A058187 -id:A058187 - cp's OEIS frontend

A189976 a(n) is the number of incongruent two-color bracelets of n beads, 8 of them black (A005514), having a diameter of symmetry.

1, 1, 5, 5, 15, 15, 35, 35, 70, 70, 126, 126, 210, 210, 330, 330, 495, 495, 715, 715, 1001, 1001, 1365, 1365, 1820, 1820, 2380, 2380, 3060, 3060, 3876, 3876, 4845, 4845, 5985, 5985, 7315, 7315, 8855, 8855, 10626

Offset: 8

Vladimir Shevelev, May 03 2011

For n >= 9, a(n-1) is the number of incongruent two-color bracelets of n beads, 9 from them are black (A032281), having a diameter of symmetry.

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A005514, A032281, A008805, A058187.

[Binomial(Floor(n/2),4): n in[8..60]]; // Vincenzo Librandi, Aug 10 2014

A189976 :=proc(n): binomial(floor(n/2),4) end: seq(A189976(n), n=8..48); # Johannes W. Meijer, Aug 15 2011

Module[{c=Binomial[Range[4,30],4]},Riffle[c,c]] (* Harvey P. Dale, Aug 09 2014 *)
Table[(Binomial[Floor[n/2], 4]), {n, 8, 40}] (* Vincenzo Librandi, Aug 10 2014 *)

a(n) = C(floor(n/2),4).
a(n+5) = A194005(n,n-4). [Johannes W. Meijer, Aug 15 2011]
G.f.: -x^8/((x-1)^5*(x+1)^4). [Colin Barker, Feb 06 2013]

Data added and link corrected by Johannes W. Meijer, Aug 15 2011

A190717 Triplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 10, 10, 10, 20, 20, 20, 35, 35, 35, 56, 56, 56, 84, 84, 84, 120, 120, 120, 165, 165, 165, 220, 220, 220, 286, 286, 286, 364, 364, 364, 455, 455, 455, 560, 560, 560, 680, 680, 680, 816, 816, 816, 969, 969, 969

Offset: 0

Johannes W. Meijer, May 18 2011

The Ca1 and Ze3 triangle sums, see A180662 for their definitions, of the triangle A159797 are linear sums of shifted versions of the triplicated tetrahedral numbers, e.g. Ca1(n) = a(n-1) + a(n-2) + 2*a(n-3) + a(n-6).

The Ca1, Ca2, Ze3 and Ze4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

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

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), this sequence (triplicated), A190718 (quadruplicated), A049347, A144677.

A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A190717(n),n=0..50);

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,1,1,4,4,4,10,10,10,20},60] (* Harvey P. Dale, Mar 09 2018 *)

a(n) = binomial(floor(n/3)+3,3).
a(n) + a(n-1) + a(n-2) = A144677(n).
a(n) = Sum_{k=0..n} (A144677(n-k)*A049347(k)).
G.f.: 1/((x-1)^4*(x^2+x+1)^3).
Sum_{n>=0} 1/a(n) = 9/2. - Amiram Eldar, Aug 18 2022

A190718 Quadruplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 4, 4, 10, 10, 10, 10, 20, 20, 20, 20, 35, 35, 35, 35, 56, 56, 56, 56, 84, 84, 84, 84, 120, 120, 120, 120, 165, 165, 165, 165, 220, 220, 220, 220, 286, 286, 286, 286, 364, 364, 364, 364, 455, 455, 455, 455

Offset: 0

Johannes W. Meijer, May 18 2011

The Gi1 triangle sums, for the definitions of these and other triangle sums see A180662, of the triangle A159797 are linear sums of shifted versions of the quadruplicated tetrahedral numbers A000292, i.e., Gi1(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-8).

The Gi1 and Gi2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

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

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), A190717 (triplicated).

A190718:= proc(n) binomial(floor(n/4)+3,3) end:
seq(A190718(n),n=0..52);

LinearRecurrence[{1,0,0,3,-3,0,0,-3,3,0,0,1,-1},{1,1,1,1,4,4,4,4,10,10,10,10,20},60] (* Harvey P. Dale, Oct 20 2012 *)

a(n) = binomial(floor(n/4)+3,3).
a(n-3) + a(n-2) + a(n-1) + a(n) = A144678(n).
a(n) = +a(n-1) +3*a(n-4) -3*a(n-5) -3*a(n-8) +3*a(n-9) +a(n-12) -a(n-13).
G.f.: 1 / ( (1+x)^3*(1+x^2)^3*(x-1)^4 ).
Sum_{n>=0} 1/a(n) = 6. - Amiram Eldar, Aug 18 2022

A023857 a(n) = 1(n+3-1) + 2(n+3-2) + .... + k*(n+3-k), where k=floor((n+1)/2).

