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%I #44 Mar 06 2020 09:19:56
%S 1,1,2,2,4,4,7,8,11,14,19,22,30,36,46,55,70,83,104,123,151,179,218,
%T 256,309,363,433,507,602,701,828,961,1127,1306,1525,1759,2046,2355,
%U 2725,3129,3609,4131,4750,5424,6214,7081,8090,9195,10478,11886,13506,15290,17335,19583,22154,24981,28197,31741,35757,40176,45176
%N Partitions with parts repeated at most twice and repetition only allowed if first part has an odd index (first index = 1).
%H Alois P. Heinz, <a href="/A227134/b227134.txt">Table of n, a(n) for n = 0..10000</a>
%F Conjecture: A227134(n) + A227135(n) = A182372(n) for n >= 0, see comment in A182372.
%F 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
%F 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
%e 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 + ...
%e 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))) + ...
%e 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:
%e 01: [ 1:0 1:1 2:0 2:1 3:0 4:1 ]
%e 02: [ 1:0 1:1 2:0 2:1 7:0 ]
%e 03: [ 1:0 1:1 2:0 3:1 6:0 ]
%e 04: [ 1:0 1:1 2:0 4:1 5:0 ]
%e 05: [ 1:0 1:1 2:0 9:1 ]
%e 06: [ 1:0 1:1 3:0 3:1 5:0 ]
%e 07: [ 1:0 1:1 3:0 8:1 ]
%e 08: [ 1:0 1:1 4:0 7:1 ]
%e 09: [ 1:0 1:1 5:0 6:1 ]
%e 10: [ 1:0 1:1 11:0 ]
%e 11: [ 1:0 2:1 3:0 3:1 4:0 ]
%e 12: [ 1:0 2:1 3:0 7:1 ]
%e 13: [ 1:0 2:1 4:0 6:1 ]
%e 14: [ 1:0 2:1 5:0 5:1 ]
%e 15: [ 1:0 2:1 10:0 ]
%e 16: [ 1:0 3:1 4:0 5:1 ]
%e 17: [ 1:0 3:1 9:0 ]
%e 18: [ 1:0 4:1 8:0 ]
%e 19: [ 1:0 5:1 7:0 ]
%e 20: [ 1:0 12:1 ]
%e 21: [ 2:0 2:1 3:0 6:1 ]
%e 22: [ 2:0 2:1 4:0 5:1 ]
%e 23: [ 2:0 2:1 9:0 ]
%e 24: [ 2:0 3:1 4:0 4:1 ]
%e 25: [ 2:0 3:1 8:0 ]
%e 26: [ 2:0 4:1 7:0 ]
%e 27: [ 2:0 5:1 6:0 ]
%e 28: [ 2:0 11:1 ]
%e 29: [ 3:0 3:1 7:0 ]
%e 30: [ 3:0 4:1 6:0 ]
%e 31: [ 3:0 10:1 ]
%e 32: [ 4:0 4:1 5:0 ]
%e 33: [ 4:0 9:1 ]
%e 34: [ 5:0 8:1 ]
%e 35: [ 6:0 7:1 ]
%e 36: [13:0 ]
%p ## Computes A227134 and A227135 in order n^2 time and order n^2 memory:
%p a34:=proc(n) # n-th term of A227134
%p return oddMin(n,1):
%p end proc:
%p a35:=proc(n) # n-th term of A227135
%p return evenMin(n,1):
%p end proc:
%p # oddMin(n,m) finds number of partitions of n (as in A227134) but with the
%p # minimum part AT LEAST m
%p oddMin:=proc(n, m) option remember:
%p if(n=0) then return 1: fi: ## Start base cases
%p if((n<0) or (m>n)) then return 0: fi:
%p if(n=m) then return 1: fi: ## End base cases
%p 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
%p end proc:
%p # evenMin(n,m) finds number of partitions of n (as in A227135) but with the
%p # minimum part AT LEAST m
%p evenMin:=proc(n, m) option remember:
%p if(n=0) then return 1: fi: ## Start base cases
%p if((n<0) or (m>n)) then return 0: fi:
%p if(n=m) then return 1: fi: ## End base cases
%p return evenMin(n, m+1) + oddMin(n-m, m+1): ## Is the element m in the partition
%p end proc:
%p ## _Patrick Devlin_, Jul 02 2013
%p # second Maple program:
%p b:= proc(n, i, t) option remember; `if`(n=0, t,
%p `if`(i*(i+1)<n, 0, add(b(n-i*j, i-1,
%p irem(t+j, 2)), j=0..min(t+1, n/i))))
%p end:
%p a:= n-> add(b(n$2, t), t=0..1):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Feb 15 2017
%t 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_ *)
%o (PARI) {A002620(n)=floor(n/2)*ceil(n/2)}
%o {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)}
%o for(n=0,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Jul 06 2013
%Y Cf. A227135 (parts may repeat after even index).
%K nonn
%O 0,3
%A _Joerg Arndt_, Jul 02 2013