A008747 -id:A008747 - cp's OEIS frontend

A284249 Number T(n,k) of k-element subsets of [n] whose sum is a triangular number; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 3, 4, 5, 3, 1, 1, 1, 3, 5, 8, 6, 4, 1, 1, 1, 3, 7, 12, 11, 9, 4, 1, 1, 1, 3, 9, 16, 20, 18, 11, 5, 1, 1, 1, 4, 10, 22, 32, 35, 26, 14, 5, 1, 1, 1, 4, 12, 29, 48, 61, 55, 36, 17, 6, 1, 1, 1, 4, 14, 37, 70, 100, 106, 84, 48, 21, 6, 1, 1

Offset: 0

Alois P. Heinz, Mar 23 2017

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 2,  3,  3,  1,   1;
  1, 3,  4,  5,  3,   1,   1;
  1, 3,  5,  8,  6,   4,   1,  1;
  1, 3,  7, 12, 11,   9,   4,  1,  1;
  1, 3,  9, 16, 20,  18,  11,  5,  1,  1;
  1, 4, 10, 22, 32,  35,  26, 14,  5,  1, 1;
  1, 4, 12, 29, 48,  61,  55, 36, 17,  6, 1, 1;
  1, 4, 14, 37, 70, 100, 106, 84, 48, 21, 6, 1, 1;

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Triangular number

Columns k=0-10 give: A000012, A003056, A320848, A320849, A320850, A320851, A320852, A320853, A320854, A320855, A320856.
Second and third lower diagonals give: A008619(n+1), A008747(n+1).
Row sums give A284250.
T(2n,n) gives A284251.
Cf. A000217, A281871.

b:= proc(n, s) option remember; expand(`if`(n=0,
      `if`(issqr(8*s+1), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..16);

b[n_, s_] := b[n, s] = Expand[If[n == 0, If[IntegerQ @ Sqrt[8*s + 1], 1, 0], b[n - 1, s] + x*b[n - 1, s + n]]];
T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 16}] // Flatten (*Jean-François Alcover, May 29 2018, from Maple *)

A277619 Number of aperiodic necklaces (Lyndon words) with k<=4 black beads and n-k white beads.

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 8, 14, 19, 28, 37, 51, 64, 84, 103, 129, 155, 189, 222, 265, 307, 359, 411, 474, 536, 611, 685, 772, 859, 960, 1060, 1176, 1291, 1422, 1553, 1701, 1848, 2014, 2179, 2363, 2547, 2751, 2954, 3179, 3403, 3649, 3895, 4164, 4432, 4725, 5017

Offset: 0

Herbert Kociemba, Oct 24 2016

			a(6)=8. The aperiodic necklaces are BWWWWW, BBWWWW, BWBWWW, BBBWWW, BBWBWW, BBWWBW, BBBBWW, and BBBWBW.

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

Cf. A001037 (k arbitrary), A008747 (k<=3).

Mathematica section of A032168 gives g.f. for k=m black beads and n-k white beads.

(* The g.f. for the number of aperiodic necklaces (Lyndon words) with k<=m black beads and n-k white beads is *)
gf[x_,m_]:=Sum[x^i/i Plus@@(MoebiusMu[#](1-x^#)^(-(i/#))&/@Divisors[i]),{i,1,m}]+x+1
(* Here we have the case m=4 *)

Vec((1+x-3*x^2-2*x^3+3*x^4+5*x^5-3*x^7+x^9)/((-1+x)^4*(1+x)^2*(1+x+x^2)) + O(x^60)) \\ Colin Barker, Oct 29 2016

G.f.: (1+x-3*x^2-2*x^3+3*x^4+5*x^5-3*x^7+x^9)/((-1+x)^4*(1+x)^2*(1+x+x^2)).

a(n) = a(n-1)+2*a(n-2)-a(n-3)-2*a(n-4)-a(n-5)+2*a(n-6)+a(n-7)-a(n-8) for n>7. - Colin Barker, Oct 29 2016

A277629 Number of aperiodic necklaces (Lyndon words) with k<=5 black beads and n-k white beads.

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 9, 17, 26, 42, 62, 93, 130, 183, 246, 329, 428, 553, 698, 877, 1082, 1328, 1608, 1937, 2307, 2736, 3215, 3762, 4369, 5055, 5810, 6657, 7584, 8614, 9737, 10976, 12320, 13795, 15388, 17126, 18997, 21029, 23208, 25565, 28085, 30799, 33694, 36801, 40105, 43641, 47392

Offset: 0

Herbert Kociemba, Oct 24 2016

			a(5)=6: The aperiodic necklaces are BWWWW, BBWWW, BWBWW, BBBWW, BBWBW, BBBBW.

