A046171 -id:A046171 - cp's OEIS frontend

A046170 Number of self-avoiding walks on a 2-D lattice of length n which start at the origin, take first step in the {+1,0} direction and whose vertices are always nonnegative in x and y.

1, 2, 5, 12, 30, 73, 183, 456, 1151, 2900, 7361, 18684, 47652, 121584, 311259, 797311, 2047384, 5260692, 13542718, 34884239, 89991344, 232282110, 600281932, 1552096361, 4017128206, 10401997092, 26957667445, 69892976538, 181340757857, 470680630478, 1222433229262, 3175981845982

Offset: 1

Eric W. Weisstein

Siqi Wang, Table of n, a(n) for n = 1..40
Stephen A. Silver, C program
Siqi Wang, C++ program used to generate the sequence.
Eric Weisstein's World of Mathematics, Self-Avoiding Walk

Cf. A038373, A046171.

a(n) = A038373(n)/2. - Siqi Wang, Jul 15 2022

More terms from Stephen A. Silver

More terms from Siqi Wang, Jul 15 2022

A331030 Divide the terms of the harmonic series into groups sequentially so that the sum of each group is minimally greater than 1. a(n) is the number of terms in the n-th group.

Original entry on oeis.org

2, 5, 13, 36, 98, 266, 723, 1965, 5342, 14521, 39472, 107296, 291661, 792817, 2155100, 5858169, 15924154, 43286339, 117664468, 319845186, 869429357, 2363354022, 6424262292, 17462955450, 47469234471, 129034757473, 350752836478, 953445061679, 2591732385596

Offset: 1

Pablo Hueso Merino and Alejandro Argüelles Trujillo, Jan 07 2020

nonn

a(n) = A046171(n+1) through a(5), and grows similarly for n > 5.

Let b(n) = Sum_{j=1..n} a(n); then for n >= 2 it appears that b(n) = round((b(n-1) + 1/2)*e). Verified through n = 10000 (using the approximation Sum_{j=1..k} 1/j = log(k) + gamma + 1/(2*k) - 1/(12*k^2) + 1/(120*k^4) - 1/(252*k^6) + 1/(240*k^8) - ..., where gamma is the Euler-Mascheroni constant, A001620). Cf. A081881. - Jon E. Schoenfield, Jan 10 2020

			a(1)=2 because 1 + 1/2 = 1.5 > 1,
a(2)=5 because 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 1.0928... > 1,
etc.

Cf. A002387, A046171, A081881, A096005, A180224, A295572, A331028.

lista(lim=oo)={my(s=0, p=0); for(i=1, lim, s+=1/i; if(s>1, print1(i-p, ", "); s=0; p=i))} \\ Andrew Howroyd, Jan 08 2020

x = 0.0
y = 0.0
z = 0.0
for i in range(1,100000000000000000000000):
  y += 1
  x = x + 1/i
  z = z + 1/i
  if x > 1:
    print(y)
    y = 0
    x = 0

a(1)=2, a(n) = (min(p) : Sum_{s=r..p} 1/s > 1)-r+1, r=Sum_{k=1..n-1} a(k).

a(25)-a(29) from Jon E. Schoenfield, Jan 10 2020

cp's OEIS Frontend

A046170 Number of self-avoiding walks on a 2-D lattice of length n which start at the origin, take first step in the {+1,0} direction and whose vertices are always nonnegative in x and y.

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A331030 Divide the terms of the harmonic series into groups sequentially so that the sum of each group is minimally greater than 1. a(n) is the number of terms in the n-th group.

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