OFFSET
0,2
COMMENTS
Also number of n X 2 0..1 arrays with rows and columns unimodal (cf. A223620, column 2). - Georg Fischer, Nov 03 2021
Let R(n) be the set of lattice points in Z^2 reachable from (0, 0) by a sequence of n moves where, for k=1..n, the k-th move is one of (+k, 0), (0, +k), (-k, 0), (0, -k). Then a(n) = |R(n)|. - Maksim Kita, Nov 29 2025
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
R. J. Mathar, The number of binary n x m matrices with at most k 1's in each row or column, (2014), Table 2, column 2.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = (A000217(n) + 1)^2.
a(n) = (binomial(2+n,2) - binomial(n,1))^2. - Zerinvary Lajos, May 30 2006, corrected by R. J. Mathar, May 14 2014
a(n) = A000124(n)^2. - Omar E. Pol, Oct 30 2007
a(n) = 1 + A061316(n). Zerinvary Lajos, Apr 25 2008
G.f.: ( -1+x-6*x^2+x^3-x^4 ) / (x-1)^5. - R. J. Mathar, May 14 2014
Sum_{n>=0} 1/a(n) = (4/49) * Pi * sech(sqrt(7)*Pi/2)^2 * (sqrt(7)*sinh(sqrt(7)*Pi)- 7*Pi). - Amiram Eldar, Dec 04 2025
EXAMPLE
a(5) = (t(5)+1)^2 = 16^2 = 256.
MAPLE
A086601 := proc(n)
(n+2+n^2)^2 /4 ;
end proc:
seq(A086601(n), n=0..20) ; # R. J. Mathar, May 14 2014
MATHEMATICA
(Accumulate[Range[0, 40]]+1)^2 (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 4, 16, 49, 121}, 40] (* Harvey P. Dale, Jan 14 2020 *)
PROG
(PARI) w=vector(40, i, (t(i)+1)^2)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jon Perry, Jul 23 2003
STATUS
approved