cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-26 of 26 results.

A162940 a(n) = binomial(n+1,2)*6^2.

Original entry on oeis.org

0, 36, 108, 216, 360, 540, 756, 1008, 1296, 1620, 1980, 2376, 2808, 3276, 3780, 4320, 4896, 5508, 6156, 6840, 7560, 8316, 9108, 9936, 10800, 11700, 12636, 13608, 14616, 15660, 16740, 17856, 19008, 20196, 21420, 22680, 23976, 25308, 26676, 28080, 29520, 30996
Offset: 0

Views

Author

Zerinvary Lajos, Jul 18 2009, Jul 19 2009

Keywords

Comments

Number of n permutations (n>=2) of 7 objects s, t, u, v, z, x, y with repetition allowed, containing n-2 u's. Example: If n=2 then n-2 = zero (0) u, a(1)=36 because we have ss, st, sv, sz, sx, sy, ts, tt, tv, tz, tx, ty, vs, vt, vv, vz, vx, vy, zs, zt, zv, zz, zx, zy, xs, xt, xv, xz, xx, xy, ys, yt, yv, yz, yx, yy. If n=3 then n-2 = one (1) u, a(2) = 108, >> ssu, stu, etc. If n=4 then n-2 = two (2) u, a(2)= 216, >> ssuu, stuu, ..., txuu, etc. If n=5 then n-2 = three (3) u, a(3)=360, >> ssuuu, stuuu, ..., txuuu, etc.

Links

Crossrefs

Cf. A000217, A002378, A046092, A027468, A035008, A123296.
Cf. A049598, A163758.

Programs

  • Mathematica
    Table[Binomial[n + 1, 2]*6^2, {n, 0, 58}]
  • PARI
    a(n)=18*n*(n+1) \\ Charles R Greathouse IV, Jun 16 2017

Formula

From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/9 - 1/18. (End)
From Amiram Eldar, Feb 22 2023: (Start)
a(n) = 18*n*(n+1) = 36*A000217(n) = 18*A002378(n).
Product_{n>=1} (1 - 1/a(n)) = -(18/Pi)*cos(sqrt(11)*Pi/6).
Product_{n>=1} (1 + 1/a(n)) = (18/Pi)*cos(sqrt(7)*Pi/6). (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 36*x/(1-x)^3.
E.g.f.: 18*x*(2 + x)*exp(x).
a(n) = 3*A049598(n) = 2*A163758(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A192026 Square array read by antidiagonals: W(n,m) (n >= 3, m >= 1) is the Wiener index of the graph G(n,m) obtained from an n-wheel graph by adjoining m pendant edges at each node of the cycle.

Original entry on oeis.org

36, 72, 90, 120, 180, 168, 180, 300, 336, 270, 252, 450, 560, 540, 396, 336, 630, 840, 900, 792, 546, 432, 840, 1176, 1350, 1320, 1092, 720, 540, 1080, 1568, 1890, 1980, 1820, 1440, 918, 660, 1350, 2016, 2520, 2772, 2730, 2400, 1836, 1140, 792, 1650, 2520, 3240, 3696, 3822, 3600, 3060, 2280, 1386
Offset: 3

Views

Author

Emeric Deutsch, Jun 25 2011

Keywords

Examples

			W(3,1)=36 because in the graph with vertex set {O,A,B,C,A',B',C'} and edge set {OA, OB, OC, AB, BC, CA, AA', BB', CC'} we have 9 pairs of vertices at distance 1 (the edges), 9 pairs at distance 2 (A'O, A'B, A'C, B'O, B'A, B'C, C'O, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 9*1 + 9*2 + 3*3 = 36.
The square array starts:
   36,  90, 168,  270,  396,  546,  720,  918, ...;
   72, 180, 336,  540,  792, 1092, 1440, 1836, ...;
  120, 300, 560,  900, 1320, 1820, 2400, 3060, ...;
  180, 450, 840, 1350, 1980, 2730, 3600, 4590, ...;

Links

Crossrefs

Cf. A049598.

