A277504 -id:A277504 - cp's OEIS frontend

A032122 Number of reversible strings with n beads of 5 colors.

1, 5, 15, 75, 325, 1625, 7875, 39375, 195625, 978125, 4884375, 24421875, 122078125, 610390625, 3051796875, 15258984375, 76294140625, 381470703125, 1907349609375, 9536748046875, 47683720703125

Offset: 0

Christian G. Bower

			For a(2)=15, the five achiral strings are AA, BB, CC, DD, and EE; the 10 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, BC-CB, BD-DB, BE-EB, CD-DC, CE-EC, and DE-ED.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
C. G. Bower, Transforms (2)
Index entries for linear recurrences with constant coefficients, signature (5,5,-25).

Column 5 of A277504.

Cf. A000351 (oriented), A032088(n>1) (chiral), A056451 (achiral).

I:=[1, 5, 15]; [n le 3 select I[n] else 5*Self(n-1)+ 5*Self(n-2)-25*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 31 2012

LinearRecurrence[{5, 5, -25}, {1, 5, 15}, 31] (* Vincenzo Librandi, Jan 31 2012 *)
k=5; Table[(k^n+k^Ceiling[n/2])/2,{n,0,30}] (*Robert A. Russell, Nov 25 2017*)

a(n)=(5^((n+1)\2)+5^n)/2 \\ Charles R Greathouse IV, Jan 31 2012

"BIK" (reversible, indistinct, unlabeled) transform of 5, 0, 0, 0...
a(n) = 1/2 * (5^n + 5^floor((n+1)/2)) = 5*A001447(n+1). - Ralf Stephan, Jul 07 2003
G.f.: (1-15*x^2) / ((1-5*x)*(1-5*x^2)). - Colin Barker, Jan 30 2012 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]
a(n) = 5*a(n-1) + 5*a(n-2) - 25*a(n-3). - Vincenzo Librandi, Jan 31 2012
a(n) = (A000351(n) + A056451(n)) / 2. - Robert A. Russell, Nov 10 2018

a(0)=1 prepended by Robert A. Russell, Nov 10 2018

A170801 a(n) = n^10*(n^9 + 1)/2.

Original entry on oeis.org

0, 1, 262656, 581160258, 137439477760, 9536748046875, 304679900238336, 5699447733924196, 72057594574798848, 675425860579888245, 5000000005000000000, 30579545237175985446, 159739999716270145536

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 19 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=262656, there are 2^19=524288 oriented arrangements of two colors. Of these, 2^10=1024 are achiral. That leaves (524288-1024)/2=261632 chiral pairs. Adding achiral and chiral, we get 262656. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Row 19 of A277504.
Cf. A010807 (oriented), A008454 (achiral).
Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170796 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), this sequence (m=9), A170802 (m=10).

List([0..30], n -> n^10*(n^9+1)/2); # G. C. Greubel, Nov 15 2018

[n^10*(n^9+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

seq(n^10*(n^9 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

Table[(n^19 + n^10)/2, {n,0,30}] (* Robert A. Russell, Nov 13 2018 *)

vector(30, n, n--; n^10*(n^9+1)/2) \\ G. C. Greubel, Nov 15 2018

[n^10*(n^9+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010807(n) + A008454(n)) / 2 = (n^19 + n^10) / 2.
G.f.: (Sum_{j=1..19} S2(19,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..18} A145882(19,k) * x^k / (1-x)^20.
E.g.f.: (Sum_{k=1..19} S2(19,k)*x^k + Sum_{k=1..10} S2(10,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>19, a(n) = Sum_{j=1..20} -binomial(j-21,j) * a(n-j). (End)

A170802 a(n) = n^10*(n^10 + 1)/2.

Original entry on oeis.org

0, 1, 524800, 1743421725, 549756338176, 47683720703125, 1828079250264576, 39896133290043625, 576460752840294400, 6078832731271856601, 50000000005000000000, 336374997479248716901, 1916879996254696243200

Offset: 0

N. J. A. Sloane, Dec 11 2009

By definition, all terms are triangular numbers. - Harvey P. Dale, Aug 12 2012

Number of unoriented rows of length 20 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=524800, there are 2^20=1048576 oriented arrangements of two colors. Of these, 2^10=1024 are achiral. That leaves (1048576-1024)/2=523776 chiral pairs. Adding achiral and chiral, we get 524800. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Row 20 of A277504.
Cf. A010808 (oriented), A008454 (achiral).
Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170896 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), A170801 (m=9), this sequence (m=10).

