OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
FORMULA
G.f.: 1/(1-x) + Sum_{n>=2} x^A002620(n+1) / Product_{k=1..n} (1-x^k), where A002620(n) = floor(n/2)*ceiling(n/2) forms the quarter-squares. - Paul D. Hanna, Jul 06 2013
a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = 1 / (2^(1/4)*sqrt(5*(1 + sqrt(5)))) = 0.2090492823352... - Vaclav Kotesovec, May 28 2018, updated Mar 06 2020
EXAMPLE
G.f.: 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 8*x^7 + 11*x^8 + ...
G.f.: 1/(1-x) + x^2/((1-x)*(1-x^2)) + x^4/((1-x)*(1-x^2)*(1-x^3)) + x^6/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + x^9/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) + x^12/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))) + ...
There are a(13)=36 such partitions, displayed here as partitions into two sorts of parts (format P:S for sort:part) where the first sort is 0 and sorts oscillate:
01: [ 1:0 1:1 2:0 2:1 3:0 4:1 ]
02: [ 1:0 1:1 2:0 2:1 7:0 ]
03: [ 1:0 1:1 2:0 3:1 6:0 ]
04: [ 1:0 1:1 2:0 4:1 5:0 ]
05: [ 1:0 1:1 2:0 9:1 ]
06: [ 1:0 1:1 3:0 3:1 5:0 ]
07: [ 1:0 1:1 3:0 8:1 ]
08: [ 1:0 1:1 4:0 7:1 ]
09: [ 1:0 1:1 5:0 6:1 ]
10: [ 1:0 1:1 11:0 ]
11: [ 1:0 2:1 3:0 3:1 4:0 ]
12: [ 1:0 2:1 3:0 7:1 ]
13: [ 1:0 2:1 4:0 6:1 ]
14: [ 1:0 2:1 5:0 5:1 ]
15: [ 1:0 2:1 10:0 ]
16: [ 1:0 3:1 4:0 5:1 ]
17: [ 1:0 3:1 9:0 ]
18: [ 1:0 4:1 8:0 ]
19: [ 1:0 5:1 7:0 ]
20: [ 1:0 12:1 ]
21: [ 2:0 2:1 3:0 6:1 ]
22: [ 2:0 2:1 4:0 5:1 ]
23: [ 2:0 2:1 9:0 ]
24: [ 2:0 3:1 4:0 4:1 ]
25: [ 2:0 3:1 8:0 ]
26: [ 2:0 4:1 7:0 ]
27: [ 2:0 5:1 6:0 ]
28: [ 2:0 11:1 ]
29: [ 3:0 3:1 7:0 ]
30: [ 3:0 4:1 6:0 ]
31: [ 3:0 10:1 ]
32: [ 4:0 4:1 5:0 ]
33: [ 4:0 9:1 ]
34: [ 5:0 8:1 ]
35: [ 6:0 7:1 ]
36: [13:0 ]
MAPLE
a34:=proc(n) # n-th term of A227134
return oddMin(n, 1):
end proc:
a35:=proc(n) # n-th term of A227135
return evenMin(n, 1):
end proc:
# oddMin(n, m) finds number of partitions of n (as in A227134) but with the
# minimum part AT LEAST m
oddMin:=proc(n, m) option remember:
if(n=0) then return 1: fi: ## Start base cases
if((n<0) or (m>n)) then return 0: fi:
if(n=m) then return 1: fi: ## End base cases
return oddMin(n, m+1) + evenMin(n-m, m+1) + oddMin(n-2*m, m+1): ## How many times is the element m in the partition
end proc:
# evenMin(n, m) finds number of partitions of n (as in A227135) but with the
# minimum part AT LEAST m
evenMin:=proc(n, m) option remember:
if(n=0) then return 1: fi: ## Start base cases
if((n<0) or (m>n)) then return 0: fi:
if(n=m) then return 1: fi: ## End base cases
return evenMin(n, m+1) + oddMin(n-m, m+1): ## Is the element m in the partition
end proc:
## Patrick Devlin, Jul 02 2013
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, t,
`if`(i*(i+1)<n, 0, add(b(n-i*j, i-1,
irem(t+j, 2)), j=0..min(t+1, n/i))))
end:
a:= n-> add(b(n$2, t), t=0..1):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 15 2017
MATHEMATICA
nMax = 60; 1/(1-x) + Sum[x^Floor[(n+1)^2/4]/Product[1-x^k, {k, 1, n}], {n, 2, Ceiling @ Sqrt[4*nMax]}] + O[x]^(nMax+1) // CoefficientList[#, x]& (* Jean-François Alcover, Feb 15 2017, after Paul D. Hanna *)
PROG
(PARI) {A002620(n)=floor(n/2)*ceil(n/2)}
{a(n)=polcoeff(1/(1-x+x*O(x^n)) + sum(m=2, sqrtint(4*n), x^A002620(m+1)/prod(k=1, m, 1-x^k+x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 06 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jul 02 2013
STATUS
approved