A027383 -id:A027383 - cp's OEIS frontend

A305540 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

1, 1, 1, 1, 2, 1, 4, 3, 1, 6, 6, 1, 10, 21, 12, 1, 14, 36, 24, 1, 22, 93, 132, 60, 1, 30, 150, 240, 120, 1, 46, 345, 900, 960, 360, 1, 62, 540, 1560, 1800, 720, 1, 94, 1173, 4980, 9300, 7920, 2520, 1, 126, 1806, 8400, 16800, 15120, 5040, 1, 190, 3801, 24612, 71400, 103320, 73080, 20160, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320

Offset: 1

Robert A. Russell, Jun 04 2018

The number of achiral necklaces is equivalent to the number of achiral bracelets.

			The triangle begins with T(1,1):
1;
1,   1;
1,   2;
1,   4,     3;
1,   6,     6;
1,  10,    21,     12;
1,  14,    36,     24;
1,  22,    93,    132,     60;
1,  30,   150,    240,    120;
1,  46,   345,    900,    960,     360;
1,  62,   540,   1560,   1800,     720;
1,  94,  1173,   4980,   9300,    7920,    2520;
1, 126,  1806,   8400,  16800,   15120,    5040;
1, 190,  3801,  24612,  71400,  103320,   73080,   20160;
1, 254,  5796,  40824, 126000,  191520,  141120,   40320;
1, 382, 11973, 113652, 480060, 1048320, 1234800,  745920, 181440;
1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;
For a(4,2)=4, the achiral loops are AAAB, AABB, ABAB, and ABBB.

Odd rows are A019538.
Even rows are A172106.
Columns 1-6 are A057427, A027383, A056489, A056490, A056491, and A056492.

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

T(n, k) = (k!/2)*(stirling(floor((n+1)/2), k, 2)+stirling(ceil((n+1)/2), k, 2));
tabf(nn) = for(n=1, nn, for (k=1, ceil((n+1)/2), print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 02 2018

T(n,k) = (k!/2) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), where S2(n,k) is the Stirling subset number A008277.
T(n,k) = 2*A273891(n,k) - A087854(n,k).
G.f. for column k>1: (k!/2) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2). - Robert A. Russell, Sep 26 2018

A329862 Positive integers whose binary expansion has cuts-resistance 2.

Original entry on oeis.org

3, 4, 6, 9, 11, 12, 13, 18, 19, 20, 22, 25, 26, 37, 38, 41, 43, 44, 45, 50, 51, 52, 53, 74, 75, 76, 77, 82, 83, 84, 86, 89, 90, 101, 102, 105, 106, 149, 150, 153, 154, 165, 166, 169, 171, 172, 173, 178, 179, 180, 181, 202, 203, 204, 205, 210, 211, 212, 213

Offset: 1

Gus Wiseman, Nov 23 2019

nonn

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The sequence of terms together with their binary expansions begins:
   3:      11
   4:     100
   6:     110
   9:    1001
  11:    1011
  12:    1100
  13:    1101
  18:   10010
  19:   10011
  20:   10100
  22:   10110
  25:   11001
  26:   11010
  37:  100101
  38:  100110
  41:  101001
  43:  101011
  44:  101100
  45:  101101
  50:  110010

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of 2's in A319416.
Numbers whose binary expansion has cuts-resistance 1 are A000975.
Binary words with cuts-resistance 2 are conjectured to be A027383.
Compositions with cuts-resistance 2 are A329863.
Cuts-resistance of binary expansion without first digit is A319420.
Binary words counted by cuts-resistance are A319421 and A329860.
Compositions counted by cuts-resistance are A329861.
Cf. A107907, A114901, A164707, A318928, A319411, A329745, A329865, A329866.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Select[Range[100],degdep[IntegerDigits[#,2]]==2&]

A144464 Triangle T(n,m) read by rows: T(n,m) = 2^min(m,n-m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4, 8, 4, 2, 1, 1, 2, 4, 8, 8, 4, 2, 1, 1, 2, 4, 8, 16, 8, 4, 2, 1, 1, 2, 4, 8, 16, 16, 8, 4, 2, 1, 1, 2, 4, 8, 16, 32, 16, 8, 4, 2, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 09 2008

