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 A005561 Number of walks on square lattice. (Formerly M3596) 4
 1, 4, 24, 84, 392, 1344, 5760, 19800, 81675, 283140, 1145144, 4008004, 16032016, 56632576, 225059328, 801773856, 3173688180, 11392726800, 44986664800, 162594659920, 641087516256, 2331227331840, 9183622822400, 33577620944400, 132211882468575, 485773975404900 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,2 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 3..1000 R. K. Guy, Letter to N. J. A. Sloane, May 1990 R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6. See w_n'(3). FORMULA 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 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 MAPLE wnprime := proc(n, y) local k; if type(n-y, 'even') then k := (n-y)/2 ; binomial(n+1, k)*(binomial(n, k)-binomial(n, k-1)) ; else k := (n-y-1)/2 ; binomial(n+1, k)*binomial(n, k+1)-binomial(n+1, k+1)*binomial(n, k-1) ; end if; end proc: A005561 := proc(n) wnprime(n, 3) ; end proc: seq(A005561(n), n=3..30) ; # R. J. Mathar, Apr 02 2017 MATHEMATICA 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 *) PROG (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))} (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 CROSSREFS Cf. A005558-A005562, A093768. Sequence in context: A334581 A341688 A341877 * A061612 A097875 A212681 Adjacent sequences: A005558 A005559 A005560 * A005562 A005563 A005564 KEYWORD nonn,walk AUTHOR STATUS approved

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Last modified December 3 23:05 EST 2022. Contains 358543 sequences. (Running on oeis4.)