This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227134 #48 Feb 23 2025 07:04:59
%S A227134 1,1,2,2,4,4,7,8,11,14,19,22,30,36,46,55,70,83,104,123,151,179,218,
%T A227134 256,309,363,433,507,602,701,828,961,1127,1306,1525,1759,2046,2355,
%U A227134 2725,3129,3609,4131,4750,5424,6214,7081,8090,9195,10478,11886,13506,15290,17335,19583,22154,24981,28197,31741,35757,40176,45176
%N A227134 Partitions with parts repeated at most twice and repetition only allowed if first part has an odd index (first index = 1).
%H A227134 Alois P. Heinz, <a href="/A227134/b227134.txt">Table of n, a(n) for n = 0..10000</a>
%F A227134 Conjecture: A227134(n) + A227135(n) = A182372(n) for n >= 0, see comment in A182372.
%F A227134 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 A227134 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 A227134 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 A227134 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 A227134 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 A227134 01: [ 1:0 1:1 2:0 2:1 3:0 4:1 ]
%e A227134 02: [ 1:0 1:1 2:0 2:1 7:0 ]
%e A227134 03: [ 1:0 1:1 2:0 3:1 6:0 ]
%e A227134 04: [ 1:0 1:1 2:0 4:1 5:0 ]
%e A227134 05: [ 1:0 1:1 2:0 9:1 ]
%e A227134 06: [ 1:0 1:1 3:0 3:1 5:0 ]
%e A227134 07: [ 1:0 1:1 3:0 8:1 ]
%e A227134 08: [ 1:0 1:1 4:0 7:1 ]
%e A227134 09: [ 1:0 1:1 5:0 6:1 ]
%e A227134 10: [ 1:0 1:1 11:0 ]
%e A227134 11: [ 1:0 2:1 3:0 3:1 4:0 ]
%e A227134 12: [ 1:0 2:1 3:0 7:1 ]
%e A227134 13: [ 1:0 2:1 4:0 6:1 ]
%e A227134 14: [ 1:0 2:1 5:0 5:1 ]
%e A227134 15: [ 1:0 2:1 10:0 ]
%e A227134 16: [ 1:0 3:1 4:0 5:1 ]
%e A227134 17: [ 1:0 3:1 9:0 ]
%e A227134 18: [ 1:0 4:1 8:0 ]
%e A227134 19: [ 1:0 5:1 7:0 ]
%e A227134 20: [ 1:0 12:1 ]
%e A227134 21: [ 2:0 2:1 3:0 6:1 ]
%e A227134 22: [ 2:0 2:1 4:0 5:1 ]
%e A227134 23: [ 2:0 2:1 9:0 ]
%e A227134 24: [ 2:0 3:1 4:0 4:1 ]
%e A227134 25: [ 2:0 3:1 8:0 ]
%e A227134 26: [ 2:0 4:1 7:0 ]
%e A227134 27: [ 2:0 5:1 6:0 ]
%e A227134 28: [ 2:0 11:1 ]
%e A227134 29: [ 3:0 3:1 7:0 ]
%e A227134 30: [ 3:0 4:1 6:0 ]
%e A227134 31: [ 3:0 10:1 ]
%e A227134 32: [ 4:0 4:1 5:0 ]
%e A227134 33: [ 4:0 9:1 ]
%e A227134 34: [ 5:0 8:1 ]
%e A227134 35: [ 6:0 7:1 ]
%e A227134 36: [13:0 ]
%p A227134 ## Computes A227134 and A227135 in order n^2 time and order n^2 memory:
%p A227134 a34:=proc(n) # n-th term of A227134
%p A227134 return oddMin(n,1):
%p A227134 end proc:
%p A227134 a35:=proc(n) # n-th term of A227135
%p A227134 return evenMin(n,1):
%p A227134 end proc:
%p A227134 # oddMin(n,m) finds number of partitions of n (as in A227134) but with the
%p A227134 # minimum part AT LEAST m
%p A227134 oddMin:=proc(n, m) option remember:
%p A227134 if(n=0) then return 1: fi: ## Start base cases
%p A227134 if((n<0) or (m>n)) then return 0: fi:
%p A227134 if(n=m) then return 1: fi: ## End base cases
%p A227134 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 A227134 end proc:
%p A227134 # evenMin(n,m) finds number of partitions of n (as in A227135) but with the
%p A227134 # minimum part AT LEAST m
%p A227134 evenMin:=proc(n, m) option remember:
%p A227134 if(n=0) then return 1: fi: ## Start base cases
%p A227134 if((n<0) or (m>n)) then return 0: fi:
%p A227134 if(n=m) then return 1: fi: ## End base cases
%p A227134 return evenMin(n, m+1) + oddMin(n-m, m+1): ## Is the element m in the partition
%p A227134 end proc:
%p A227134 ## _Patrick Devlin_, Jul 02 2013
%p A227134 # second Maple program:
%p A227134 b:= proc(n, i, t) option remember; `if`(n=0, t,
%p A227134 `if`(i*(i+1)<n, 0, add(b(n-i*j, i-1,
%p A227134 irem(t+j, 2)), j=0..min(t+1, n/i))))
%p A227134 end:
%p A227134 a:= n-> add(b(n$2, t), t=0..1):
%p A227134 seq(a(n), n=0..60); # _Alois P. Heinz_, Feb 15 2017
%t A227134 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 A227134 (PARI) {A002620(n)=floor(n/2)*ceil(n/2)}
%o A227134 {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 A227134 for(n=0,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Jul 06 2013
%Y A227134 Cf. A227135 (parts may repeat after even index).
%K A227134 nonn
%O A227134 0,3
%A A227134 _Joerg Arndt_, Jul 02 2013