			The triangle starts in row n=0 as:
{1},
{1, 1},
{1, 2, 1},
{1, 2, 2, 1},
{1, 2, 4, 2, 1},
{1, 2, 4, 4, 2, 1},
{1, 2, 4, 8, 4, 2, 1},
{1, 2, 4, 8, 8, 4, 2, 1},
{1, 2, 4, 8, 16, 8, 4, 2, 1},
{1, 2, 4, 8, 16, 16, 8, 4, 2, 1},
{1, 2, 4, 8, 16, 32, 16, 8, 4, 2, 1}

Cf. A004197, A152714, A152716, A152717.

Clear[f, t]; f[n_, m_] = If[m <= Floor[n/2], m, n - m]; Table[Table[f[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

T(n,m)=1<Charles R Greathouse IV, Jan 15 2012

Row sums: sum_{m=0..n} T(n,m) = A027383(n).

T(n,k) = 2^A004197(n,k). - Philippe Deléham, Feb 25 2014

Offset corrected by the Associate Editors of the OEIS, Sep 11 2009

Better name by Philippe Deléham, Feb 25 2014

A329863 Number of compositions of n with cuts-resistance 2.

Original entry on oeis.org

0, 0, 1, 0, 3, 6, 9, 22, 47, 88, 179, 354, 691, 1344, 2617, 5042, 9709, 18632, 35639, 68010, 129556, 246202, 467188, 885036, 1674211, 3163094, 5969022, 11251676, 21189382, 39867970, 74950464, 140798302, 264313039, 495861874, 929709687, 1742193854, 3263069271, 6108762316

Offset: 0

Gus Wiseman, Nov 23 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(2) = 1 through a(7) = 22 compositions (empty column not shown):
  (1,1)  (2,2)    (1,1,3)    (3,3)      (1,1,5)
         (1,1,2)  (1,2,2)    (1,1,4)    (1,3,3)
         (2,1,1)  (2,2,1)    (4,1,1)    (2,2,3)
                  (3,1,1)    (1,1,2,2)  (3,2,2)
                  (1,1,2,1)  (1,1,3,1)  (3,3,1)
                  (1,2,1,1)  (1,2,2,1)  (5,1,1)
                             (1,3,1,1)  (1,1,2,3)
                             (2,1,1,2)  (1,1,3,2)
                             (2,2,1,1)  (1,1,4,1)
                                        (1,4,1,1)
                                        (2,1,1,3)
                                        (2,1,2,2)
                                        (2,2,1,2)
                                        (2,3,1,1)
                                        (3,1,1,2)
                                        (3,2,1,1)
                                        (1,1,2,1,2)
                                        (1,1,2,2,1)
                                        (1,2,1,1,2)
                                        (1,2,2,1,1)
                                        (2,1,1,2,1)
                                        (2,1,2,1,1)

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Column k = 2 of A329861.
Compositions with cuts-resistance 1 are A003242.
Compositions with runs-resistance 2 are A329745.
Numbers whose binary expansion has cuts-resistance 2 are A329862.
Binary words with cuts-resistance 2 are conjectured to be A027383.
Cuts-resistance of binary expansion is A319416.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]==2&]],{n,0,10}]

Ca(N) = {1/(1-sum(k=1, N, x^k/(1+x^k)))}
A_x(N) = {my(x='x+O('x^N)); concat([0,0],Vec(-1+(1+sum(m=1,N, Ca(N)*x^(2*m)*(Ca(N)-1)/(1+x^m*(2+x^m*(1+Ca(N))))))/(1-sum(m=1,N, Ca(N)*x^(2*m)/(1+x^m*(2+x^m*(1+Ca(N))))))))}
A_x(38) \\ John Tyler Rascoe, Feb 20 2025

G.f.: -1 + (1 + Ca(x) * Sum_{m>0} x^(2*m) * (Ca(x)-1)/(1 + x^m * (2 + x^m * (1+Ca(x)))))/(1 - Ca(x) * Sum_{m>0} x^(2*m)/(1 + x^m * (2 + x^m * (1+Ca(x))))) where Ca(x) is the g.f. for A003242. - John Tyler Rascoe, Feb 20 2025

a(21) onwards from John Tyler Rascoe, Feb 20 2025

A056342 Number of bracelets of length n using exactly two different colored beads.

