OFFSET
0,2
COMMENTS
From R. J. Mathar, Oct 12 2010: (Start)
The top row, n=2, of an array that counts chess king walks with k >= 0 steps on an n X n board, starting at one of the four corners:
1,3,12, 39,120, 363, 1092, 3279, 9840, 29523, 88572, 265719, 797160,
1,3,21,101,501,2405,11653, 56197, 271493, 1310597, 6328709, 30556549,
1,3,21,126,741,4341,25416,148791, 871041, 5099166,29851041,174751041,
1,3,21,126,810,5169,33447,215796,1395588, 9018255,58302057,376845978,
1,3,21,126,810,5360,36167,246034,1680313,11495503,78705226,539048956,
1,3,21,126,810,5360,36700,254756,1788468,12617828,89338116,633604564,
1,3,21,126,810,5360,36700,256255,1816090,12993280,93566653,676648735,
1,3,21,126,810,5360,36700,256255,1820335,13080120,94845670,692120270,
1,3,21,126,810,5360,36700,256255,1820335,13092211,95117374,696421066,
1,3,21,126,810,5360,36700,256255,1820335,13092211,95151979,697268152,
1,3,21,126,810,5360,36700,256255,1820335,13092211,95151979,697367593,
These are partial sums along rows of the array described in A086346. (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See p. 17.
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and Ziggu Puzzles, LIPIcs 366, 13th Int'l Conf. Fun Algor. (FUN 2026), Art. 22. See p. 15 (Table 6).
Index entries for linear recurrences with constant coefficients, signature (4,-3).
FORMULA
a(0) = 1 and a(n) = 3*A003462(n) for n > 0.
G.f.: (1-x+3*x^2)/(1-4*x+3*x^2). [Corrected by Georg Fischer, May 24 2019]
a(n) = Sum_{k=0..n} 3^k*A123110(n,k). - Philippe Deléham, Feb 09 2007
a(n) = A029858(n+1), n > 0. - R. J. Mathar, Jun 18 2008
a(n+1) - a(n) = 3^n, n >= 2. - R. J. Mathar, Aug 18 2011
E.g.f.: 1 + 3*(exp(3*x) - exp(x))/2. - G. C. Greubel, May 24 2019
MATHEMATICA
LinearRecurrence[{4, -3}, {1, 3, 12}, 30] (* Georg Fischer, May 24 2019 *)
(* Alternative: *)
Join[{1}, NestList[3#+3&, 3, 30]] (* Harvey P. Dale, Aug 16 2020 *)
PROG
(Magma) I:=[1, 3, 12]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 27 2012
(PARI) my(x='x+O('x^30)); Vec((1-x+3*x^2)/(1-4*x+3*x^2)) \\ G. C. Greubel, May 24 2019
(SageMath) ((1-x+3*x^2)/(1-4*x+3*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
(GAP) a:=[1, 3, 12];; for n in [4..30] do a[n]:=4*a[n-1]-3*a[n-2]; od; a; # G. C. Greubel, May 24 2019
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Philippe Deléham, Sep 28 2006
STATUS
approved
