) = n; T(n, k) = 1 + n * (n - k + 1).

cp's OEIS Frontend

A107909 Numbers having no consecutive zeros or no consecutive ones in binary representation.

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Perl

A026646 a(n) = Sum_{i=0..n} Sum_{j=0..n} A026637(i,j).

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A145111 Square array A(n,k) of numbers of length n binary words with fewer than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.

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A111639 Expansion of (3+8*x-3*x^2-2*x^3)/((x^2+4*x+1)*(x^2-2*x-1)).

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A111640 Expansion of (-1+3*x+x^2-x^3)/((x^2+4*x+1)*(x^2-2*x-1)).

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A111641 Expansion of -(1+x+3*x^2+x^3)/((x^2+4*x+1)*(x^2-2*x-1)).

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A111642 Expansion of 2*(x-1)*(x+1)/((x^2+4*x+1)*(x^2-2*x-1)).

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A111643 Expansion of 2*(x+1)^2/((x^2+4*x+1)*(x^2-2*x-1)).

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A111639 Expansion of (3+8x-3x^2-2x^3)/((x^2+4x+1)(x^2-2x-1)).

A111640 Expansion of (-1+3x+x^2-x^3)/((x^2+4x+1)(x^2-2x-1)).

A111641 Expansion of -(1+x+3x^2+x^3)/((x^2+4x+1)(x^2-2x-1)).

A111642 Expansion of 2(x-1)(x+1)/((x^2+4x+1)(x^2-2*x-1)).

A111643 Expansion of 2(x+1)^2/((x^2+4x+1)(x^2-2x-1)).