Original entry on oeis.org

0, 1, 2, 4, 6, 11, 16, 28, 44, 76, 124, 222, 378, 685, 1222, 2248, 4110, 7683, 14308, 27010, 50962, 96907, 184408, 352696, 675186, 1296856, 2493724, 4806076, 9272778, 17920858, 34669600, 67159048, 130216122, 252745366, 490984486, 954637556, 1857545298, 3617214679, 7048675958, 13744694926, 26818405350

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet.

			For a(6)=11, the arrangements are AAAAAB, AAAABB, AAABAB, AAABBB, AABAAB, AABBBB, ABABAB, ABABBB, ABBABB, ABBBBB, and AABABB, the last being chiral. Its reverse is AABBAB. - _Robert A. Russell_, Sep 26 2018

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]

G. C. Greubel, Table of n, a(n) for n = 1..3000

Column 2 of A273891.
Equals A052823 - A059076.
Cf. A008277, A027383.

a[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n) - 2; Array[a, 41] (* Jean-François Alcover, Nov 05 2017 *)
k=2; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = my(k=2); (k!/4)*(stirling(floor((n+1)/2),k,2) + stirling(ceil((n+1)/2),k,2)) + (k!/(2*n))*sumdiv(n,d,eulerphi(d)*stirling(n/d,k,2)); \\ Michel Marcus, Sep 28 2018

a(n) = A000029(n) - 2.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (A052823(n) + A027383(n-2)) / 2 = A059076(n) + A027383(n-2).
a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where k=2 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=2 is the number of colors. (End)

More terms from Joerg Arndt, Jun 10 2016

A133628 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, else a(n) = a(n-1) + p^((n-1)/2), where p=4.

Original entry on oeis.org

1, 4, 8, 20, 36, 84, 148, 340, 596, 1364, 2388, 5460, 9556, 21844, 38228, 87380, 152916, 349524, 611668, 1398100, 2446676, 5592404, 9786708, 22369620, 39146836, 89478484, 156587348, 357913940, 626349396, 1431655764, 2505397588

Offset: 1

Hieronymus Fischer, Sep 19 2007

nonn

This is essentially a duplicate of A097164. - R. J. Mathar, Jun 08 2008

Partial sums of A084221.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Index entries for linear recurrences with constant coefficients, signature (1, 4, -4).

Sequences with similar recurrence rules: A027383(p=2), A087503(p=3), A133629(p=5).
See A133629 for general formulas with respect to the recurrence rule parameter p.
Related sequences: A132666, A132667, A132668, A132669.
Other related sequences for different p: A016116(p=2), A038754(p=3), A084221(p=4), A133632(p=5).

[4^Floor(n/2)+4^Floor((n+1)/2)/3-4/3: n in [1..40]]; // Vincenzo Librandi, Aug 17 2011

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=4*a[n-2]+4 od: seq(a[n], n=1..31); # Zerinvary Lajos, Mar 17 2008

nxt[{n_,a_}]:={n+1,If[OddQ[n],a+3*4^((n+1)/2-1),a+4^(n/2)]}; Transpose[ NestList[ nxt,{1,1},30]][[2]] (* Harvey P. Dale, Mar 31 2013 *)

vector(40, n, (3*4^floor(n/2) + 4^floor((n+1)/2) - 4)/3) \\ G. C. Greubel, Nov 08 2018

a(n) = Sum_{k=1..n} A084221(k).
G.f.: x*(1+3*x)/((1-4*x^2)*(1-x)).
a(n) = (4/3)*(4^(n/2)-1) if n is even, otherwise a(n) = (4/3)*(7*4^((n-3)/2)-1).
a(n) = (4/3)*(4^floor(n/2) + 4^floor((n-1)/2) - 4^floor((n-2)/2) - 1).
a(n) = 4^floor(n/2) + 4^floor((n+1)/2)/3 - 4/3.
a(n) = A132668(a(n+1)) - 1.
a(n) = A132668(a(n-1) + 1) for n > 0.
A132668(a(n)) = a(n-1) + 1 for n > 0.

