A227134 Partitions with parts repeated at most twice and repetition only allowed if first part has an odd index (first index = 1).

1, 1, 2, 2, 4, 4, 7, 8, 11, 14, 19, 22, 30, 36, 46, 55, 70, 83, 104, 123, 151, 179, 218, 256, 309, 363, 433, 507, 602, 701, 828, 961, 1127, 1306, 1525, 1759, 2046, 2355, 2725, 3129, 3609, 4131, 4750, 5424, 6214, 7081, 8090, 9195, 10478, 11886, 13506, 15290, 17335, 19583, 22154, 24981, 28197, 31741, 35757, 40176, 45176

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

Conjecture: A227134(n) + A227135(n) = A182372(n) for n >= 0, see comment in A182372.

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

## Computes A227134 and A227135 in order n^2 time and order n^2 memory:
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

Cf. A227135 (parts may repeat after even index).

Sequence in context: A095700 A339404 A035944 * A240013 A050366 A332753

Adjacent sequences: A227131 A227132 A227133 * A227135 A227136 A227137

Keyword

nonn

Author

Joerg Arndt, Jul 02 2013

Status

approved