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.

Showing 1-6 of 6 results.

A182467 a(n) = 3a(n-1) - 2a(n-2) with a(0)=36 and a(1)=90.

Original entry on oeis.org

36, 90, 198, 414, 846, 1710, 3438, 6894, 13806, 27630, 55278, 110574, 221166, 442350, 884718, 1769454, 3538926, 7077870, 14155758, 28311534, 56623086, 113246190, 226492398, 452984814, 905969646, 1811939310, 3623878638, 7247757294, 14495514606, 28991029230
Offset: 0

Views

Author

Odimar Fabeny, Apr 30 2012

Keywords

Comments

Number of vertices into building blocks of 3d objects with 9 vertices.

Examples

			a(0) = 9+18+9;
a(1) = 9+18+36+18+9;
a(2) = 9+18+36+72+36+18+9;
a(3) = 9+18+36+72+144+72+36+18+9.

Links

Crossrefs

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182464, A182465, A182466.

Programs

  • Mathematica
    LinearRecurrence[{3,-2},{36,90},30] (* or *) CoefficientList[Series[(-18(x-2))/(1-3x+2x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 29 2013 *)

Formula

a(n) = a(n-1)*2 + 18
G.f.: -((18*(x-2))/(2*x^2-3*x+1)). - Harvey P. Dale, Apr 29 2013
a(n) = 18*A153893(n). - Michel Marcus, Jun 01 2014

A182461 a(n) = 3*a(n-1) - 2*a(n-2) with a(0)=16 and a(1)=40.

Original entry on oeis.org

16, 40, 88, 184, 376, 760, 1528, 3064, 6136, 12280, 24568, 49144, 98296, 196600, 393208, 786424, 1572856, 3145720, 6291448, 12582904, 25165816, 50331640, 100663288, 201326584, 402653176, 805306360, 1610612728, 3221225464, 6442450936, 12884901880
Offset: 0

Views

Author

Odimar Fabeny, Apr 30 2012

Keywords

Comments

Number of vertices into building blocks of 3d objects with 4 vertices.

Examples

			a(0) = 4+8+4;
a(1) = 4+8+16+8+4;
a(2) = 4+8+16+32+16+8+4;
a(3) = 4+8+16+32+64+32+16+8+4.

Links

Crossrefs

Cf. A000045, A028399, A038578, A089143, A173033, A182462, A182464, A182465, A182466, A182467.

Programs

  • Mathematica
    CoefficientList[Series[-((8 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 02 2014 *)

Formula

a(n) = a(n-1)*2 + 8.
G.f.: 16 + 40*x + 88*x^2 + 184*x^3 + 376*x^4 + 760*x^5 + 1528*x^6 + ...
a(n) = 8 * A055010(n+1). [Joerg Arndt, Jun 01 2014]
G.f.: -((8*(x - 2))/(2*x^2 - 3*x + 1)). - Vincenzo Librandi, Jun 02 2014

A182462 a(n) = 3a(n-1) - 2a(n-2) with a(0)=20 and a(1)=50.

Original entry on oeis.org

20, 50, 110, 230, 470, 950, 1910, 3830, 7670, 15350, 30710, 61430, 122870, 245750, 491510, 983030, 1966070, 3932150, 7864310, 15728630, 31457270, 62914550, 125829110, 251658230, 503316470, 1006632950, 2013265910, 4026531830, 8053063670, 16106127350
Offset: 0

Views

Author

Odimar Fabeny, Apr 30 2012

Keywords

Comments

Number of vertices into building blocks of 3d objects with 5 vertices.

Examples

			a(0) = 5+10+5;
a(1) = 5+10+20+10+5;
a(2) = 5+10+20+40+20+10+5;
a(3) = 5+10+20+40+80+40+20+10+5.

Links

Crossrefs

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182464, A182465, A182466, A182467.

Programs

  • Mathematica
    CoefficientList[Series[-((10 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 02 2014 *)

Formula

a(n) = a(n-1)*2 + 10.
a(n) = 10*A153893(n). - Michel Marcus, Jun 01 2014
G.f.: -((10*(x - 2))/(2*x^2 - 3*x + 1)). - Vincenzo Librandi, Jun 02 2014

A182465 a(n) = 3a(n-1) - 2a(n-2) with a(0)=28 and a(1)=70.

Original entry on oeis.org

28, 70, 154, 322, 658, 1330, 2674, 5362, 10738, 21490, 42994, 86002, 172018, 344050, 688114, 1376242, 2752498, 5505010, 11010034, 22020082, 44040178, 88080370, 176160754, 352321522, 704643058, 1409286130, 2818572274, 5637144562, 11274289138, 22548578290
Offset: 0

Views

Author

Odimar Fabeny, Apr 30 2012

Keywords

Comments

Number of vertices into building blocks of 3d objects with 7 vertices.

