A056309 -id:A056309 - cp's OEIS frontend

A122746 G.f.: 1/((1-2x)(1-2*x^2)).

1, 2, 6, 12, 28, 56, 120, 240, 496, 992, 2016, 4032, 8128, 16256, 32640, 65280, 130816, 261632, 523776, 1047552, 2096128, 4192256, 8386560, 16773120, 33550336, 67100672, 134209536, 268419072, 536854528, 1073709056, 2147450880, 4294901760, 8589869056

Offset: 0

N. J. A. Sloane, Sep 24 2006

Equals row sums of triangle A156665. - Gary W. Adamson, Feb 12 2009
a(n) is the number of subsets of {1,2,...,n+1} that contain at least one odd integer. - Geoffrey Critzer, Mar 03 2009
a(n-3) is the number of chiral pairs of color patterns of length n using two colors. Two color patterns are equivalent if the colors are permuted. For example, a string of five colors using exactly two different colors has six chiral pairs: AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB. The number of color patterns of length n using exactly k colors when chiral pairs are counted twice is the Stirling subset number S2(n,k). The number of achiral color patterns of length n using exactly 2 colors is S2(floor(n/2)+1,2). The value of a(n-3) is half the difference of these two. - Robert A. Russell, Feb 01 2018
a(n-2) is the number of chiral pairs for a row of n colors with exactly 2 different colors. If the reverse of a sequence is different, the combination of the two is a chiral pair. For a row of 4 colors using exactly 2 different colors, the chiral pairs are AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, and BABB-BBAB. Thus a(4-2) = a(2) = 6. - Robert A. Russell, Jun 10 2018

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 28*x^4 + 56*x^5 + 120*x^6 + 240*x^7 + 496*x^8 + ... - _Michael Somos_, Jul 01 2018

Harvey P. Dale, Table of n, a(n) for n = 0..1000
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, Lect. Not. Comp. Sci 1403 (1998), 475-488, NL(F_n).
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Essentially the same as A032085.
Cf. A000079, A060546.
Cf. A006516, A020522, A156665.
Cf. A000918, A056309, A056453.

List([0..35],n->2^(n+1)-2^(QuoInt(n+1,2))); # Muniru A Asiru, Sep 27 2018

seq(coeff(series(((1-2*x)*(1-2*x^2))^(-1),x,n+1), x, n), n = 0..35); # Muniru A Asiru, Sep 27 2018

RecurrenceTable[{a[n] == 2 (BitOr[a[n - 1], a[n - 2]]), a[0] == 1, a[1] == 2}, a, {n, 0, 32}] (* Geoffrey Critzer, Jan 09 2011 *)
CoefficientList[Series[1/((1-2x)(1-2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-4},{1,2,6},40] (* Harvey P. Dale, Jun 25 2013 *)
Table[(StirlingS2[n,2] - StirlingS2[Floor[n/2]+1,2])/2, {n,3,30}] (* Robert A. Russell, Jan 29 2018 *)
a[ n_] := 2^(n + 1) - 2^Quotient[n + 1, 2]; (* Michael Somos, Jul 01 2018 *)

{a(n) = 2^(n+1) - 2^((n+1)\2)}; /* Michael Somos, Jul 01 2018 */

From Alexander Adamchuk, Sep 25 2006: (Start)
a(2k) = A006516(k+1) = 2^k*(2^(k+1) - 1) = A020522(k+1) /2.
a(2k+1) = 2*A006516(k+1) = 2^(k+1)*(2^(k+1) - 1) = A020522(k+1). (End)
a(n) = 2^(n+1) - 2^(floor((n+1)/2)). - Geoffrey Critzer, Mar 03 2009
a(n) = 2*(a(n-1) bitwiseOR a(n-2)), a(0)=1, a(1)=2. - Pierre Charland, Dec 12 2010
G.f.: (1+x*Q(0))/(1-x)^2, where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
a(0)=1, a(1)=2, a(2)=6, a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Harvey P. Dale, Jun 25 2013
a(n) = (A000079(n+2) - A060546(n+2))/ 2. - Robert A. Russell, Jun 19 2018
a(n) = -a(-3-n) * 2^(n+2 + floor((n+1)/2)) for all n in Z. - Michael Somos, Jul 01 2018
a(n) = (A000918(n+2) - A056453(n+2)) / 2 = A000918(n+2) - A056309(n+2) = A056309(n+2) - A056453(n+2). - Robert A. Russell, Sep 26 2018

A056326 Number of reversible string structures with n beads using exactly two different colors.