Index entries for linear recurrences with constant coefficients, signature (1, 2, -1, -2, 0, 1, -1, 0, 2, 1, -2, -1, 1).

Cf. A001037 (k arbitrary), A008747 (k<=3), A277619 (k<=4). Mathematica section of A032168 gives g.f. for k=m black beads and n-k white beads.

(* The g.f. for the number of aperiodic necklaces (Lyndon words) with k<=m black beads and n-k white beads is *)
gf[x_, m_]:=Sum[x^i/i Plus@@(MoebiusMu[#](1-x^#)^(-(i/#))&/@Divisors[i]), {i, 1, m}]+x+1
(* Here we have the case m=5 *)

G.f.: (-1 - x + 3*x^2 + 2*x^3 - 3*x^4 - 4*x^5 - 2*x^7 - 4*x^8 + 3*x^10 - 2*x^11 - 4*x^12 + x^14)/( (-1+x)^5*(1+x)^2*(1+x+x^2)*(1+x+x^2+x^3+x^4) ).

A008751 Expansion of (1+x^8)/((1-x)(1-x^2)(1-x^3)).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 19, 23, 26, 31, 35, 40, 45, 51, 56, 63, 69, 76, 83, 91, 98, 107, 115, 124, 133, 143, 152, 163, 173, 184, 195, 207, 218, 231, 243, 256, 269, 283, 296, 311, 325, 340, 355

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A008747.

Concatenation([1,1,2], List([3..50], n-> Int(((n-1)^2 +17)/6))); # G. C. Greubel, Aug 04 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x^8)/((1-x)*(1-x^2)*(1-x^3)) )); // G. C. Greubel, Aug 04 2019

CoefficientList[Series[(1+x^8)/((1-x)(1-x^2)(1-x^3)),{x,0,50}],x] (* Vincenzo Librandi, Feb 25 2012 *)
Join[{1,1,2}, Floor[((Range[3, 50] -1)^2 +17)/6]] (* G. C. Greubel, Aug 04 2019 *)

my(x='x+O('x^50)); Vec((1+x^8)/((1-x)*(1-x^2)*(1-x^3))) \\ G. C. Greubel, Aug 04 2019

((1+x^8)/((1-x)*(1-x^2)*(1-x^3))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Aug 04 2019

From Henry Bottomley, Sep 05 2000: (Start)
a(n) = floor((n^2 - 2*n + 18)/6) for n>2.
a(n) = a(n-2) + a(n-3) - a(n-5) + 2.
a(n) = A001399(n) + A001399(n-8).
a(n) = A008747(n-2) + 2 for n>2. (End)

A175812 Partial sums of ceiling(n^2/6).

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 18, 27, 38, 52, 69, 90, 114, 143, 176, 214, 257, 306, 360, 421, 488, 562, 643, 732, 828, 933, 1046, 1168, 1299, 1440, 1590, 1751, 1922, 2104, 2297, 2502, 2718, 2947, 3188, 3442, 3709, 3990, 4284, 4593, 4916, 5254, 5607, 5976, 6360, 6761, 7178

Offset: 0

Mircea Merca, Dec 05 2010

Partial sums of A008747.

There are several sequences of integers of the form ceiling(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,8,10,11,12, 14,15,16,19,20,23,24).

			a(6) = 0 + 1 + 1 + 2 + 3 + 5 + 6 = 18.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

Cf. A008747.

[Round((2*n+1)*(2*n^2+2*n+17)/72): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

seq(floor((n+1)*(2*n^2+n+17)/36),n=0..50)

Accumulate[Ceiling[Range[0,50]^2/6]] (* Harvey P. Dale, Jan 17 2016 *)

a(n) = (n+1)*(2*n^2+n+17)\36; \\ Altug Alkan, Sep 21 2018

a(n) = round((2*n+1)*(2*n^2 + 2*n + 17)/72).
a(n) = floor((n+1)*(2*n^2 + n + 17)/36).
a(n) = ceiling((2*n^3 + 3*n^2 + 18*n)/36).
a(n) = round((2*n^3 + 3*n^2 + 18*n)/36).
a(n) = a(n-6) + (n+1)*(n-6) + 18, n > 5.
From Mircea Merca, Jan 09 2011: (Start)
a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7), n > 6.
G.f.: x*(x^4+1) / ( (x+1)*(x^2+x+1)*(x-1)^4 ). (End)

A277631 Number of aperiodic necklaces (Lyndon words) with k<=6 black beads and n-k white beads.