Examples

			a(0) = 7+14+7;
a(0) = 7+14+28+14+7;
a(0) = 7+14+28+56+28+14+7;
a(0) = 7+14+28+56+112+56+28+14+7.

Links

Crossrefs

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182464, A182466, A182467.

Programs

  • Mathematica
    CoefficientList[Series[-((14 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2014 *)
LinearRecurrence[{3,-2},{28,70},30] (* Harvey P. Dale, Oct 05 2015 *)

Formula

a(n) = a(n-1)*2 + 14.
a(n) = 14*A153893(n). - Michel Marcus, Jun 01 2014
G.f.: -((14*(x-2))/(2*x^2-3*x+1)). - Vincenzo Librandi, Jun 01 2014

A182466 a(n) = 3a(n-1) - 2a(n-2) with a(0)=32 and a(1)=80.

Original entry on oeis.org

32, 80, 176, 368, 752, 1520, 3056, 6128, 12272, 24560, 49136, 98288, 196592, 393200, 786416, 1572848, 3145712, 6291440, 12582896, 25165808, 50331632, 100663280, 201326576, 402653168, 805306352, 1610612720, 3221225456, 6442450928, 12884901872, 25769803760, 51539607536
Offset: 0

Views

Author

Odimar Fabeny, Apr 30 2012

Keywords

Comments

Number of vertices into building blocks of 3d objects with 8 vertices.

Examples

			a(0) = 8+16+8;
a(1) = 8+16+32+16+8;
a(2) = 8+16+32+64+32+16+8;
a(3) = 8+16+32+64+128+64+32+16+8.

Links

Crossrefs

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182464, A182465, A182467.

Programs

  • Mathematica
    LinearRecurrence[{3,-2},{32,80},40] (* or *) Table[8(3*2^n-2),{n,40}] (* Harvey P. Dale, Aug 23 2012 *)
CoefficientList[Series[-((16 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 02 2014 *)

Formula

a(n) = a(n-1)*2 + 16.
a(n) = 8*(3*2^n-2). - Harvey P. Dale, Aug 23 2012
G.f.: -((16(x-2))/(2*x^2-3*x+1)). - Harvey P. Dale, Aug 23 2012

A212221 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is 1/(2*n) times the number of n-colorings of the complete tripartite graph K_(k,k,k).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 12, 6, 0, 0, 1, 30, 78, 10, 0, 0, 1, 66, 474, 340, 15, 0, 0, 1, 138, 2238, 4780, 1095, 21, 0, 0, 1, 282, 9546, 46420, 32955, 2856, 28, 0, 0, 1, 570, 38958, 385660, 617775, 168546, 6412, 36
Offset: 1

Views

Author

Alois P. Heinz, May 06 2012

Keywords

Comments

The complete tripartite graph K_(n,n,n) has 3*n vertices and 3*n^2 = A033428(n) edges; see A212220 for example. The chromatic polynomial of K_(n,n,n) has 3*n+1 = A016777(n) coefficients.

Examples

			Square array A(n,k) begins:
   0,    0,     0,      0,       0,         0,          0, ...
   0,    0,     0,      0,       0,         0,          0, ...
   1,    1,     1,      1,       1,         1,          1, ...
   3,   12,    30,     66,     138,       282,        570, ...
   6,   78,   474,   2238,    9546,     38958,     155994, ...
  10,  340,  4780,  46420,  385660,   2995540,   22666780, ...
  15, 1095, 32955, 617775, 9248595, 123920295, 1569542955, ...

Links

Crossrefs

Rows 1+2,3-4 give: A000004, A000012, A089143(n-1) = 1/2*A182464(n-2) = 1/3*A182467(n-2).
Columns 1-2 give: A000217(n-2), 1/(2*n)*A115400(n).
Cf. A008277, A016777, A033428, A212220, A282247.

Programs

  • Maple
    P:= proc(n) option remember;
      unapply(expand(add(add(Stirling2(n, k) *Stirling2(n, m)
       *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)), q)
    end:
A:= (n, k)-> P(k)(n)/(2*n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
  • Mathematica
    p[n_] := p[n] = Function[q, Expand[Sum[Sum[StirlingS2[n, k] * StirlingS2[n, m] * Product[q-i, {i, 0, k+m-1}]*(q-k-m)^n, {m, 1, n}], {k, 1, n}]]]; a[n_, k_] := p[k][n]/(2*n); Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Formula

A(n,k) = 1/(2*n) * Sum_{j,m=1..k} S2(k,j) * S2(k,m) * (n-j-m)^k * Product_{i=0..j+m-1} (n-i) with S2 = A008277.
A(n,n) = A282247(n).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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