Emely Hanna Li Lobnig has authored 5 sequences.
A379463
a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -1.
Original entry on oeis.org
1, 1, 1, 1, 3, 11, 31, 71, 150, 334, 826, 2146, 5498, 13690, 33762, 84306, 214451, 551107, 1417291, 3637627, 9343555, 24096675, 62439587, 162331747, 422773098, 1102422546, 2879207046, 7534606366, 19756893196, 51894005428, 136496647696, 359478351816, 947912008073
Offset: 0
For n = 4, the a(4)=3 paths are HHHH, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).
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A379463 := proc(n)
add(2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2),k=0..floor(n/4)) ;
end proc:
seq(A379463(n),n=0..50) ; # R. J. Mathar, Jan 29 2025
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a(n) = sum(k=0, floor(n/4), 2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2)) \\ Thomas Scheuerle, Jan 07 2025
A379462
a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -3.
Original entry on oeis.org
1, 1, 1, 4, 13, 31, 75, 204, 561, 1499, 4001, 10814, 29364, 79704, 216672, 590764, 1614421, 4419049, 12116139, 33277722, 91546143, 252209535, 695803659, 1922166420, 5316714156, 14723570406, 40820144106, 113293243636, 314759548879, 875342190283, 2436582442381
Offset: 0
For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0). An example of a path with these steps, but not staying on or above y = -3, is for n=6: DDUUUU.
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lista(nn) = my(v=vector(nn+5), w); print1(v[4]=1); for(n=1, nn, w=v; for(i=1, n+3, w[i]+=v[i+2]; w[i+1]+=v[i]); v=w; print1(", ", v[4])); \\ Jinyuan Wang, Jan 07 2025
A379464
a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -2.
Original entry on oeis.org
1, 1, 1, 1, 4, 16, 46, 106, 226, 514, 1306, 3466, 9002, 22634, 56330, 142026, 364743, 945303, 2448511, 6323695, 16336885, 42363693, 110340297, 288229377, 753920796, 1973799396, 5174280216, 13588243696, 35748326836, 94188788164, 248464963876, 656148369796
Offset: 0
For n = 4, the a(4)=4 paths are HHHH, UDUU, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).
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f:= proc(n,y) option remember;
if n = 0 then if y = 0 then return 1 else return 0 fi fi;
if y > n then return 0 fi;
if y >= -1 then procname(n-1,y-1) + procname(n-1,y) + procname(n-1,y+3)
else procname(n-1,y) + procname(n-1,y+3)
fi;
end proc:
map(f, [$0..40],0); # Robert Israel, Jan 23 2025
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a(n) = sum(k=0, floor(n/4), 3*binomial(n, k*4)*binomial(4*k+3, k)/(4*k+3)) \\ Thomas Scheuerle, Jan 07 2025
A378849
a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.
Original entry on oeis.org
1, 1, 1, 3, 9, 21, 48, 120, 309, 787, 2011, 5215, 13652, 35894, 94823, 251889, 672285, 1801185, 4842757, 13064059, 35349463, 95912989, 260896318, 711338596, 1943690464, 5321704006, 14597781706, 40112702176, 110404515703, 304338523999, 840140172621, 2322386563353
Offset: 0
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a:= proc(n) option remember; `if`(n<4, [1$3, 3][n+1],
(2*(8*n^3+3*n^2-25*n-6)*a(n-1)-2*(n-1)*(12*n^2-9*n-10)*
a(n-2)+(43*n+13)*(n-1)*(n-2)*a(n-3)-31*(n-1)*(n-2)*
(n-3)*a(n-4))/(2*(2*n+3)*(n+3)*(n-2)))
end:
seq(a(n), n=0..31); # Alois P. Heinz, Dec 09 2024
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a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+1, k)/(k+1)) \\ Thomas Scheuerle, Dec 09 2024
A378850
a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -2.
Original entry on oeis.org
1, 1, 1, 4, 13, 31, 73, 190, 505, 1316, 3431, 9065, 24122, 64325, 172082, 462344, 1246685, 3371135, 9140289, 24847422, 67708743, 184906614, 505986933, 1387240401, 3810083424, 10481797131, 28880894706, 79692785251, 220203155689, 609242057143, 1687666776031
Offset: 0
For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0).
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a := n -> hypergeom([4/3, 5/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 2/3, 5/2, 2], -27/4):
seq(simplify(a(n)), n = 0..30); # Peter Luschny, Dec 18 2024
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a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+3, k+1)/(2*k+3)) \\ Thomas Scheuerle, Dec 09 2024