List([0..30], n -> n^10*(n^10+1)/2); # G. C. Greubel, Nov 15 2018

[n^10*(n^10+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

seq(n^10*(n^10 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

n10[n_]:=Module[{c=n^10},(c(c+1))/2];Array[n10,15,0] (* Harvey P. Dale, Jul 17 2012 *)

vector(30, n, n--; n^10*(n^10+1)/2) \\ G. C. Greubel, Nov 15 2018

for n in range(0,20): print(int(n**10*(n**10 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^10*(n^10+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010808(n) + A008454(n)) / 2 = (n^20 + n^10) / 2.
G.f.: (Sum_{j=1..20} S2(20,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..19} A145882(20,k) * x^k / (1-x)^21.
E.g.f.: (Sum_{k=1..20} S2(20,k)*x^k + Sum_{k=1..10} S2(10,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>20, a(n) = Sum_{j=1..21} -binomial(j-22,j) * a(n-j). (End)

A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 3, 4, 0, 5, 6, 15, 7, 0, 6, 10, 36, 39, 18, 0, 7, 15, 70, 126, 132, 29, 0, 8, 21, 120, 310, 540, 357, 70, 0, 9, 28, 189, 645, 1620, 2034, 1131, 126, 0, 10, 36, 280, 1197, 3990, 7790, 8316, 3276, 266, 0

Offset: 1

Andrew Howroyd, Apr 04 2017

A string and its reverse are considered to be equivalent.

			Table starts:
1   2    3     4      5      6       7       8 ...
0   1    3     6     10     15      21      28 ...
0   4   15    36     70    120     189     280 ...
0   7   39   126    310    645    1197    2044 ...
0  18  132   540   1620   3990    8568   16632 ...
0  29  357  2034   7790  23295   58779  131012 ...
0  70 1131  8316  39370 140610  412965 1050616 ...
0 126 3276 32760 195300 839790 2882376 8388576 ...
...

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]

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

Columns 2-6 are A045625, A056314, A056315, A056316, A056317.

Cf. A277504, A143324.

b[n_, k_] := (k^n + k^Ceiling[n/2])/2;
a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k]&];
Table[a[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017, translated from PARI *)

b(n,k) = (k^n + k^(ceil(n/2))) / 2;
a(n,k) = sumdiv(n,d, moebius(n/d) * b(d,k));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print(););

T(n, k) = Sum_{d | n} mu(n/d) * (k^n + k^(ceiling(n/2))) / 2.

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

A056308 Number of reversible strings with n beads using a maximum of six different colors.

Original entry on oeis.org

1, 6, 21, 126, 666, 3996, 23436, 140616, 840456, 5042736, 30236976, 181421856, 1088414496, 6530486976, 39182222016, 235093332096, 1410555793536, 8463334761216, 50779983373056, 304679900238336, 1828079250264576, 10968475501587456, 65810852102532096

Offset: 0

Marks R. Nester

A string and its reverse are considered to be equivalent. Thus aabc and cbaa are considered to be identical, but abca is a different string.

			For a(2)=21, the six achiral strings are AA, BB, CC, DD, EE, and FF; the 15 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, AF-FA, BC-CB, BD-DB, BE-EB, BF-FB, CD-DC, CE-EC, CF-FC, DE-ED, DF-FD, and EF-FE.

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]

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,6,-36).

Column 6 of A277504.

Cf. A000400 (oriented), A320524 (chiral), A056452 (achiral).

I:=[1,6,21]; [n le 3 select I[n] else 6*Self(n-1) +6*Self(n-2) - 36*Self(n-3): n in [1..30]]; // G. C. Greubel, Nov 10 2018

k=6; Table[(k^n+k^Ceiling[n/2])/2,{n,0,30}] (* Robert A. Russell, Nov 25 2017 *)
a[ n_] := (6^n + 6^Quotient[n + 1, 2]) / 2; (* Michael Somos, Jul 10 2018 *)
LinearRecurrence[{6, 6, -36}, {1, 6, 21}, 31] (* Robert A. Russell, Nov 10 2018 *)

Vec((1-21*x^2) / ((1 - 6*x)*(1 - 6*x^2)) + O(x^30)) \\ Colin Barker, Mar 20 2017 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]

{a(n) = (6^n + 6^((n+1)\2)) / 2}; \\ Michael Somos, Jul 10 2018

a(n) = (6^n + 6^floor((n+1)/2))/2.
G.f.: (1-21*x^2) / ((1-6*x)*(1-6*x^2)). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]
a(n) = 6*a(n-1) + 6*a(n-2) - 36*a(n-3) for n > 3. - Colin Barker, Mar 20 2017
a(n) = (A000400(n) + A056452(n)) / 2. - Robert A. Russell, Jun 19 2018
a(n) = 6^(n + floor((n-1)/2)) * a(1-n) for all n in Z. - Michael Somos, Jul 10 2018

a(0)=1 prepended by Robert A. Russell, Nov 10 2018

A321391 Array read by antidiagonals: T(n,k) is the number of achiral rows of n colors using up to k colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 4, 1, 0, 1, 6, 5, 16, 9, 8, 1, 0, 1, 7, 6, 25, 16, 27, 8, 1, 0, 1, 8, 7, 36, 25, 64, 27, 16, 1, 0, 1, 9, 8, 49, 36, 125, 64, 81, 16, 1, 0, 1, 10, 9, 64, 49, 216, 125, 256, 81, 32, 1, 0

Offset: 0

Robert A. Russell, Nov 08 2018

The antidiagonals go from top-right to bottom-left.

