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 A227134 Partitions with parts repeated at most twice and repetition only allowed if first part has an odd index (first index = 1). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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) 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

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Last modified January 26 06:32 EST 2021. Contains 340434 sequences. (Running on oeis4.)