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 1-3 of 3 results.

A090018 a(n) = 6*a(n-1) + 3*a(n-2) for n > 2, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 39, 252, 1629, 10530, 68067, 439992, 2844153, 18384894, 118841823, 768205620, 4965759189, 32099171994, 207492309531, 1341251373168, 8669985167601, 56043665125110, 362271946253463, 2341762672896108, 15137391876137037, 97849639275510546, 632510011281474387
Offset: 0

Views

Author

Paul Barry, Nov 19 2003

Keywords

Comments

From Johannes W. Meijer, Aug 09 2010: (Start)
a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner or side square on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032. The central square leads to A180028. (End)

Links

Crossrefs

Cf. A141041, A180028, A180032.
Sequences with g.f. of the form 1/(1 - 6*x - k*x^2): A106392 (k=-10), A027471 (k=-9), A006516 (k=-8), A081179 (k=-7), A030192 (k=-6), A003463 (k=-5), A084326 (k=-4), A138395 (k=-3), A154244 (k=-2), A001109 (k=-1), A000400 (k=0), A005668 (k=1), A135030 (k=2), this sequence (k=3), A135032 (k=4), A015551 (k=5), A057089 (k=6), A015552 (k=7), A189800 (k=8), A189801 (k=9), A190005 (k=10), A015553 (k=11).

Programs

  • Magma
    [n le 2 select 6^(n-1) else 6*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011
  • Maple
    a:= n-> (<<0|1>, <3|6>>^n. <<1,6>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 17 2011
  • Mathematica
    Join[{a=1,b=6},Table[c=6*b+3*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,3}, {1,6}, 41] (* G. C. Greubel, Oct 10 2022 *)
  • PARI
    my(x='x+O('x^30)); Vec(1/(1-6*x-3*x^2)) \\ G. C. Greubel, Jan 24 2018
  • Sage
    [lucas_number1(n,6,-3) for n in range(1, 31)] # Zerinvary Lajos, Apr 24 2009

Formula

a(n) = (3+2*sqrt(3))^n*(sqrt(3)/4+1/2) + (1/2-sqrt(3)/4)*(3-2*sqrt(3))^n.
a(n) = (-i*sqrt(3))^n * ChebyshevU(n, isqrt(3)), i^2=-1.
From Johannes W. Meijer, Aug 09 2010: (Start)
G.f.: 1/(1 - 6*x - 3*x^2).
Limit_{k->oo} a(n+k)/a(k) = A141041(n) + A090018(n-1)*sqrt(12) for n >= 1.
Limit_{n->oo} A141041(n)/A090018(n-1) = sqrt(12). (End)
a(n) = Sum_{k=0..n} A099089(n,k)*3^k. - Philippe Deléham, Nov 21 2011
E.g.f.: exp(3*x)*(2*cosh(2*sqrt(3)*x) + sqrt(3)*sinh(2*sqrt(3)*x))/2. - Stefano Spezia, Apr 23 2025

Extensions

Typo in Mathematica program corrected by Vincenzo Librandi, Nov 15 2011

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

Links

Crossrefs

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
1..A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
2..A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
3..A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
4..A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
5..A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
6..A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
7..A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
8..A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
9..A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
10.A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
  • Mathematica
    LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)
  • PARI
    a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A223212 3X3X3 triangular graph coloring a rectangular array: number of nX2 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

18, 126, 918, 6642, 48114, 348462, 2523798, 18278946, 132387858, 958837662, 6944516694, 50296639122, 364280484978, 2638352661966, 19108640336598, 138397015977282, 1002359858893074, 7259732297153982, 52579632512961558
Offset: 1

Views

Author

R. H. Hardin Mar 18 2013

Keywords

Comments

Column 2 of A223218

Examples

			Some solutions for n=3
..0..1....1..4....0..1....0..2....4..1....4..1....4..2....2..4....0..2....5..2
..2..4....2..1....1..0....1..4....1..2....2..4....2..4....4..1....2..0....2..5
..0..2....1..2....4..1....3..1....4..1....4..2....0..2....1..2....0..1....0..2

Links

Formula

Empirical: a(n) = 6*a(n-1) +9*a(n-2).
Empirical g.f.: -18*x*(1+x)/(-1+6*x+9*x^2) . a(n) = 18*(A189801(n)+A189801(n-1)). - R. J. Mathar, May 21 2018
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.