LATTICE SELF-RETURN RESTRICTED WALK COUNTS
Annotated computer output, Warren D.Smith, Nov 2014.

W=wedge angle in degrees.  
If W=0, return to 00 only possible if #steps even, so only every 2nd count given.
H=1 if all 6 neighbors allowed in eq.tri. lattice as walk steps.
H=2 if only the 3 even numbered neighbors allowed.  When H=2, return to 00 only is possible if 
#steps divisible by 3, hence only every 3rd sequence member is given.
Number of returning to 00 walks given in sequence.
For many of these, the sequence happens to be in OEIS, but also for many it is not.

W=0, H=1: http://oeis.org/A000108
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
2*w[n]*( (-4+12*n)*n-5 ) = w[n+1]*( (7+6*n)*n-10 )
1*w[n]*( (-8+16*n)*n-8 ) = w[n+1]*( (4+4*n)*n-8 )

W=30, H=1: unknown to OEIS:  
1,0,1,0,3,2,16,28,130,358,1432,4900,19314,73706,298267,1212212,5075717,21528440,93069979,407787376,
1812466904,8153337204,37106793736,170651907678,792569501641,3714453391034,17556196588970,
83635548269590,401384489558378,1939712828700004,9434982573962381,46174843688639676,
227289495212760658,

W=60, H=1: http://oeis.org/A151366:
1, 0, 2, 2, 12, 30, 130, 462, 1946, 7980, 34776, 153120, 694056, 3194334, 14971242, 71133062
gf known.

W=90, H=1: http://oeis.org/A057648:
1, 0, 2, 2, 13, 34, 158, 594, 2665, 11558, 53320, 247488, 1181266, 5708884, 28049474, 139417402

W=120, H=1: http://oeis.org/A151372:
1, 0, 3, 4, 26, 80, 387, 1596, 7518, 34656, 167310, 813384, 4040212, 20285408, 103195235
No quadratic(n)*w[n]=quadratic(n)*w[n+1] recurrence found.

W=150, H=1: unknown to OEIS:  
1,0,3,4,27,84,419,1756,8473,39752,195760,968104,4892378,24957278,128936196,671983900,3533365192,
18714554170,99796938574,535352099024,2887534929785,15651152169808,85215057908750,465873973561938,
2556583654159437,14078689046794064,77778973017819575,
No quadratic(n)*w[n]=quadratic(n)*w[n+1] recurrence found.

W=180, H=1: unknown to OEIS:  
1,0,4,6,42,140,720,3150,15610,75432,380016,1921920,9901584,51438816,270160176,1429890462,
7626646170,40939712528,221074926072,1200080565684,6545819168460,35859031869960,197222301254160,
1088653176365040,6029402078784240,33496355260791360,186620994454720000,1042495547306341440,
5837935436913591360,
Alin Bostan: this sequence appears to satisfy the linear recurrence
u(N+3) = 36*(N+2)*(N+1)/((N+3)*(N+5))*u(N) 
  +6*(N+2)*(4*N^2+25*N+37)/((N+5)*(N+4)*(N+3))*u(N+1)
  +(N+3)*(N+2)/((N+4)*(N+5)) *u(N+2)
rewrite:
(N+3)*(N+4)*(N+5))*u(N+3) = 36*(N+4)*(N+2)*(N+1)*u(N) 
  +6*(N+2)*(4*N^2+25*N+37)*u(N+1)
  +(N+3)^2*(N+2) *u(N+2)

W=60, H=2: http://oeis.org/A005789 = http://oeis.org/A151334:
1, 1, 5, 42, 462, 6006, 87516, 1385670, 23371634, 414315330, 7646001090, 145862174640, 
recurrence:  3*w[n]*( (9+9*n)*n+2 ) = w[n+1]*( (5+1*n)*n+6 )
a(n) = 2*(3*n)!/(n!*(n+1)!*(n+2)!);

W=90, H=2: http://oeis.org/A218441: 
1, 1, 6, 60, 770, 11466, 188496, 3325608, 61866090, 1199333850, 24030289140, 494663027040, 
binomial(2*n,n)/(n+1) * binomial(3*n,n)/(2*n+1)

W=120, H=2: http://oeis.org/A006335:  Kreweras walks
1, 2, 16, 192, 2816, 46592, 835584, 15876096, 315031552, 6466437120, 136383037440
4^n*(3*n)!/((n+1)!*(2*n+1)!)
(2+n)*(2*n+3)*a(n+1) = 6*(3*n+2)*(3*n+1)*a(n).
recurrence: 6*w[n]*( (9+9*n)*n+2 ) = w[n+1]*( (7+2*n)*n+6 )
Germain Kreweras: Sur une classe de problemes de denombrement lies au treillis 
des partitions des entiers, 
Cahiers du Bureau Universitaire de Recherche Operationnelle, 
Institut de Statistique, Universite de Paris 6 (1965), circa p. 82.

W=150, H=2: unknown to OEIS:  
1,2,17,216,3330,57545,1072326,21082108,431406658,9106370280,197047243752,4351014294393,
97708108984581,2225656274729522,51319743620874018
No quadratic(n)*w[n]=quadratic(n)*w[n+1] recurrence found.

W=180, H=2: http://oeis.org/A007004:
1, 3, 30, 420, 6930, 126126, 2450448, 49884120, 1051723530, 22787343150, 504636071940
 (3*n)! / ((n+1)*(n!)^3)
n*(n+1)*a(n) = 3*(3*n-1)*(3*n-2)*a(n-1).
recurrence: 3*w[n]*( (9+9*n)*n+2 ) = w[n+1]*( (3+1*n)*n+2 )

W=360: H=1: http://oeis.org/A002898
1, 0, 6, 12, 90, 360, 2040, 10080, 54810, 290640, 1588356, 8676360, 47977776, 266378112, 
1488801600, 8355739392, 47104393050, 266482019232, 1512589408044, 8610448069080,
49144928795820, 281164160225520
a(0)=1,  a(1)=0,  a(2)=6, 
(9+6*n+n^2)*a(n+3) = (108*n+72+36*n^2)*a(n)+(24*n^2+96*n+96)*a(n+1)+(n^2+5*n+6)*a(n+2).
Number of n-step closed paths on hexagonal lattice

W=360: H=2, http://oeis.org/A006480
1, 6, 90, 1680, 34650, 756756, 17153136, 399072960, 9465511770, 227873431500, 5550996791340, 
136526995463040, 3384731762521200, 84478098072866400, 2120572665910728000, 53494979785374631680,
1355345464406015082330
(3n)!/(n!)^3. 
Number of paths of length 3n in an n X n X n grid from (0,0,0) to (n,n,n).
OEIS says this arises from "Ramanujan elliptic function of signature 3"
which I presume means this Green function is transcendental.
  
also arose:  http://oeis.org/A098272:
2, 8, 96, 1536, 28160, 559104, 11698176, 254017536, 5670567936, 129328742400
2^(2*n+1) * binomial(3*n,n) / (2*n+1)