A335942 -id:A335942 - cp's OEIS frontend

A335941 Number of partitions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.

1, 1, 2, 1, 1, 4, 2, 5, 5, 9, 8, 15, 11, 14, 22, 28, 30, 36, 37, 53, 60, 80, 83, 104, 114, 148, 157, 201, 218, 283, 284, 362, 400, 455, 518, 624, 697, 807, 907, 1036, 1181, 1368, 1531, 1727, 1990, 2197, 2563, 2849, 3182, 3568, 4095, 4548, 5143, 5720, 6420

Offset: 0

Alois P. Heinz, Jun 30 2020

nonn

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 2: 11, 2.
a(3) = 1: 21.
a(4) = 1: 211.
a(5) = 4: 2111, 221, 311, 32.
a(6) = 2: 2211, 321.
a(7) = 5: 22111, 2221, 3211, 322, 331.
a(8) = 5: 22211, 32111, 3221, 3311, 332.
a(9) = 9: 222111, 321111, 32211, 3222, 33111, 3321, 42111, 4311, 432.
a(10) = 8: 2221111, 322111, 32221, 331111, 33211, 4222, 4321, 433.

Chai Wah Wu, Table of n, a(n) for n = 0..158 (n = 0..120 from Alois P. Heinz)

Cf. A317081, A317088, A335942.

b:= proc(n,i,s) option remember;
     `if`(n=0, `if`(s={$0..max(s)}, 1, 0), `if`(i<1, 0, add(
        b(n-i*j, i-1, {s[], j, `if`(j=0, 0, i)}), j=0..n/i)))
    end:
a:= n-> b(n, floor((sqrt(1+8*(n+1))-1)/2), {0}):
seq(a(n), n=0..55);

b[n_, i_, s_] := b[n, i, s] =
     If[n == 0, If[s == Range[0, Max[s]], 1, 0], If[i < 1, 0, Sum[
     b[n-i*j, i-1, Union@Flatten@{s, j, If[j == 0, 0, i]}], {j, 0, n/i}]]];
a[n_] := b[n, Floor[(Sqrt[1 + 8*(n + 1)] - 1)/2], {0}];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)

A338271 a(n) is the number of compositions of n, b_1 + ... + b_t = n such that sqrt(b_1 + sqrt(b_2 + ... + sqrt(b_t)...)) is an integer.

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 2, 2, 4, 2, 6, 2, 8, 4, 14, 6, 20, 8, 28, 14, 44, 20, 66, 30, 96, 46, 146, 70, 220, 102, 326, 154, 490, 232, 740, 346, 1102, 520, 1652, 782, 2484, 1166, 3716, 1750, 5568, 2628, 8358, 3936, 12518, 5900, 18760, 8848, 28138, 13256, 42170

Offset: 1

Peter Kagey, Oct 19 2020

nonn

a(n) <= Sum_{k=1..floor(sqrt(n)/2)} A338286(floor((n-4*k^2)/2)) when n is even.

a(n) <= Sum_{k=1..floor((sqrt(n) - 1)/2)} A338286(floor((n-4*k^2-4*k-1)/2)) when n is odd and greater than 1.

			(Let s(k) = sqrt(k) for brevity.)
For n = 14, the a(14) = 8 valid compositions are:
14 = 2+2+2+2+2+3+1 and 2 = s(2+s(2+s(2+s(2+s(2+s(3+s(1)))))))
14 = 1+7+2+3+1     and 2 = s(1+s(7+s(2+s(3+s(1)))))
14 = 2+1+7+3+1     and 2 = s(2+s(1+s(7+s(3+s(1)))))
14 = 2+2+1+8+1     and 2 = s(2+s(2+s(1+s(8+s(1)))))
14 = 2+2+2+2+2+4   and 2 = s(2+s(2+s(2+s(2+s(2+s(4))))))
14 = 1+7+2+4       and 2 = s(1+s(7+s(2+s(4))))
14 = 2+1+7+4       and 2 = s(2+s(1+s(7+s(4))))
14 = 2+2+1+9       and 2 = s(2+s(2+s(1+s(9))))

Peter Kagey, Table of n, a(n) for n = 1..500
Code Golf Stack Exchange, The square root of the square root of the square root of the...

Cf. A000196, A338268.

Cf. A098504, A101391, A143862, A238351, A238686, A325676. A335942.

a(n) = Sum_{i=k..A000196(n)} A338268(n,k).

A335443 Number of compositions of n where neighboring runs have different lengths.

Original entry on oeis.org

1, 1, 2, 2, 5, 8, 13, 24, 42, 68, 122, 210, 360, 622, 1077, 1858, 3198, 5519, 9549, 16460, 28386, 49031, 84595, 145988, 251956, 434805, 750418, 1294998, 2234971, 3857106, 6656383, 11487641, 19825318, 34214136, 59046458, 101901743, 175860875, 303498779

Offset: 0

Alois P. Heinz, Jul 06 2020

nonn

			a(0) = 1: the empty composition.
a(1) = 1: 1.
a(2) = 2: 2, 11.
a(3) = 2: 3, 111.
a(4) = 5: 4, 22, 112, 211, 1111.
a(5) = 8: 5, 113, 122, 221, 311, 1112, 2111, 11111.
a(6) = 13: 6, 33, 114, 222, 411, 1113, 1221, 2112, 3111, 11112, 11211, 21111, 111111.
a(7) = 24: 7, 115, 133, 223, 322, 331, 511, 1114, 1222, 2113, 2221, 3112, 4111, 11113, 11122, 11311, 21112, 22111, 31111, 111112, 111211, 112111, 211111, 1111111.
a(8) = 42: 8, 44, 116, 224, 233, 332, 422, 611, 1115, 1223, 1331, 2114, 2222, 3113, 3221, 4112, 5111, 11114, 11222, 11411, 12221, 21113, 22211, 31112, 41111, 111113, 111122, 111221, 111311, 112112, 113111, 122111, 211112, 211211, 221111, 311111, 1111112, 1111211, 1112111, 1121111, 2111111, 11111111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A003242, A011782, A329738, A329739, A329748, A329749, A329766, A335942.

b:= proc(n, l, t) option remember; `if`(n=0, 1, add(add(
      `if`(j=t, 0, b(n-i*j, i, j)), j=1..n/i), i={$1..n} minus {l}))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);

b[n_, l_, t_] := b[n, l, t] = If[n == 0, 1, Sum[Sum[If[j == t, 0,
     b[n-i*j, i, j]], {j, 1, n/i}], {i, Range[n]~Complement~{l}}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 13 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A335941 Number of partitions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.

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A338271 a(n) is the number of compositions of n, b_1 + ... + b_t = n such that sqrt(b_1 + sqrt(b_2 + ... + sqrt(b_t)...)) is an integer.

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A335443 Number of compositions of n where neighboring runs have different lengths.

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