Original entry on oeis.org

0, 1, 2, 5, 9, 19, 35, 71, 135, 271, 527, 1055, 2079, 4159, 8255, 16511, 32895, 65791, 131327, 262655, 524799, 1049599, 2098175, 4196351, 8390655, 16781311, 33558527, 67117055, 134225919, 268451839, 536887295, 1073774591, 2147516415, 4295032831, 8590000127

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, 0, -6, 4).

Column k=2 of A284949 and of A291883.

Cf. A056309.

Table[(StirlingS2[n,2] + StirlingS2[Floor[n/2]+1,2])/2, {n,1,30}] (* Robert A. Russell, Jan 29 2018 *)
LinearRecurrence[{3, 0, -6, 4}, {0, 1, 2, 5}, 35] (* or *)
Rest@ CoefficientList[Series[x^2*(x^2 + x - 1)/((x - 1) (2 x - 1) (2 x^2 - 1)), {x, 0, 35}], x] (* Michael De Vlieger, Jan 31 2018 *)

a(n) = A005418(n) - 1.
From Colin Barker, Nov 25 2012: (Start)
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
G.f.: x^2*(x^2+x-1)/((x-1)*(2*x-1)*(2*x^2-1)). (End)

A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

Original entry on oeis.org

2, 9, 35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135, 140737496743935, 562949970198527

Offset: 1

Reinhard Zumkeller, Mar 23 2004

Smallest numbers having in binary representation n 0's and n 1's: a(n) = Min{m: A023416(m)=A000120(m)=n}.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Binary
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007088, A001700.
Subsequence of A031443.
Cf. A000120, A023416.
Cf. A007582, A056326, A005367, A063376, A032125, A056309, A028403, A001576, A028400, A049775.

LinearRecurrence[{7, -14, 8}, {2, 9, 35}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
Table[FromDigits[Join[PadRight[{1},n,0],PadRight[{},n-2,1]],2],{n,2,30}]//Sort (* or *) Rest[CoefficientList[Series[x (-2+5x)/((x-1)(2x-1)(4x-1)),{x,0,30}],x]] (* Harvey P. Dale, Jul 30 2021 *)

a(n+1) = 2*a(n) + 4^n + 1.
a(n) = 2^(2*n-1) + 2^(n-1) - 1.
a(n) = A007582(n)-1 = A056326(2n+1) = A005367(n-1)/2 = A063376(n)/2-1 = A032125(n+1)/3-1 = A056309(2n+1)/2 = A028403(n+1)/4-1 = (A001576(n)-3)/2 = (A028400(n+1)-9)/8 = Sum_{k=2..n+1} A049775(k). - Ralf Stephan, Mar 24 2004
G.f.: x*(-2+5*x) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
E.g.f.: exp(x)*(exp(3*x) + exp(x) - 2)/2. - Stefano Spezia, Sep 27 2023

A305621 Triangle read by rows: T(n,k) is the number of rows of n colors with exactly k different colors counting chiral pairs as equivalent, i.e., the rows are reversible.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 8, 18, 12, 1, 18, 78, 120, 60, 1, 34, 273, 780, 900, 360, 1, 70, 921, 4212, 8400, 7560, 2520, 1, 134, 2916, 20424, 63000, 95760, 70560, 20160, 1, 270, 9150, 93360, 417120, 952560, 1164240, 725760, 181440, 1, 526, 28065, 409380, 2551560, 8217720, 14817600, 15120000, 8164800, 1814400, 1, 1054, 85773, 1749780, 14804700, 64615680, 161247240, 239500800, 209563200, 99792000, 19958400

Offset: 1

Robert A. Russell, Jun 06 2018

			The triangle begins:
  1;
  1,   1;
  1,   4,    3;
  1,   8,   18,      12;
  1,  18,   78,     120,      60;
  1,  34,  273,     780,     900,     360;
  1,  70,  921,    4212,    8400,    7560,     2520;
  1, 134, 2916,   20424,   63000,   95760,    70560,    20160;
  1, 270, 9150,   93360,  417120,  952560,  1164240,   725760,  181440;
  ...
For T(3,2)=4, the achiral color rows are ABA and BAB, while the chiral pairs are AAB-BAA and ABB-BBA. For T(3,3)=3, the color rows are all chiral pairs: ABC-CBA, ACB-BCA, and BAC-CAB.

