A083920 -id:A083920 - cp's OEIS frontend

A111138 Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.

0, 0, 1, 1, 2, 4, 4, 5, 7, 10, 10, 11, 13, 16, 20, 20, 21, 23, 26, 30, 35, 35, 36, 38, 41, 45, 50, 56, 56, 57, 59, 62, 66, 71, 77, 84, 84, 85, 87, 90, 94, 99, 105, 112, 120, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 165, 166, 168, 171, 175, 180, 186, 193, 201

Offset: 1

Arturo Magidin, Oct 17 2005; definition corrected Feb 01 2006

For a subgroup H of order p^n (p an odd prime) of the subgroup generated by all commutators [x_j,x_i] in the relatively free group F of class three and exponent p, freely generated by x_1, x_2,..., x_k, (k sufficiently large) the minimum size of the subgroup of [H,F] of F_3 is p^{kn - a(n)}.
The sequence arises when finding a purely numerical sufficient condition for the capability of p-groups of class two and exponent p, where p is an odd prime.
Partial sums of A002262. - Gionata Neri, Sep 04 2015

			a(31) = b(30) + a(b(30)) = 23 + a(23) = 23 + b(22) + a(b(22)) = 23 + 16 + a(16) = 39 + b(15) + a(b(15)) = 39 + 10 + a(10) = 49 + b(9) + a(b(9)) = 49 + 6 + a(6) = 55 + b(5) + a(b(5)) = 55 + 3 + a(3) = 58 + 1 = 59.

Arturo Magidin, Capable groups of prime exponent and class two II, arXiv:math/0506578 [math.GR], 2005.

Cf. A083920.

a[1] = a[2] = 0; a[3] = 1; a[n_] := a[n] = b[n - 1] + a[b[n - 1]]; b[n_] := n - Floor[(Sqrt[8n + 1] - 1)/2]; Array[a, 64] (* Robert G. Wilson v, Feb 01 2006 *)

a(n) = my(r,m=sqrtint(n<<1,&r)); if(r>1,2); \\ Kevin Ryde, Oct 26 2024

If we write n = (m choose 2) + s, 0<=s<=m, then a(n)=(m choose 3) + (s choose 2).

a(N) = Comb(T,2)+Comb(R,3) where R:=Round(Sqrt(2*N)) and T:=N-Comb(R,2). - Gerald Hillier, Nov 18 2017

More terms from Robert G. Wilson v, Feb 01 2006

A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(2a(n-1)), a(nontriangular(n)) = 2n, where triangular = A000217, nontriangular = A014132.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 16, 24, 20, 14, 9, 32, 48, 40, 28, 18, 13, 64, 96, 80, 56, 36, 26, 11, 128, 192, 160, 112, 72, 52, 22, 17, 256, 384, 320, 224, 144, 104, 44, 34, 25, 512, 768, 640, 448, 288, 208, 88, 68, 50, 21, 1024, 1536, 1280, 896, 576, 416, 176, 136, 100, 42, 15

Offset: 1

Antti Karttunen, May 18 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10440
Index entries for sequences that are permutations of the natural numbers

Inverse: A257798.
Cf. A000217, A002024, A010054, A083920.
Cf. also a similar permutation A220347 from which this differs for the first time at n=15, where a(15) = 9, while A220347(15) = 11.
a(n) differs from A054582(n-1) for the first time at n=21, where a(21) = 13, while A054582(20) = 11.

a(1) = 1; for n > 1: if A010054(n) = 1 [when n is triangular], then a(n) = 1+(2*a(A002024(n)-1)), otherwise a(n) = 2*a(A083920(n)).

A270967 Number of set partitions of [n] having the maximal possible number of pairs (m,m+1) such that m+1 is in some block b and m is in block b+1.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 21, 10, 4, 1, 76, 36, 15, 5, 1, 281, 133, 57, 21, 6, 1, 1052, 499, 218, 85, 28, 7, 1, 3974, 1890, 838, 339, 121, 36, 8, 1, 15113, 7207, 3233, 1343, 505, 166, 45, 9, 1, 57774, 27622, 12509, 5302, 2069, 726, 221, 55, 10, 1, 221781, 106289

Offset: 0

Alois P. Heinz, Mar 26 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A083920, A270953.

a(n) = A270953(n,n-floor((1+sqrt(max(0,8*n-7)))/2)).

cp's OEIS Frontend

A111138 Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.

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A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(2a(n-1)), a(nontriangular(n)) = 2n, where triangular = A000217, nontriangular = A014132.

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A270967 Number of set partitions of [n] having the maximal possible number of pairs (m,m+1) such that m+1 is in some block b and m is in block b+1.

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cp's OEIS Frontend

A111138 Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.

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Examples

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Programs

Mathematica

PARI

Formula

Extensions

A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(2*a(n-1)), a(nontriangular(n)) = 2*n, where triangular = A000217, nontriangular = A014132.

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Formula

A270967 Number of set partitions of [n] having the maximal possible number of pairs (m,m+1) such that m+1 is in some block b and m is in block b+1.

Views

Author

Keywords

Links

Crossrefs

Formula

A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(2a(n-1)), a(nontriangular(n)) = 2n, where triangular = A000217, nontriangular = A014132.