login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005561 Number of walks on square lattice.
(Formerly M3596)
4

%I M3596

%S 1,4,24,84,392,1344,5760,19800,81675,283140,1145144,4008004,16032016,

%T 56632576,225059328,801773856,3173688180,11392726800,44986664800,

%U 162594659920,641087516256,2331227331840,9183622822400,33577620944400,132211882468575,485773975404900

%N Number of walks on square lattice.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005561/b005561.txt">Table of n, a(n) for n = 3..1000</a>

%H R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>

%H R. K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6. See w_n'(3).

%F a(n) = C(n+4, ceiling(n/2))*C(n+3, floor(n/2)) - C(n+4, ceiling((n-1)/2))*C(n+3, floor((n-1)/2)). - _Paul D. Hanna_, Apr 16 2004

%F Conjecture: (n-2)*(n-3)*(2*n+1)*(n+6)*(n+5)*a(n) - 4*n*(n+1)*(2*n^2+4*n+33)*a(n-1) - 16*n^2*(n-1)*(2*n+3)*(n+1)*a(n-2) = 0. - _R. J. Mathar_, Apr 02 2017

%p wnprime := proc(n,y)

%p local k;

%p if type(n-y,'even') then

%p k := (n-y)/2 ;

%p binomial(n+1,k)*(binomial(n,k)-binomial(n,k-1)) ;

%p else

%p k := (n-y-1)/2 ;

%p binomial(n+1,k)*binomial(n,k+1)-binomial(n+1,k+1)*binomial(n,k-1) ;

%p end if;

%p end proc:

%p A005561 := proc(n)

%p wnprime(n,3) ;

%p end proc:

%p seq(A005561(n),n=3..30) ; # _R. J. Mathar_, Apr 02 2017

%t Table[Binomial[n+4, Ceiling[n/2]] Binomial[n+3, Floor[n/2]]-Binomial[n+4, Ceiling[(n-1)/2]] Binomial[n+3, Floor[(n-1)/2]], {n, 0, 30}] (* _Vincenzo Librandi_, Apr 03 2017 *)

%o (PARI) {a(n)=binomial(n+4,ceil(n/2))*binomial(n+3,floor(n/2)) - binomial(n+4,ceil((n-1)/2))*binomial(n+3,floor((n-1)/2))}

%o (MAGMA) [Binomial(n+4, Ceiling(n/2))*Binomial(n+3, Floor(n/2)) - Binomial(n+4, Ceiling((n-1)/2))*Binomial(n+3, Floor((n-1)/2)): n in [0..30]]; // _Vincenzo Librandi_, Apr 03 2017

%Y Cf. A005558-A005562, A093768.

%K nonn,walk

%O 3,2

%A _N. J. A. Sloane_

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 21:00 EST 2019. Contains 329779 sequences. (Running on oeis4.)