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%I #43 Mar 06 2020 09:25:03
%S 1,1,1,2,2,4,4,6,8,10,12,17,20,25,31,39,47,58,69,85,102,123,145,175,
%T 207,246,290,343,401,473,551,646,751,875,1012,1177,1358,1570,1807,
%U 2083,2389,2746,3140,3597,4106,4690,5337,6082,6907,7848,8895,10085,11404,12902,14561,16438,18520,20864,23460,26385,29619
%N Partitions with parts repeated at most twice and repetition only allowed if first part has an even index (first index = 1).
%H Alois P. Heinz, <a href="/A227135/b227135.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+2)-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 = 2^(3/4) / (sqrt(5)*(1 + sqrt(5))^(3/2)) = 0.1291995618069... - _Vaclav Kotesovec_, May 28 2018, updated Mar 06 2020
%e G.f.: 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 8*x^8 +...
%e G.f.: 1/(1-x) + x^3/((1-x)*(1-x^2)) + x^5/((1-x)*(1-x^2)*(1-x^3)) + x^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + x^11/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) + x^15/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))) +...
%e There are a(13)=25 such partitions, displayed here as partitions into two sorts of parts (format P:S for sort:part) where the first sort is 1 and sorts oscillate:
%e 01: [ 1:1 2:0 2:1 3:0 5:1 ]
%e 02: [ 1:1 2:0 2:1 4:0 4:1 ]
%e 03: [ 1:1 2:0 2:1 8:0 ]
%e 04: [ 1:1 2:0 3:1 7:0 ]
%e 05: [ 1:1 2:0 4:1 6:0 ]
%e 06: [ 1:1 2:0 10:1 ]
%e 07: [ 1:1 3:0 3:1 6:0 ]
%e 08: [ 1:1 3:0 4:1 5:0 ]
%e 09: [ 1:1 3:0 9:1 ]
%e 10: [ 1:1 4:0 8:1 ]
%e 11: [ 1:1 5:0 7:1 ]
%e 12: [ 1:1 6:0 6:1 ]
%e 13: [ 1:1 12:0 ]
%e 14: [ 2:1 3:0 3:1 5:0 ]
%e 15: [ 2:1 3:0 8:1 ]
%e 16: [ 2:1 4:0 7:1 ]
%e 17: [ 2:1 5:0 6:1 ]
%e 18: [ 2:1 11:0 ]
%e 19: [ 3:1 4:0 6:1 ]
%e 20: [ 3:1 5:0 5:1 ]
%e 21: [ 3:1 10:0 ]
%e 22: [ 4:1 9:0 ]
%e 23: [ 5:1 8:0 ]
%e 24: [ 6:1 7:0 ]
%e 25: [13:1 ]
%p ## See A227134
%p # second Maple program:
%p b:= proc(n, i, t) option remember; `if`(n=0, 1-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 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1 - t, If[i*(i + 1) < n, 0, Sum[ b[n - i*j, i - 1, Mod[t + j, 2]], {j, 0, Min[t + 1, n/i]}]]];
%t a[n_] := Sum[b[n, n, t], {t, 0, 1}];
%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 21 2018, after _Alois P. Heinz_ *)
%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+2)-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. A227134 (parts may repeat after odd index).
%K nonn
%O 0,4
%A _Joerg Arndt_, Jul 02 2013
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