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<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,13,-3).
a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3), for n >= 5.
LinearRecurrence[{1, 1, 13, -3}, {1, 2, 3, 6, 11}, 61]
(Magma) I:=[3, 6, 11]; [1, 2] cat [n le 3 select I[n ] else Self(n-1) +3*Self(n-2) +-3*Self(n-3): n in [1..60]];
@CachedFunction
def A254006(n): return 3^(n/2)*(1 + (-1)^n)/2
def aA358027(n): return (n+1/3) if *( 4*A254006(n<3) else a+ 7*A254006(n-1) +a2*int(n-==0) + 2) +a*int(n==1) - 3 ) # a = A047081
[aA358027(n) for n in (0..60)]
allocated for G. C. Greubel
Expansion of g.f.: (1 + x - 2*x^2 + 2*x^4)/((1-x)*(1-3*x^2)).
1, 2, 3, 6, 11, 20, 35, 62, 107, 188, 323, 566, 971, 1700, 2915, 5102, 8747, 15308, 26243, 45926, 78731, 137780, 236195, 413342, 708587, 1240028, 2125763, 3720086, 6377291, 11160260, 19131875, 33480782, 57395627
0,2
<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1).
a(n) = (1/3)*(2*[n=0] + 2*[n=1] - 3 + 4*A254006(n) + 7*A254006(n-1))).
E.g.f.: (1/3)*( 2 + 2*x - 3*exp(x) + 4*cosh(sqrt(3)*x) + (7/sqrt(3))*sinh(sqrt(3)*x) ).
G.f.: (1 +x -2*x^2 +2*x^4)/((1-x)*(1-3*x^2)). - Clark Kimberling, Oct 31 2022
LinearRecurrence[{1, 1, 1}, {1, 2, 3}, 61]
(Magma) [n le 3 select n else Self(n-1) +Self(n-2) +Self(n-3): n in [1..60]];
(SageMath)
@CachedFunction
def a(n): return (n+1) if (n<3) else a(n-1) +a(n-2) +a(n-3) # a = A047081
[a(n) for n in (0..60)]
allocated
easy,nonn
G. C. Greubel, Oct 31 2022
approved
editing
allocated for G. C. Greubel
recycled
allocated
reviewed
approved
proposed
reviewed
editing
proposed
Distinct digital permutations of the first m terms of A033307 arranged lexicographically, for m = 1, 2, 3, ...
1, 12, 21, 123, 132, 213, 231, 312, 321, 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321, 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524
1,2
Permutations starting with the digit 0 are omitted.
(Python)
from itertools import islice, count
from sympy.utilities.iterables import multiset_permutations as mp
def bgen(): yield from (c for n in count(1) for c in str(n))
def agen():
s, g = "", bgen()
while True:
s += next(g)
yield from (int("".join(p)) for p in mp(s) if p[0] != "0")
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 25 2022
(Python)
def toint(l):
a=0; L=len(l); p=1
for i in range(L-1, -1, -1):
a+=l[i]*p; p*=10
return a
def f(n):
tot=0; L=0; ans=[]
while 1:
L+=1; cont=0; cur=1; d=[]
while cont<L:
cur1=cur; cur+=1
cur1=str(cur1)
while cur1:
lc=cur1[0]; cur1=cur1[1:]; d.append(int(lc)); cont+=1
if cont==L:
break
d=sorted(d)
if d[0]==0:
d.remove(1);
d=[1]+d
while 1:
ans.append(toint([*d])); tot+=1
if tot==n:
return ans
j=L-2
while j>-1:
if d[j]<d[j+1]:
for k in range(j+1, L):
if k==L-1:
d[j], d[k]=d[k], d[j]; d=d[:j+1]+sorted(d[j+1:])
elif d[k+1]<=d[j]:
d[j], d[k]=d[k], d[j]; d=d[:j+1]+sorted(d[j+1:]); break
break
j-=1
if j==-1:
break
print(f(10000)) # Michele Botta, Oct 25 2022
easy,nonn,base,changed
recycled
Michele Botta and Marco Ripà, Oct 25 2022
proposed
editing