cp's OEIS Frontend

A035008 Total number of possible knight moves on an (n+2) X (n+2) chessboard, if the knight is placed anywhere.

%I A035008 #95 Feb 16 2025 08:32:37
%S A035008 0,16,48,96,160,240,336,448,576,720,880,1056,1248,1456,1680,1920,2176,
%T A035008 2448,2736,3040,3360,3696,4048,4416,4800,5200,5616,6048,6496,6960,
%U A035008 7440,7936,8448,8976,9520,10080,10656,11248,11856,12480,13120,13776
%N A035008 Total number of possible knight moves on an (n+2) X (n+2) chessboard, if the knight is placed anywhere.
%C A035008 16 times the triangular numbers A000217.
%C A035008 Centered 16-gonal numbers A069129, minus 1. Also, sequence found by reading the segment (0, 16) together with the line from 16, in the direction 16, 48, ..., in the square spiral whose vertices are the triangular numbers A000217. - _Omar E. Pol_, Apr 26 2008, Nov 20 2008
%C A035008 For n >= 1, number of permutations of n+1 objects selected from 5 objects v, w, x, y, z with repetition allowed, containing n-1 v's. Examples: at n=1, n-1=0 (i.e., zero v's), and a(1)=16 because we have ww, wx, wy, wz, xw, xx, xy, xz, yw, yx, yy, yz, zw, zx, zy, zz; at n=2, n-1=1 (i.e., one v), and there are 3 permutations corresponding to each one in the n=1 case (e.g., the single v can be inserted in any of three places in the 2-object permutation xy, yielding vxy, xvy, and xyv), so a(2) = 3*a(1) = 3*16 = 48; at n=3, n-1=2 (i.e., two v's), and a(3) = C(4,2)*a(1) = 6*16 = 96; etc. - _Zerinvary Lajos_, Aug 07 2008 (this needs clarification, _Joerg Arndt_, Feb 23 2014)
%C A035008 Sequence found by reading the line from 0, in the direction 0, 16, ... and the same line from 0, in the direction 0, 48, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - _Omar E. Pol_, Oct 03 2011
%C A035008 For n > 0, a(n) is the area of the triangle with vertices at ((n-1)^2, n^2), ((n+1)^2, (n+2)^2), and ((n+3)^2, (n+2)^2). - _J. M. Bergot_, May 22 2014
%C A035008 For n > 0, a(n) is the number of self-intersecting points in star polygon {4*(n+1)/(2*n+1)}. - _Bui Quang Tuan_, Mar 28 2015
%C A035008 Equivalently: integers k such that k$ / (k/2)! and k$ / (k/2+1)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information). - _Bernard Schott_, Dec 02 2021
%H A035008 Vincenzo Librandi, <a href="/A035008/b035008.txt">Table of n, a(n) for n = 0..1000</a>
%H A035008 Omar E. Pol, <a href="http://polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A035008 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StarPolygon.html">Star Polygon</a>.
%H A035008 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A035008 a(n) = 8*n*(n+1).
%F A035008 G.f.: 16*x/(1-x)^3.
%F A035008 a(n) = A069129(n+1) - 1. - _Omar E. Pol_, Apr 26 2008
%F A035008 a(n) = binomial(n+1,2)*4^2, n >= 0. - _Zerinvary Lajos_, Aug 07 2008
%F A035008 a(n) = 8*n^2 + 8*n = 16*A000217(n) = 8*A002378(n) = 4*A046092(n) = 2*A033996(n). - _Omar E. Pol_, Dec 12 2008
%F A035008 a(n) = a(n-1) + 16*n, with a(0)=0. - _Vincenzo Librandi_, Nov 17 2010
%F A035008 E.g.f.: 8*exp(x)*x*(2 + x). - _Stefano Spezia_, May 19 2021
%F A035008 From _Amiram Eldar_, Feb 22 2023: (Start)
%F A035008 Sum_{n>=1} 1/a(n) = 1/8.
%F A035008 Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/8.
%F A035008 Product_{n>=1} (1 - 1/a(n)) = -(8/Pi)*cos(sqrt(3/2)*Pi/2).
%F A035008 Product_{n>=1} (1 + 1/a(n)) = (8/Pi)*cos(Pi/(2*sqrt(2))). (End)
%e A035008 3 X 3-Board: knight can be placed in 8 positions with 2 moves from each, so a(1) = 16.
%p A035008 seq(binomial(n+1,2)*4^2, n=0..33); # _Zerinvary Lajos_, Aug 07 2008
%t A035008 CoefficientList[Series[16 x/(1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 24 2014 *)
%t A035008 LinearRecurrence[{3,-3,1},{0,16,48},50] (* or *) 16*Accumulate[ Range[ 0,50]] (* _Harvey P. Dale_, Aug 05 2018 *)
%o A035008 (Magma) [8*n*(n+1): n in [0..50]]; // _Wesley Ivan Hurt_, May 22 2014
%o A035008 (PARI) a(n)=8*n*(n+1) \\ _Charles R Greathouse IV_, Sep 30 2015
%Y A035008 Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A002492 (Bishop) and A049450 (Pawn).
%Y A035008 Cf. A000217, A069129, A027468, A008586 A038231, A002378, A033996, A046092.
%Y A035008 Cf. A348692.
%Y A035008 Subsequence of A008586 and of A349081.
%K A035008 easy,nonn,nice
%O A035008 0,2
%A A035008 Ulrich Schimke (ulrschimke(AT)aol.com), Dec 11 1999
%E A035008 More terms from _Erich Friedman_
%E A035008 Minor errors corrected and edited by _Johannes W. Meijer_, Feb 04 2010