| Displaying 1-6 of 6 results found.
page
1
Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-2,-1,0,1,2}.
+10
7
1, 2, 7, 29, 126, 565, 2583, 11971, 56038, 264345, 1254579, 5983628, 28655047, 137697549, 663621741, 3206344672, 15525816066, 75324830665, 366071485943, 1781794374016, 8684511754535, 42381025041490, 207055067487165, 1012617403658500, 4956924278927910
COMMENTS
By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
MATHEMATICA
frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -2, 2}]; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
PROG
(PARI) seq(n)={my(v=vector(n), m=sum(i=-2, 2, x^i), p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p, x, 1)); v} \\ Andrew Howroyd, Jun 27 2018
Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-2,-1,1,2}.
+10
7
1, 2, 5, 17, 58, 209, 761, 2823, 10557, 39833, 151147, 576564, 2208163, 8486987, 32714813, 126430229, 489685674, 1900350201, 7387530575, 28763059410, 112142791763, 437774109384, 1710883748796, 6693281604018, 26210038447737, 102724200946467, 402925631267151
COMMENTS
By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
MATHEMATICA
frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -2, 2}] - 1; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
PROG
(PARI) seq(n)={my(v=vector(n), m=sum(i=-2, 2, x^i)-1, p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p, x, 1)); v} \\ Andrew Howroyd, Jun 27 2018
Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,0,1,2,3}.
+10
6
1, 3, 15, 87, 530, 3329, 21316, 138345, 906853, 5989967, 39804817, 265812731, 1782288408, 11991201709, 80911836411, 547334588037, 3710610424765, 25204313298581, 171492983631249, 1168638213247713, 7974592724571446, 54484621312318007, 372671912259214487
COMMENTS
By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
MATHEMATICA
frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}]; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
PROG
(PARI) seq(n)={my(v=vector(n), m=sum(i=-3, 3, x^i), p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p, x, 1)); v} \\ Andrew Howroyd, Jun 27 2018
Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-4,-3,-2,-1,0,1,2,3,4}.
+10
6
1, 4, 26, 194, 1521, 12289, 101205, 844711, 7120398, 60477913, 516774114, 4437360897, 38256405777, 330948944639, 2871299293535, 24973776734091, 217690276938940, 1901204163460913, 16632544424086901, 145730139895667887, 1278596503973570665, 11231908572986043199
COMMENTS
By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
MATHEMATICA
frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -4, 4}]; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
PROG
(PARI) seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i), p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p, x, 1)); v} \\ Andrew Howroyd, Jun 27 2018
Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}.
+10
6
1, 3, 12, 60, 311, 1674, 9173, 51002, 286384, 1620776, 9228724, 52810792, 303447096, 1749612736, 10117583749, 58656027314, 340806249367, 1984018271850, 11569932938192, 67574451148408, 395214184047366, 2314315680481252, 13567587349336459, 79621279809031310
COMMENTS
By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
MATHEMATICA
frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}] - 1; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
PROG
(PARI) seq(n)={my(v=vector(n), m=sum(i=-3, 3, x^i)-1, p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p, x, 1)); v} \\ Andrew Howroyd, Jun 27 2018
Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-4,-3,-2,-1,1,2,3,4}.
+10
1
1, 4, 26, 174, 1231, 8899, 65492, 487646, 3664123, 27723979, 210946444, 1612394958, 12371547879, 95230159650, 735060394986, 5687343753535, 44096482961189, 342530654187820, 2665058975987628, 20765913987073659, 162019898098364055, 1265622208055843635
MATHEMATICA
seq[n_] := Module[{v = Table[1, n], m = Sum[ x^i, {i, -4, 4}] - 1, p = 1}, For[i = 2, i <= n, i++, p = Expand[p*m]; p = p - Select[p, Exponent[#, x] < 0&]; v[[i]] = (p /. x -> 1)]; v];
PROG
(PARI) seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i)-1, p=1); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p, x, 1)); v} \\ Andrew Howroyd, Jun 27 2018
Search completed in 0.006 seconds
|