A299918 Motzkin numbers (A001006) mod 8.

1, 1, 2, 4, 1, 5, 3, 7, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 7, 7, 5, 5, 4, 2, 1, 5, 3, 7, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 4, 6, 5, 5, 2, 4, 5, 1, 1, 1, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 6, 4, 1, 1, 2, 4, 1

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..10000

S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Catalan and Motzkin numbers modulo 4 and 8, Europ. J. Combin. 29 (2008), 1449-1466.

Christian Krattenthaler and Thomas W. Müller, Motzkin numbers and related sequences modulo powers of 2, arXiv:1608.05657 [math.CO], 2016-2018.

E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, J. Théorie Nombres Bordeaux 27 (2015), 245-288.

Ying Wang and Guoce Xin, A Classification of Motzkin Numbers Modulo 8, Electron. J. Combin., 25(1) (2018), #P1.54.

Maple

f:= rectoproc({(3+3*n)*a(n)+(5+2*n)*a(1+n)+(-4-n)*a(n+2), a(0) = 1, a(1) = 1}, a(n), remember): seq(f(n) mod 8, n=0..200); # Robert Israel, Mar 16 2018

Mathematica

Table[Mod[GegenbauerC[n, -n - 1, -1/2] / (n + 1), 8], {n, 0, 100}] (* Vincenzo Librandi, Sep 08 2018 *)

Prog

(PARI) catalan(n) = binomial(2*n, n)/(n+1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*catalan(k+1)) % 8; \\ Michel Marcus, May 23 2022

Crossrefs

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

Sequence in context: A060370 A318704 A165064 * A021418 A283741 A094640

Adjacent sequences: A299915 A299916 A299917 * A299919 A299920 A299921

Keyword

nonn,hear

Author

N. J. A. Sloane, Mar 16 2018

Status

approved