Original entry on oeis.org

3, 4, 13, 16, 34, 40, 70, 80, 125, 140, 203, 224, 308, 336, 444, 480, 615, 660, 825, 880, 1078, 1144, 1378, 1456, 1729, 1820, 2135, 2240, 2600, 2720, 3128, 3264, 3723, 3876, 4389, 4560, 5130, 5320, 5950, 6160, 6853, 7084, 7843, 8096, 8924, 9200, 10100, 10400, 11375, 11700

Offset: 1

Clark Kimberling

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

Cf. A023855, A023856, A024305, A024854.

List([1..60], n-> (4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48) # G. C. Greubel, Jun 12 2019

[(4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48: n in [1..60]]; // G. C. Greubel, Jun 12 2019

seq(sum(i*(n-i+3), i=1..ceil(n/2)), n=1..60); # Wesley Ivan Hurt, Sep 20 2013

Table[-Ceiling[n/2]*(Ceiling[n/2]+1)*(2*Ceiling[n/2]-3n-8)/6, {n,60}] (* Wesley Ivan Hurt, Sep 20 2013 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{3,4,13,16,34,40,70},60] (* Harvey P. Dale, Feb 13 2018 *)

a(n) = (4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48; \\ G. C. Greubel, Jun 12 2019

[(4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48 for n in (1..60)] # G. C. Greubel, Jun 12 2019

a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+3) = -ceiling(n/2)*(ceiling(n/2)+1)*(2*ceiling(n/2) - 3*n - 8)/6. - Wesley Ivan Hurt, Sep 20 2013
G.f. x*(3+x) / ( (1+x)^3*(1-x)^4 ). - R. J. Mathar, Sep 25 2013
a(n) = 3*A058187(n-1) + A058187(n-2). - R. J. Mathar, Sep 25 2013
a(n) = (4*n^3 + 27*n^2 + 50*n + 21 - 3*(n^2 + 6*n + 7)*(-1)^n)/48. - Luce ETIENNE, Nov 21 2014
E.g.f.: (x*(51 + 18*x + 2*x^2)*cosh(x) + (21 + 30*x + 21*x^2 + 2*x^3)*sinh(x))/24. - G. C. Greubel, Jun 12 2019

Title simplified by Sean A. Irvine, Jun 12 2019

A024854 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (natural numbers), t = (natural numbers >= 3).

Original entry on oeis.org

4, 5, 16, 19, 40, 46, 80, 90, 140, 155, 224, 245, 336, 364, 480, 516, 660, 705, 880, 935, 1144, 1210, 1456, 1534, 1820, 1911, 2240, 2345, 2720, 2840, 3264, 3400, 3876, 4029, 4560, 4731, 5320, 5510, 6160, 6370, 7084, 7315, 8096, 8349, 9200, 9476, 10400, 10700, 11700

Offset: 2

Clark Kimberling

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

Cf. A023855, A023856, A023857, A024305, A058187.

[(4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48: n in [2..60]]; // Vincenzo Librandi, Oct 31 2014

seq(sum(i*(k-i+3), i=1..floor(k/2)), k=2..70); # Wesley Ivan Hurt, Sep 20 2013

Table[-Floor[n/2] * (Floor[n/2] + 1) * (2 * Floor[n/2] - 3n - 8)/6, {n, 2, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)
CoefficientList[Series[- (- 4 - x + x^2)/((1 + x)^3 (x - 1)^4), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 31 2014 *)

[(4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48 for n in (2..60)] # G. C. Greubel, Jul 13 2022

a(n) = Sum_{i=1..floor(n/2)} i*(n-i+3) = -floor(n/2)*(floor(n/2)+1)*(2*floor(n/2)-3n-8)/6. - Wesley Ivan Hurt, Sep 20 2013
G.f. x^2*(4 + x - x^2) / ( (1+x)^3*(1-x)^4 ). - R. J. Mathar, Sep 25 2013
a(n) = 4*A058187(n-2) + A058187(n-3) - A058187(n-4). - R. J. Mathar, Sep 25 2013
a(n) = (4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48. - Luce ETIENNE, Nov 14 2014
E.g.f.: (1/24)*( x*(9 + 18*x + 2*x^2)*cosh(x) + (-9 + 30*x + 15*x^2 + 2*x^3)*sinh(x) ). - G. C. Greubel, Jul 13 2022

A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 4, 1, 1, 0, 4, 2, 1, 10, 4, 5, 3, 1, 0, 10, 8, 7, 4, 1, 20, 10, 14, 13, 10, 5, 1, 0, 20, 20, 22, 20, 14, 6, 1, 35, 20, 30, 34, 35, 30, 19, 7, 1, 0, 35, 40, 50, 56, 55, 44, 25, 8, 1, 56, 35, 55, 70, 84, 91, 85, 63, 32, 9, 1, 0, 56, 70, 95, 120, 140, 146, 129, 88, 40, 10, 1

Offset: 0

Henry Bottomley, Nov 20 2000

			Rows are (1,0,4,0,10,0,20,...), (1,1,4,4,10,10,20,...), (1,2,5,8,14,20,30,...), (1,3,7,13,22,34,50,...), (1,4,10,20,35,56,84,...) etc.