			The array begins with T(0,0):
1 1  1   1    1     1     1      1      1      1       1       1 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 ...
For T(3,3)=9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC.

Columns 0-6 are A000007, A000012, A060546, A056449, A056450, A056451, A056452.

Cf. A003992 (oriented), A277504 (unoriented), A293500 (chiral).

Table[If[n>0, (n-k)^Ceiling[k/2], 1], {n, 0, 12}, {k, 0, n}] // Flatten

T(n,k) = [n==0] + [n>0] * k^ceiling(n/2).

The generating function for column k is (1+k*x) / (1-k*x^2).

A168178 a(n) = n^3*(n^2 + 1)/2.

Original entry on oeis.org

0, 1, 20, 135, 544, 1625, 3996, 8575, 16640, 29889, 50500, 81191, 125280, 186745, 270284, 381375, 526336, 712385, 947700, 1241479, 1604000, 2046681, 2582140, 3224255, 3988224, 4890625, 5949476, 7184295, 8616160, 10267769, 12163500

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 5 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=20, there are 8 achiral (AAAAA, AABAA, ABABA, ABBBA, BAAAB, BABAB, BBABB, BBBBB) and 12 chiral pairs (AAAAB-BAAAA, AAABA-ABAAA, AAABB-BBAAA, AABAB-BABAA, AABBA-ABBAA, AABBB-BBBAA, ABAAB-BAABA, ABABB-BBABA, ABBAB-BABBA, ABBBB-BBBBA, BAABB-BBAAB, BABBB-BBBAB). - Robert A. Russell, Nov 14 2018

For n > 0, a(2n+1) is the number of non-isomorphic kC_m-snakes, where m = 2n+1 or m = 2n (for n>=2). A kC_n-snake is a connected graph in which the k>=2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 16 2019

C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60(2001), 85-96.

Vincenzo Librandi, Table of n, a(n) for n = 0..595
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A155977.
Row 5 of A277504.
Cf. A000584 (oriented), A000578 (achiral).

List([0..40], n -> n^3*(1 +n^2)/2); # G. C. Greubel, Nov 15 2018

[n^3*(n^2+1)/2: n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Table[(n^5+n^3)/2,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1},{0, 1, 20, 135, 544, 1625},40] (* Robert A. Russell, Nov 14 2018 *)

vector(40, n, n--; n^3*(1+n^2)/2) \\ G. C. Greubel, Nov 15 2018

[n^3*(1 + n^2)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

From Robert A. Russell, Nov 14 2018: (Start)
a(n) = (A000584(n) + A000578(n)) / 2 = (n^5 + n^3) / 2.
G.f.: (Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..3} S2(3,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..4} A145882(5,k) * x^k / (1-x)^6.
E.g.f.: (Sum_{k=1..5} S2(5,k)*x^k + Sum_{k=1..3} S2(3,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>5, a(n) = Sum_{j=1..6} -binomial(j-7,j) * a(n-j). (End)
From G. C. Greubel, Nov 15 2018: (Start)
G.f.: x*(1 + 14*x + 30*x^2 + 14*x^3 + x^4)/(1-x)^6.
E.g.f.: x*(2 + 18*x + 26*x^2 + 10*x^3 + x^4)*exp(x)/2. (End)

A071231 a(n) = (n^8 + n^4)/2.

Original entry on oeis.org

0, 1, 136, 3321, 32896, 195625, 840456, 2883601, 8390656, 21526641, 50005000, 107186761, 215001216, 407879641, 737913736, 1281470625, 2147516416, 3487920481, 5510032776, 8491846681, 12800080000, 18911526921, 27438053896, 39155632561, 55037822976

Offset: 0

N. J. A. Sloane, Jun 11 2002

Number of unoriented rows of length 8 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=136, there are 2^8=256 oriented arrangements of two colors. Of these, 2^4=16 are achiral. That leaves (256-16)/2=120 chiral pairs. Adding achiral and chiral, we get 136. - Robert A. Russell, Nov 13 2018

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row 8 of A277504.
Cf. A001016 (oriented), A000583 (achiral).
Cf. A000583, A001016.
Cf. A008277, A145882.
Subsequence of A000217 (triangular numbers).