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 9, 18, 29, 51, 82, 135, 205, 315, 458, 662, 925, 1281, 1724, 2305, 3014, 3911, 4992, 6326, 7905, 9820, 12059, 14724, 17811, 21435, 25586, 30408, 35885, 42175, 49273, 57352, 66401, 76627, 88012, 100781, 114928, 130697, 148074, 167343, 188483, 211798, 237282, 265260, 295717, 329025, 365160

Offset: 0

Herbert Kociemba, Oct 24 2016

			a(6)=9. The aperiodic necklaces are BWWWWW, BBWWWW, BWBWWW, BBBWWW, BBWBWW, BBWWBW, BBBBWW, BBBWBW and BBBBBW.

Index entries for linear recurrences with constant coefficients, signature (1, 2, -1, -2, 0, 2, -2, -2, 3, 3, -2, -2, 2, 0, -2, -1, 2, 1, -1).

Cf. A001037 (k arbitrary), A008747 (k<=3), A277619 (k<=4), A277629 (k<=5). Mathematica section of A032168 gives g.f. for k=m black beads and n-k white beads.

(* The g.f. for the number of aperiodic necklaces (Lyndon words) with k<=m black beads and n-k white beads is *)
gf[x_, m_]:=Sum[x^i/i Plus@@(MoebiusMu[#](1-x^#)^(-(i/#))&/@Divisors[i]), {i, 1, m}]+x+1
(* Here we have the case m=6 *)

G.f.: (1 + x - 3*x^2 - 2*x^3 + 3*x^4 + 4*x^5-x^6 + 2*x^7 + 9*x^8 + 6*x^9 + 7*x^11 + 12*x^12 + 7*x^13 + 3*x^14 + 6*x^15 + 6*x^16 + x^17-3*x^18 + x^20)/( (-1+x)^6*(1+x)^3*(1-x+x^2)*(1+x+x^2)^2*(1+x+x^2+x^3+x^4) ).

A277633 Number of aperiodic necklaces (Lyndon words) with k<=8 black beads and n-k white beads.

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 98, 180, 311, 546, 915, 1520, 2440, 3855, 5916, 8935, 13178, 19138, 27264, 38303, 52950, 72311, 97419, 129839, 171066, 223260, 288498, 369708, 469708, 592363, 741433, 921933, 1138761, 1398343, 1706956, 2072696, 2503513, 3009482, 3600515, 4289032, 5087253, 6010305, 7073122, 8293962

Offset: 0

Herbert Kociemba, Oct 24 2016

Index entries for linear recurrences with constant coefficients, signature (1, 3, -2, -4, 1, 4, -1, -5, 2, 7, -1, -6, 0, 4, 0, -6, -1, 7, 2, -5, -1, 4, 1, -4, -2, 3, 1, -1).

Cf. A001037 (k arbitrary), A008747 (k<=3), A277619 (k<=4), A277629 (k<=5), A277631 (k<=6).

The Mathematica section of A032168 gives the g.f. for k=m black beads and n-k white beads.

(* The g.f. for the number of aperiodic necklaces (Lyndon words) with k<=m black beads and n-k white beads. Here we have the case m=8 *)
gf[x_, m_]:=Sum[x^i/i Plus@@(MoebiusMu[#](1-x^#)^(-(i/#))&/@Divisors[i]), {i, 1, m}]+x+1

G.f.: 1 + x + x/(1-x) + 1/2*x^2*(1/(1-x)^2 - 1/(1-x^2)) + 1/3*x^3*(1/(1-x)^3 - 1/(1-x^3)) + 1/4*x^4*(1/(1-x)^4 - 1/(1-x^2)^2) + 1/5*x^5*(1/(1-x)^5 - 1/(1-x^5)) + 1/6*x^6*(1/(1-x)^6 - 1/(1-x^2)^3 - 1/(1-x^3)^2 + 1/(1-x^6)) + 1/7*x^7*(1/(1-x)^7 - 1/(1-x^7)) + 1/8*x^8*(1/(1-x)^8 - 1/(1-x^2)^4).

cp's OEIS Frontend

A284249 Number T(n,k) of k-element subsets of [n] whose sum is a triangular number; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A277619 Number of aperiodic necklaces (Lyndon words) with k<=4 black beads and n-k white beads.

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A277629 Number of aperiodic necklaces (Lyndon words) with k<=5 black beads and n-k white beads.

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A008751 Expansion of (1+x^8)/((1-x)*(1-x^2)*(1-x^3)).

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A175812 Partial sums of ceiling(n^2/6).

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A277631 Number of aperiodic necklaces (Lyndon words) with k<=6 black beads and n-k white beads.

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Mathematica

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A277633 Number of aperiodic necklaces (Lyndon words) with k<=8 black beads and n-k white beads.

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Mathematica

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A008751 Expansion of (1+x^8)/((1-x)(1-x^2)(1-x^3)).