Programs

  • Maple
    W := proc (n, m) options operator, arrow: n*(n-1)*(m+1)*(2*m+1) end proc: for n from 3 to 12 do seq(W(n-i, i+1), i = 0 .. n-3) end do; # yields the antidiagonals in triangular form
W := proc (n, m) options operator, arrow: n*(n-1)*(m+1)*(2*m+1) end proc: for n from 3 to 12 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 3,4,...,12.

Formula

W(n,1) = A049598(n-1).
W(n,m) = n*(n-1)*(m+1)*(2*m+1) (n >= 3, m >= 1).
The Wiener polynomial of the graph G(n,m) is P(n,m;t) = n*(m+2)*t + (1/2)*n*(m^2+n+5*m-3)*t^2 + n*m*(m+n-3)*t^3 + (1/2)*n*m^2*(n-3)*t^4.

A322462 Numbers on the 0-1-12 line in a spiral on a grid of equilateral triangles.

Original entry on oeis.org

0, 1, 12, 13, 36, 37, 72, 73, 120, 121, 180, 181, 252, 253, 336, 337, 432, 433, 540, 541, 660, 661, 792, 793, 936, 937, 1092, 1093, 1260, 1261, 1440, 1441, 1632, 1633, 1836, 1837, 2052, 2053, 2280, 2281, 2520, 2521, 2772, 2773, 3036, 3037, 3312, 3313, 3600
Offset: 0

Views

Author

Hans G. Oberlack, Dec 09 2018

Keywords

Comments

Sequence found by reading the line from 0, in the direction 0, 1, 12, ... in the triangle spiral.

Examples

			a(0) = 0
a(1) = a(1 - 1) + 1 = 0 + 1
a(2) = (3/2) * 2 * (2 + 2) = 3 * 4 = 12
a(3) = a(3 - 1) + 1 = 12 + 1 = 13
a(4) = (3/2) * 4*(4 + 2) = 3 * 2 * 6 = 6 * 6 = 36
a(5) = a(4) + 1 = 36 + 1 = 37.

Links

Crossrefs

Cf. A049598.

Programs

  • Maple
    seq(coeff(series(-x*(x^3-x^2+11*x+1)/((x+1)^2*(x-1)^3),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Dec 19 2018
  • Mathematica
    a[0] = 0; a[n_] := a[n] = If[OddQ[n], a[n - 1] + 1, 3/2*n*(n + 2)]; Array[a, 50, 0] (* Amiram Eldar, Dec 09 2018 *)
  • PARI
    concat(0, Vec(x*(1 + 11*x - x^2 + x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Dec 09 2018

Formula

a(n) = (3/2)*n*(n+2) = A049598(n/2) if n even, a(n) = a(n-1)+1 if n odd.
G.f.: -x*(x^3-x^2+11*x+1)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 09 2018
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4. - Colin Barker, Dec 09 2018

Extensions

Examples added by Hans G. Oberlack, Dec 20 2018

A192029 Square array read by antidiagonals: W(n,m) (n >= 3, m >= 1) is the Wiener index of the graph G(n,m) obtained from an n-wheel graph by adjoining at each of its n rim nodes a path with m nodes (if m=1, then the n-wheel is not modified).

Original entry on oeis.org

6, 12, 36, 20, 72, 111, 30, 120, 220, 252, 42, 180, 365, 496, 480, 56, 252, 546, 820, 940, 816, 72, 336, 763, 1224, 1550, 1592, 1281, 90, 432, 1016, 1708, 2310, 2620, 2492, 1896, 110, 540, 1305, 2272, 3220, 3900, 4095, 3680, 2682, 132, 660, 1630, 2916, 4280, 5432, 6090, 6040, 5196, 3660
Offset: 3

