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Number of paths on square lattice.
(Formerly M1490)
2

%I M1490 #30 Mar 21 2021 17:28:11

%S 1,2,5,16,54,180,595,1964,6485,21418,70740,233640,771661,2548622,

%T 8417525,27801196,91821114,303264540,1001614735,3308108744,

%U 10925940965,36085931638,119183735880,393637139280,1300095153721,4293922600442

%N Number of paths on square lattice.

%D H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.

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

%H Michael De Vlieger, <a href="/A006191/b006191.txt">Table of n, a(n) for n = 1..1928</a>

%H H. L. Abbott and D. Hanson, <a href="/A006189/a006189.pdf">A lattice path problem</a>, Ars Combin., 6 (1978), 163-178. (Annotated scanned copy)

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,2,1).

%F a(n) = 1 + Sum_{k=1..n-1} A006189(k). - _Sean A. Irvine_, Jan 20 2017

%F From _Colin Barker_, Jan 20 2017: (Start)

%F a(n) = 4*a(n-1) - 3*a(n-2) + 2*a(n-3) + a(n-4) for n>4.

%F G.f.: x*(1 - 2*x) / ((1 - x + x^2)*(1 - 3*x - x^2)).

%F (End)

%t LinearRecurrence[{4,-3,2,1},{1,2,5,16},30] (* _Harvey P. Dale_, Mar 22 2018 *)

%K nonn,walk,easy

%O 1,2

%A _N. J. A. Sloane_

%E Offset corrected and more terms from _Sean A. Irvine_, Jan 20 2017