A133632 a(1)=1, a(n) = (p-1)a(n-1), if n is even, otherwise a(n) = pa(n-2), where p = 5.

Original entry on oeis.org

1, 4, 5, 20, 25, 100, 125, 500, 625, 2500, 3125, 12500, 15625, 62500, 78125, 312500, 390625, 1562500, 1953125, 7812500, 9765625, 39062500, 48828125, 195312500, 244140625, 976562500, 1220703125, 4882812500, 6103515625, 24414062500

Offset: 1

Hieronymus Fischer, Sep 19 2007

nonn

Binomial transform = A134418: (1, 5, 14, 48, 152, 496, 1600, ...). Double binomial transform = A048875: (1, 6, 25, 106, 449, 1902, ...) - Gary W. Adamson, Oct 24 2007

Index entries for linear recurrences with constant coefficients, signature (0, 5).

For the partial sums see A133629.
Sequences with similar recurrence rules: A016116(p=2), A038754(p=3), A084221(p=4).
Partial sums for other p: A027383(p=2), A087503(p=3), A133628(p=4).
Other related sequences: A132666, A132667, A132668, A132669.
Cf. A134418, A048875.

RecurrenceTable[{a[1]==1,a[2]==4,a[n]==If[EvenQ[n],4a[n-1],5a[n-2]]},a,{n,30}] (* Harvey P. Dale, Jan 14 2013 *)

The following formulas are given for a general natural parameter p > 1 (p = 5 for this sequence).
G.f.: g(x) = x(1+(p-1)x)/(1-px^2).
a(n) = p^floor((n-1)/2)*(p+(p-2)*(-1)^n)/2.
a(n) = A133629(n) - A133629(n-1) for n > 1.
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

A191595 Order of smallest n-regular graph of girth 5.

Original entry on oeis.org

5, 10, 19, 30, 40, 50

Offset: 2

N. J. A. Sloane, Jun 07 2011

Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012

Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

M. Abreu et al., A family of regular graphs of girth 5, Discrete Math., 308 (2008), 1810-1815.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
G. Royle, Cages of higher valency

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).

Moore lower bound on the orders of (k,g) cages: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306(k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10),A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 02 2011

a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013

A209727 T(n,k) = 1/4 the number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Original entry on oeis.org

2, 3, 3, 4, 4, 4, 6, 5, 5, 6, 8, 7, 6, 7, 8, 12, 9, 8, 8, 9, 12, 16, 13, 10, 10, 10, 13, 16, 24, 17, 14, 12, 12, 14, 17, 24, 32, 25, 18, 16, 14, 16, 18, 25, 32, 48, 33, 26, 20, 18, 18, 20, 26, 33, 48, 64, 49, 34, 28, 22, 22, 22, 28, 34, 49, 64, 96, 65, 50, 36, 30, 26, 26, 30, 36, 50, 65, 96

Offset: 1

R. H. Hardin, Mar 12 2012

Table starts
..2..3..4..6..8.12.16.24.32.48.64..96.128.192.256.384.512.768.1024.1536.2048
..3..4..5..7..9.13.17.25.33.49.65..97.129.193.257.385.513.769.1025.1537.2049
..4..5..6..8.10.14.18.26.34.50.66..98.130.194.258.386.514.770.1026.1538.2050
..6..7..8.10.12.16.20.28.36.52.68.100.132.196.260.388.516.772.1028.1540.2052
..8..9.10.12.14.18.22.30.38.54.70.102.134.198.262.390.518.774.1030.1542.2054
.12.13.14.16.18.22.26.34.42.58.74.106.138.202.266.394.522.778.1034.1546.2058
.16.17.18.20.22.26.30.38.46.62.78.110.142.206.270.398.526.782.1038.1550.2062
.24.25.26.28.30.34.38.46.54.70.86.118.150.214.278.406.534.790.1046.1558.2070