Views

Author

Emeric Deutsch, Jun 27 2011

Keywords

Examples

			W(3,2)=36 because in the graph with vertex set {O,A,B,C,A',B',C'} and edge set {OA, OB, OC, AB, BC, CA, AA', BB', CC'} we have 9 pairs of vertices at distance 1 (the edges), 9 pairs at distance 2 (A'O, A'B, A'C, B'O, B'A, B'C, C'O, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 9*1 + 9*2 + 3*3 = 36.
The square array starts:
   6,  36, 111,  252,  480,  816, 1281, ...;
  12,  72, 220,  496,  940, 1592, 2492, ...;
  20, 120, 365,  820, 1550, 2620, 4095, ...;
  30, 180, 546, 1224, 2310, 3900, 6090, ...;

Links

Crossrefs

Cf. A002378, A049598.

Programs

  • Maple
    W := proc (n, m) options operator, arrow: (1/6)*n*m*(3*n*m+3*n*m^2+2-6*m-2*m^2) end proc: for n from 3 to 12 do seq(W(n-i, i+1), i = 0 .. n-3) end do; # yields the antidiagonals in triangular form
W := proc (n, m) options operator, arrow: (1/6)*n*m*(3*n*m+3*n*m^2+2-6*m-2*m^2) end proc: for n from 3 to 12 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 3,4,...,12.
P := proc (n, m) options operator, arrow: sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+n*t*(1-t^m)/(1-t)+n*t*(1-t^m)^2/(1-t)^2+(1/2)*n*(n-3)*t^2*(1-t^m)^2/(1-t)^2))) end proc; P(4, 3);

Formula

W(n,1) = A002378(n-1) = n(n-1).
W(n,2) = A049598(n-1).
W(n,m) = (1/6)*n*m*(3*n*m + 3*n*m^2 + 2 - 6*m - 2*m^2) (n >= 3, m >= 1).
The Wiener polynomial P(n,m;t) of the graph G(n,m) is given in the 3rd Maple program. It gives, for example, P(3,4) = 16*t + 18*t^2 + 20*t^3 + 14*t^4 + 8*t^5 + 2*t^6. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(4,3)=220.

A192031 Rectangular array read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the helm graph G(n) obtained from a wheel graph with n spokes by adjoining a pendant edge at each node of the cycle (n>=3, k>=1). The entries in row n are the coefficients of the corresponding Wiener polynomial.

Original entry on oeis.org

9, 9, 3, 12, 14, 8, 2, 15, 20, 15, 5, 18, 27, 24, 9, 21, 35, 35, 14, 24, 44, 48, 20, 27, 54, 63, 27, 30, 65, 80, 35, 33, 77, 99, 44, 36, 90, 120, 54, 39, 104, 143, 65, 42, 119, 168, 77, 45, 135, 195, 90, 48, 152, 224, 104, 51, 170, 255, 119, 54, 189, 288, 135, 57, 209, 323, 152, 60, 230, 360, 170
Offset: 3

Views

Author

Emeric Deutsch, Jun 30 2011

Keywords

Comments

The graph G(n) is a special case of the graph G(n,m) defined in A192026 (m=1).
T(n,k) is also the number of unordered pairs of nodes at distance k in the gear graph G(n) obtained from a wheel graph with n spokes by adding a node between each pair of adjacent nodes of the cycle (n>=3, k>=1). Example: T(3,3)=3 because in the graph G(3) with vertex set {A,B,C,D,B',C',D'} and edge set {BD', D'C, CB', B'D, DC', C'B,AB,AC,AD} there are exactly 3 pairs of vertices at distance 3: BB', CC', and DD'.
Row 3 contains 3 entries; row n>=4 contains 4 entries.
Sum of entries in row n is n*(2n+1)=A014105(n).
Sum(k*T(n,k),k>=1)=6*n*(n-1)=A049598(n-1) (the Wiener indices).