Rows are A000292 with zeros, A058187 (A000292 with terms duplicated), A006918, A002623, A000292, A000330, A005900, A001845, A008412.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(4, 1)=4, T(0, 2n)=T(4, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^4.

A189980 a(n) is the number of incongruent two-color bracelets of n beads, 10 from them are black (A005515), having a diameter of symmetry.

Original entry on oeis.org

1, 1, 6, 6, 21, 21, 56, 56, 126, 126, 252, 252, 462, 462, 792, 792, 1287, 1287, 2002, 2002, 3003, 3003, 4368, 4368, 6188, 6188, 8568, 8568, 11628, 11628, 15504, 15504, 20349, 20349, 26334, 26334, 33649, 33649

Offset: 10

Vladimir Shevelev, May 03 2011

For n >= 11, a(n-1) is the number of incongruent two-color bracelets of n beads, 11 from them are black (A032282), having a diameter of symmetry.

Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Cf. A005515, A032282, A008805, A058187, A189976.

A189980 :=proc(n): binomial(floor(n/2),5) end: seq(A189980(n), n=10..47); # Johannes W. Meijer, Aug 15 2011

Table[Binomial[Floor[n/2],5],{n,10,50}] (* Harvey P. Dale, Oct 06 2017 *)

a(n) = binomial(floor(n/2), 5). [Typo fixed by Colin Barker, Feb 07 2013]
a(n+6) = A194005(n, n-5). - Johannes W. Meijer, Aug 15 2011
G.f.: x^10/((x-1)^6*(x+1)^5). - Colin Barker, Feb 07 2013

Data added and link corrected by Johannes W. Meijer, Aug 15 2011

A157898 Triangle read by rows: inverse binomial transform of A059576.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 2, 2, 4, 1, 2, 6, 4, 8, 0, 3, 6, 16, 8, 16, 1, 3, 12, 16, 40, 16, 32, 0, 4, 12, 40, 40, 96, 32, 64, 1, 4, 20, 40, 120, 96, 224, 64, 128, 0, 5, 20, 80, 120, 336, 224, 512, 128, 256

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 08 2009

The inverse binomial transform of the triangle A059576 is given by multiplying the triangle with A130595 from the left.

			First few rows of the triangle =
  1;
  0, 1;
  1, 1,  2;
  0, 2,  2,  4;
  1, 2,  6,  4,   8;
  0, 3,  6, 16,   8,  16;
  1, 3, 12, 16,  40,  16,  32;
  0, 4, 12, 40,  40,  96,  32,  64;
  1, 4, 20, 40, 120,  96, 224,  64, 128;
  0, 5, 20, 80, 120, 336, 224, 512, 128, 256;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A059576, A097076 (row sums), A130595.

Cf. A004526, A008805, A011782, A058187, A059841, A189976.

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*t(j,k): j in [k..n]]) >;
[A157898(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 03 2022

A059576 := proc (n, k)
    if n = 0 then
        return 1;
    end if;
    if k <= n and k >= 0 then
        add((-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k), j = 0 .. min(k, n-k))
    else
        0 ;
    end if
end proc:
A157898 := proc(n,k)
    add ( A130595(n,j)*A059576(j,k),j=k..n) ;
end proc: # R. J. Mathar, Feb 13 2013

t[n_, k_]:= t[n, k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1, k-1] + 2*t[n-1, k] -(2 -Boole[n==2])*t[n-2, k-1]]; (* t= A059576 *)
A157898[n_, k_]:= A157898[n, k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
Table[A157898[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 03 2022 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n,k): return sum((-1)^(n-j)*binomial(n,j)*t(j,k) for j in (k..n))
flatten([[A157898(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 03 2022

Sum_{k=0..n} T(n, k) = A097076(n+1).
From G. C. Greubel, Sep 03 2022: (Start)
T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(n,j)*A059576(j,k).
T(n, 0) = A059841(n).
T(n, 1) = A004526(n-1).
T(n, 2) = 2*A008805(n-2).
T(n, 3) = 4*A058187(n-3).
T(n, 4) = 8*A189976(n+4).
T(n, n) = A011782(n).
T(n, n-1) = A011782(n) - [n==0]. (End)

A176415 Periodic sequence: repeat 7,1.