			Some solutions for n=4, k=3
..2..1..2..1....0..2..0..1....1..2..0..2....0..1..0..2....2..1..2..1
..0..2..0..2....2..1..2..0....2..0..1..0....2..0..2..1....0..2..0..2
..1..0..1..0....0..2..0..1....1..2..0..2....0..1..0..2....1..0..1..0
..0..2..0..2....2..1..2..0....2..0..1..0....2..0..2..1....0..2..0..2
..1..0..1..0....0..2..0..1....1..2..0..2....0..1..0..2....2..1..2..1

R. H. Hardin, Table of n, a(n) for n = 1..1512

Column 1 is A029744(n+1).

Diagonal is A027383.

Empirical for column k:
k=1: a(n) = 2*a(n-2).
k=2..7: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).

A247903 Start with a single square; at n-th generation add a square at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

Original entry on oeis.org

1, 5, 13, 29, 53, 93, 149, 237, 357, 541, 789, 1165, 1669, 2429, 3445, 4973, 7013, 10077, 14165, 20301, 28485, 40765, 57141, 81709, 114469, 163613, 229141, 327437, 458501, 655101, 917237, 1310445, 1834725, 2621149, 3669717, 5242573, 7339717, 10485437

Offset: 0

Kival Ngaokrajang, Sep 26 2014

Refer to A247618, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e."vertex to side" expansion version. Let us assign the label "1" to the square at the origin; at n-th generation add a square at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping squares will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The squares count is A001844. See illustration ("vertex to side" is equal to "side to vertex"). For n >= 1, (a(n) - a(n-1))/4 is A027383.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms (vertex to side)
Kival Ngaokrajang, Illustration of initial terms (side to vertex)
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Vertex to vertex version: A061777, A247618, A247619, A247620.
Vertex to side version: A101946, A247904, A247905.
Cf. A001844, A027383.

[2^(n/2+1)*((7+5*Sqrt(2)) + (-1)^n*(7-5*Sqrt(2))) -(8*n+27): n in [0..50]]; // G. C. Greubel, Feb 18 2022

LinearRecurrence[{2,1,-4,2}, {1,5,13,29}, 51] (* G. C. Greubel, Feb 18 2022 *)

{
b=0; a=1; print1(1, ", ");
for (n=0, 50,
     b=b+2^floor(n/2);
     a=a+4*b;
     print1(a, ", ")
    )
}

Vec(-(2*x^3+2*x^2+3*x+1) / ((x-1)^2*(2*x^2-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

[2*2^(n/2)*((7+5*sqrt(2)) +(-1)^n*(7-5*sqrt(2))) -(8*n+27) for n in (0..50)] # G. C. Greubel, Feb 18 2022

a(0) = 1, for n >= 1, a(n) = 4*A027383(n) + a(n-1).
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +2*a(n-4). - Colin Barker, Sep 26 2014
G.f.: (1+3*x+2*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - Colin Barker, Sep 26 2014
A(n) = 2^(n/2+1)*((1+sqrt(2))^3 + (-1)^n*(1-sqrt(2))^3) - (8*n + 27). - G. C. Greubel, Feb 18 2022

More terms from Colin Barker, Sep 26 2014

cp's OEIS Frontend

A305540 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

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A329862 Positive integers whose binary expansion has cuts-resistance 2.

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A144464 Triangle T(n,m) read by rows: T(n,m) = 2^min(m,n-m).

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A329863 Number of compositions of n with cuts-resistance 2.

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A056342 Number of bracelets of length n using exactly two different colored beads.

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A133628 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, else a(n) = a(n-1) + p^((n-1)/2), where p=4.

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A133632 a(1)=1, a(n) = (p-1)*a(n-1), if n is even, otherwise a(n) = p*a(n-2), where p = 5.

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A133632 a(1)=1, a(n) = (p-1)a(n-1), if n is even, otherwise a(n) = pa(n-2), where p = 5.