Madjid Mirzavaziri has authored 3 sequences.
A188430
a(n) is the maximum of the largest elements of all n-full sets, or 0 if no such set exists.
Original entry on oeis.org
1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38
Offset: 1
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a188430 n = a188430_list !! (n-1)
a188430_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9] ++
(drop 19 a008619_list)
-- Reinhard Zumkeller, Aug 06 2015
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LinearRecurrence[{1,1,-1},{1,0,2,0,0,3,4,0,0,4,5,6,7,7,8,6,7,8,9,10,11,11},80] (* Harvey P. Dale, Jul 24 2021 *)
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Vec(x*(1 - x + x^2 - x^3 - 2*x^4 + 5*x^5 + x^6 - 7*x^7 - x^8 + 8*x^9 + x^10 - 3*x^11 - x^13 - 2*x^15 + 3*x^17 - x^21) / ((1 - x)^2*(1 + x)) + O(x^80)) \\ Colin Barker, May 11 2020
A188431
The number of n-full sets, F(n).
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 10, 13, 14, 17, 20, 25, 28, 34, 40, 46, 54, 62, 69, 80, 90, 102, 115, 131, 144, 167, 186, 213, 239, 273, 304, 349, 388, 441, 495, 563, 625, 710, 790, 890, 990, 1114, 1232, 1387, 1530, 1713, 1894, 2119, 2330, 2605, 2866, 3192, 3512, 3910, 4289, 4774, 5237, 5809, 6377, 7068, 7739
Offset: 0
a(26) = 10, because there are 10 26-full sets: {1,2,4,5,6,8}, {1,2,3,5,7,8}, {1,2,3,5,6,9}, {1,2,3,4,7,9}, {1,2,3,4,6,10}, {1,2,3,4,5,11}, {1,2,4,8,11}, {1,2,4,7,12}, {1,2,4,6,13}, {1,2,3,7,13}.
G.f.: 1 = 1/(1+x) + 1*x/((1+x)*(1+x^2)) + 0*x^2/((1+x)*(1+x^2)*(1+x^3)) + 1*x^3/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...+ a(n)*x^n / Product_{k=1..n+1} (1+x^k) +...
Cf.
A002033,
A103295,
A108917,
A276024,
A325763,
A325765,
A325781,
A325782,
A325788,
A325986,
A325790,
A325791.
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import Data.MemoCombinators (memo2, integral, Memo)
a188431 n = a188431_list !! (n-1)
a188431_list = map
(\x -> sum [fMemo x i | i <- [a188429 x .. a188430 x]]) [1..] where
fMemo = memo2 integral integral f
f _ 1 = 1
f m i = sum [fMemo (m - i) j |
j <- [a188429 (m - i) .. min (a188430 (m - i)) (i - 1)]]
-- Reinhard Zumkeller, Aug 06 2015
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sums:= proc(s) local i, m;
m:= max(s[]);
`if`(m<1, {}, {m, seq([i, i+m][], i=sums(s minus {m}))})
end:
a:= proc(n) local b;
b:= proc(i,s) local si;
if i=1 then `if`(sums(s)={$1..n}, 1, 0)
else si:= s union {i};
b(i-1, s)+ `if`(max(sums(si)[])>n, 0, b(i-1, si))
fi
end; b(n, {1})
end:
seq(a(n), n=1..40); # Alois P. Heinz, Apr 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2n or i>n-i+1, 0, b(n-i, i-1))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..80); # Alois P. Heinz, May 20 2017
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Sums[s_] := Sums[s] = With[{m = Max[s]}, If[m < 1, {}, Union @ Flatten @ Join[{m}, Table[{i, i + m}, {i, Sums[s ~Complement~ {m}]}]]]];
a[n_] := Module[{b}, b[i_, s_] := b[i, s] = Module[{si}, If[i == 1, If[Sums[s] == Range[n], 1, 0], si = s ~Union~ {i}; b[i-1, s] + If[Max[ Sums[si]] > n, 0, b[i - 1, si]]]]; b[n, {1}]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 80}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Union[Total/@Union[Subsets[#]]]==Range[0,n]&]],{n,30}] (* Gus Wiseman, May 31 2019 *)
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/* As coefficients in g.f. */
{a(n)=local(A=[1]); for(i=1, n+1, A=concat(A,0); A[#A]=polcoeff(1 - sum(m=1,#A,A[m]*x^m/prod(k=1, m, 1+x^k +x*O(x^#A) )), #A) ); A[n+1]}
for(n=0, 50, print1(a(n),", ")) /* Paul D. Hanna, Mar 06 2012 */
A188429
L(n) is the minimum of the largest elements of all n-full sets, or 0 if no such set exists.
Original entry on oeis.org
1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 12, 13, 13
Offset: 1
From _Reinhard Zumkeller_, Aug 06 2015: (Start)
Compressed table: no commas and for a and k: 10 replaced by A, 11 by B.
. -----------------------------------------------------------------------------
. n 1 5 10 15 20 25 30 35 40 45 50 55 60 65 70
. ---- .---.----.----.----.----.----.----.----.----.----.----.----.----.----.-
. t(n) 10100100010000100000100000010000000100000000100000000010000000000100000
. k(n) 1 2 3 4 5 6 7 8 9 A B
. r(n) 0101201230123401234501234560123456701234567801234567890123456789A012345
. ---- -----------------------------------------------------------------------
. a(n) 102003400455675666776777788788888998999999AA9AAAAAAABBABBBBBBBBCCBCCCCC
. -----------------------------------------------------------------------------
where t(n)=A010054(n), k(n)=A127648(n) zeros blanked, and r(n)=A002262(n). (End)
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a188429 n = a188429_list !! (n-1)
a188429_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7] ++
f [15 ..] (drop 15 a010054_list) 0 4
where f (x:xs) (t:ts) r k | t == 1 = (k + 1) : f xs ts 1 (k + 1)
| r < k - 1 = (k + 1) : f xs ts (r + 1) k
| otherwise = (k + 2) : f xs ts (r + 1) k
-- Reinhard Zumkeller, Aug 06 2015
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kr[n_] := {k, r} /. ToRules[Reduce[0 <= r <= k && n == k*((k+1)/2)+r, {k, r}, Integers]]; L[n_] := Which[{k0, r0} = kr[n]; r0 == 0, k0, 1 <= r0 <= k0-2, k0+1, k0-1 <= r0 <= k0, k0+2]; Join[{1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7}, Table[L[n], {n, 15, 80}]] (* Jean-François Alcover, Oct 10 2015 *)
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