List([0..50], n -> (n^4 + n^8)/2); # G. C. Greubel, Nov 15 2018

[(n^8 + n^4)/2: n in [0..50]]; // Vincenzo Librandi, Jun 14 2011

Table[(n^8+n^4)/2,{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126, -84,36,-9,1},{0,1,136,3321,32896,195625,840456,2883601,8390656},30] (* Harvey P. Dale, Oct 11 2011 *)

vector(50, n, n--; (n^8 + n^4)/2) \\ G. C. Greubel, Nov 15 2018

[(n^4 + n^8)/2 for n in range(50)] # G. C. Greubel, Nov 15 2018

From Harvey P. Dale, Oct 11 2011: (Start)
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9); a(0)=0, a(1)=1, a(2)=136, a(3)=3321, a(4)=32896, a(5)=195625, a(6)=840456, a(7)=2883601, a(8)=8390656.
G.f.: -((x*(x+1)*(x*(x*(x*(x*(x*(x+126)+2007)+5812)+2007)+126)+1))/ (x-1)^9). (End)
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A001016(n) + A000583(n)) / 2 = (n^8 + n^4) / 2.
G.f.: (Sum_{j=1..8} S2(8,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..7} A145882(8,k) * x^k / (1-x)^9.
E.g.f.: (Sum_{k=1..8} S2(8,k)*x^k + Sum_{k=1..4} S2(4,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>8, a(n) = Sum_{j=1..9} -binomial(j-10,j) * a(n-j). (End)
E.g.f.: x*(2 + 134*x + 972*x^2 + 1702*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7)*exp(x)/2. - G. C. Greubel, Nov 15 2018

A071232 a(n) = (n^6 + n^3)/2.

Original entry on oeis.org

0, 1, 36, 378, 2080, 7875, 23436, 58996, 131328, 266085, 500500, 886446, 1493856, 2414503, 3766140, 5697000, 8390656, 12071241, 17009028, 23526370, 32004000, 42887691, 56695276, 74024028, 95558400, 122078125, 154466676, 193720086, 240956128, 297423855, 364513500

Offset: 0

N. J. A. Sloane, Jun 11 2002

Number of unoriented rows of length 6 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=36, there are 2^6=64 oriented arrangements of two colors. Of these, 2^3=8 are achiral. That leaves (64-8)/2=28 chiral pairs. Adding achiral and chiral, we get 36. - Robert A. Russell, Nov 14 2018

For n > 0, a(2n+1) is the number of non-isomorphic 8C_m-snakes, where m = 2n+1 or m = 2n (for n>=2). A kC_n-snake is a connected graph in which the k >= 2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 16 2019

C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Row 6 of A277504.

Cf. A001014 (oriented), A085744 (chiral), A000578 (achiral).

List([0..50], n -> (n^6 + n^3)/2); # G. C. Greubel, Nov 15 2018

[(n^6 + n^3)/2: n in [0..50]]; // Vincenzo Librandi, Jun 14 2011

Table[(n^6+n^3)/2,{n,0,40}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,36,378,2080,7875,23436},40] (* Harvey P. Dale, Nov 06 2011 *)

vector(50, n, n--; (n^6 + n^3)/2) \\ G. C. Greubel, Nov 15 2018

[(n^6 + n^3)/2 for n in range(50)] # G. C. Greubel, Nov 15 2018

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=0, a(1)=1, a(2)=36, a(3)=378, a(4)=2080, a(5)=7875, a(6)=23436. - Harvey P. Dale, Nov 06 2011
G.f.: x*(28*x^4 + 155*x^3 + 147*x^2 + 29*x + 1)/(1-x)^7. - Colin Barker, Oct 12 2012
From Robert A. Russell, Nov 14 2018: (Start)
a(n) = (A001014(n) + A000578(n)) / 2 = (n^6 + n^3) / 2.
a(n) = A001014(n) - A085744(n) = A085744(n) + A000578(n).
G.f.: (Sum_{j=1..6} S2(6,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..3} S2(3,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..5} A145882(6,k) * x^k / (1-x)^7.
E.g.f.: (Sum_{k=1..6} S2(6,k)*x^k + Sum_{k=1..3} S2(3,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>6, a(n) = Sum_{j=1..7} -binomial(j-8,j) * a(n-j). (End)
E.g.f.: x*(2 +34*x +91*x^2 +65*x^3 +15*x^4 +x^5)*exp(x)/2. - G. C. Greubel, Nov 15 2018
a(n) = A000217(n^3), sum of the integers up to the n'th cube. - R. J. Mathar, Mar 11 2025

cp's OEIS Frontend

A032122 Number of reversible strings with n beads of 5 colors.

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A170801 a(n) = n^10*(n^9 + 1)/2.

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A170802 a(n) = n^10*(n^10 + 1)/2.

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A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.

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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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A056308 Number of reversible strings with n beads using a maximum of six different colors.

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