Original entry on oeis.org

7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7

Offset: 0

Klaus Brockhaus, Apr 17 2010

Interleaving of A010727 and A000012.
Also continued fraction expansion of (7+sqrt(77))/2.
Also decimal expansion of 71/99.
Essentially first differences of A047521.
Binomial transform of A176414.
Inverse binomial transform of 2*A020707 preceded by 7.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 4*x^3 + 10*x^4 + 10*x^5 + ... is the o.g.f. for A058187. - Peter Bala, Mar 13 2015

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

Cf. A010727 (all 7's sequence), A000012 (all 1's sequence), A092290 (decimal expansion of (7+sqrt(77))/2), A010688 (repeat 1, 7), A047521 (congruent to 0 or 7 mod 8), A176414 (expansion of (7+8*x)/(1+2*x)), A020707 (2^(n+2)), A058187.

&cat[ [7, 1]: n in [0..52] ];
[ 4+3*(-1)^n: n in [0..104] ];

PadRight[{},120,{7,1}] (* Harvey P. Dale, Dec 30 2018 *)

a(n)=7-n%2*6 \\ Charles R Greathouse IV, Oct 28 2011

a(n) = 4+3*(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 7, a(1) = 1.
a(n) = -a(n-1)+8 for n > 0; a(0) = 7.
a(n) = 7*((n+1) mod 2)+(n mod 2).
a(n) = A010688(n+1).
G.f.: (7+x)/(1-x^2).
Dirichglet g.f.: (1+6*2^(-s))*zeta(s). - R. J. Mathar, Apr 06 2011
Multiplicative with a(2^e) = 7, and a(p^e) = 1 for p >= 3. - Amiram Eldar, Jan 01 2023

A247976 Triangle read by rows: T(n,k) generated by m-gon expansions in the case of odd m with "vertex to vertex" version or even m with "vertex to side" version. (See comment for details.)

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7

Offset: 1

Kival Ngaokrajang, Sep 28 2014

Refer to triangle expansions in A061777 and A101946 (and their companions for m-gons) which are "vertex to vertex" and "vertex to side" versions respectively. The label values at each iteration can be arranged as triangle. Any m-gon can also be arranged as the same triangle with conditions: (i) m is odd and expansion is "vertex to vertex" version or (ii) m is even and expansion is "vertex to side" version. m*Sum_{i=1..k}T(n,k) gives the total label value in n-th iteration. See illustration.

			Triangle begins:
  1;
  1,  1;
  1,  1,  2;
  1,  2,  1,  2;
  1,  2,  1,  3,  3;
  1,  3,  3,  1,  3,  3;
  1,  3,  3,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  5, 10, 10,  5;
  1,  5, 10, 10,  5,  1,  5, 10, 10, 5;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Kival Ngaokrajang, Illustration of initial terms

Rows sum: A027383.
Column (start from 1s): c3=A008805, c4=A058187, c5=A000332 repeated, c6=A000389 repeated, c7=A000579 repeated.
Vertex to vertex: A061777, A247618, A247619, A247620.
Vertex to side: A101946, A247903, A247904, A247905.
Cf. A074909.

T[n_, k_]:= T[n, k]= If[k==1, 1, If[k==n, Floor[(n+1)/2], If[OddQ[n], If[k<=(n+ 1)/2, T[n-1, k], T[n-1, k-1] + T[n-1, k]], If[kG. C. Greubel, Feb 18 2022 *)

@CachedFunction
def T(n,k): # A247976
    if (k==1): return 1
    elif (k==n): return (n+1)//2
    elif (n%2==1): return T(n-1,k) if (k <= (n+1)/2) else T(n-1,k-1) + T(n-1,k)
    else: return T(n-1,k-1)+T(n-1,k) if (k < (n+2)/2) else T(n,k-n/2)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Feb 18 2022

T(n, k) = ( T(n-1, k) if k <= (n+1)/2 otherwise T(n-1, k-1) + T(n-1, k) ) for odd n rows, ( T(n-1, k-1) + T(n-1, k) if k < (n+2)/2 otherwise T(n, k - n/2) ) for even n rows, with T(n, 1) = 1 and T(n, n) = floor((n+1)/2). - G. C. Greubel, Feb 18 2022

cp's OEIS Frontend

A189976 a(n) is the number of incongruent two-color bracelets of n beads, 8 of them black (A005514), having a diameter of symmetry.

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A190717 Triplicated tetrahedral numbers A000292.

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A190718 Quadruplicated tetrahedral numbers A000292.

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A023857 a(n) = 1*(n+3-1) + 2*(n+3-2) + .... + k*(n+3-k), where k=floor((n+1)/2).

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A024854 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (natural numbers), t = (natural numbers >= 3).

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A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

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A189980 a(n) is the number of incongruent two-color bracelets of n beads, 10 from them are black (A005515), having a diameter of symmetry.

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A023857 a(n) = 1(n+3-1) + 2(n+3-2) + .... + k*(n+3-k), where k=floor((n+1)/2).

A024854 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (natural numbers), t = (natural numbers >= 3).