Examples

			T(3,3)=3 because in the graph G(3) with vertex set {A,B,C,D,B',C',D'} and edge set {BC,CD,DB,AB,AC,AD,BB',CC",DD'} there are exactly 3 pairs of vertices at distance 3: B'C', C'D', and D'B'.
Rectangular array starts:
9,9,3;
12,14,8,2;
15,20,15,5;
18,27,24,9;

Links

Crossrefs

Cf. A014105, A049598, A192026.

Programs

  • Maple
    P := proc (n) options operator, arrow: 3*n*t+(1/2)*n*(n+3)*t^2+n*(n-2)*t^3+(1/2)*n*(n-3)*t^4 end proc: T := proc (n, k) options operator, arrow: coeff(P(n), t, k) end proc: seq(T(3, k), k = 1 .. 3); for n from 4 to 20 do seq(T(n, k), k = 1 .. 4) end do; # yields rows 3,4,..., 20 of the rectangular array
  • Mathematica
    P[n_] := 3*n*t + (1/2)*n*(n+3)*t^2 + n*(n-2)*t^3 + (1/2)*n*(n-3)*t^4; T[n_]:=Rest@CoefficientList[P[n], t]; Table[T[n], {n, 3, 20}] // Flatten (* Jean-François Alcover, Sep 07 2024, after Maple program *)

Formula

Generating polynomial of row n (i.e. the Wiener polynomial of the graph G(n)) is P(n;t)=3*n*t+(1/2)*n*(n+3)*t^2+n*(n-2)*t^3+(1/2)*n*(n-3)*t^4.
Conjectures from Colin Barker, Mar 28 2019: (Start)
G.f.: x^3*(9 + 3*x^2 + 9*x^3 - 22*x^4 + 3*x^5 - 10*x^6 - 11*x^7 + 16*x^8 + 2*x^9 + 6*x^10 + 3*x^11 - 3*x^12 - 3*x^13) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^3).
a(n) = a(n-1) - a(n-2) + a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-6) - 2*a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11) for n>16. (End)

A267849 Triangular array: T(n,k) is the 2-row creation rook number to place k rooks on a 3 x n board.

Original entry on oeis.org

1, 1, 3, 1, 6, 12, 1, 9, 36, 60, 1, 12, 72, 240, 360, 1, 15, 120, 600, 1800, 2520, 1, 18, 180, 1200, 5400, 15120, 20160, 1, 21, 252, 2100, 12600, 52920, 141120, 181440, 1, 24, 336, 3360, 25200, 141120, 564480, 1451520, 1814400, 1, 27, 432, 5040, 45360, 317520, 1693440, 6531840, 16329600, 19958400
Offset: 0

Views

Author

Ken Joffaniel M. Gonzales, Jan 21 2016

Keywords

Comments

T(n,k) is the number of ways to place k rooks in a 3 x n Ferrers board (or diagram) under the Goldman-Haglund i-row creation rook mode for i=2. All row heights are 3.

Examples

			The triangle T(n,k) begins in row n=0 with columns 0<=k<=n:
     1
     1      3
     1      6     12
     1      9     36     60
     1     12     72    240    360
     1     15    120    600   1800   2520
     1     18    180   1200   5400  15120  20160
     1     21    252   2100  12600  52920 141120 181440
     1     24    336   3360  25200 141120 564480 1451520 1814400
     1     27    432   5040  45360 317520 1693440 6531840 16329600 19958400

Links

Crossrefs

Cf. A013610 (1-rook coefficients on the 3xn board), A121757 (2-rook coeffs. on the 2xn board), A013609 (1-rook coeffs. on the 2xn board), A013611 (1-rook coeffs. on the 4xn board), A008279 (2-rook coeffs. on the 1xn board), A082030 (row sums?), A049598 (column k=2), A007531 (column k=3 w/o factor 10), A001710 (diagonal?).

Formula

T(n,k) = T(n-1,k) + (k+2) T(n-1,k-1) subject to T(0,0)=1, T(n,k)=0 for n

Extensions

Triangle simplified (reversing rows, offset 0). - R. J. Mathar, May 03 2017
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