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

Columns 1-6 are A057427, A056309, A056310, A056311, A056312, and A056313.
Row sums are A326963.
A019538 counts chiral pairs as two, i.e., the rows are not reversible.
Cf. A277504, A305622.

Table[(k!/2) (StirlingS2[n, k] + StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten

T(n,k) = {k! * (stirling(n,k,2) + stirling((n+1)\2,k,2)) / 2} \\ Andrew Howroyd, Sep 13 2019

T(n,k) = (k!/2) * (S2(n,k) + S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
T(n,k) = (A019538(n,k) + A019538(ceiling(n/2),k)) / 2.
G.f. for column k: k! x^k / (2*Product_{i=1..k}(1-ix)) + k! (x^(2k-1)+x^(2k)) / (2*Product{i=1..k}(1-i x^2)). - Robert A. Russell, Sep 25 2018
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A277504(n, i). - Andrew Howroyd, Sep 13 2019

A001445 a(n) = (2^n + 2^[ n/2 ] )/2.

Original entry on oeis.org

3, 5, 10, 18, 36, 68, 136, 264, 528, 1040, 2080, 4128, 8256, 16448, 32896, 65664, 131328, 262400, 524800, 1049088, 2098176, 4195328, 8390656, 16779264, 33558528, 67112960, 134225920, 268443648

Offset: 2

N. J. A. Sloane

a(n) is union of A007582(n-1) and A164051(n). - Jaroslav Krizek, Aug 14 2009

Number of binary strings of length n+1, not counting strings which are the reversal, the complement, or the reversal of the complement of each other as different. - Christian Barrientos, Jun 06 2025

			G.f. = 3*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 36*x^6 + 68*x^7 + 136*x^8 + ...

James Spahlinger, Table of n, a(n) for n = 2..1001
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A056309, A052957.

Maple
```
f := n->(2^n+2^floor(n/2))/2;
```

Table[(2^n + 2^(Floor[n/2]))/2, {n, 2, 50}] (* G. C. Greubel, Sep 08 2017 *)
LinearRecurrence[{2,2,-4},{3,5,10},30] (* Harvey P. Dale, Sep 12 2021 *)

for(n=2,50, print1((2^n + 2^(n\2))/2, ", ")) \\ G. C. Greubel, Sep 08 2017

a(n) = (1/2)*A005418(n+2).
G.f.: x^2*(3-x-6*x^2)/((1-2*x)*(1-2*x^2)).
G.f.: 3*G(0) where G(k) = 1 + x*(4*2^k + 1)*(1 + 2*x*G(k+1))/(1 + 2*2^k). - Sergei N. Gladkovskii, Dec 12 2011 [Edited by Michael Somos, Sep 09 2013]
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3) for n > 4. - Chai Wah Wu, Sep 10 2020
E.g.f.: (2*cosh(2*x) + 2*cosh(sqrt(2)*x) + 2*sinh(2*x) + sqrt(2)*sinh(sqrt(2)*x) - 4 - 6*x)/4. - Stefano Spezia, Jun 14 2025

cp's OEIS Frontend

A122746 G.f.: 1/((1-2*x)*(1-2*x^2)).

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A056326 Number of reversible string structures with n beads using exactly two different colors.

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A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

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A305621 Triangle read by rows: T(n,k) is the number of rows of n colors with exactly k different colors counting chiral pairs as equivalent, i.e., the rows are reversible.

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A001445 a(n) = (2^n + 2^[ n/2 ] )/2.

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A056318 Number of primitive (aperiodic) reversible strings with n beads using exactly two different colors.

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A122746 G.f.: 1/((1-2x)(1-2*x^2)).