A083039 -id:A083039 - cp's OEIS frontend

A239140 Number of strict partitions of n having standard deviation σ < 1.

1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1

Offset: 1

Clark Kimberling, Mar 11 2014

Regarding standard deviation, see Comments at A238616.

			The standard deviations of the strict partitions of 9 are 0., 3.5, 2.5, 1.5, 2.16025, 0.5, 1.63299, 0.816497, so that a(9) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..10005
Index entries for linear recurrences with constant coefficients, signature (-1, 0, 1, 1).

Cf. A000009, A083039, A238616, A239141, A239142, A239143.

Column k=0 of A239228.

z = 30; g[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; s[t_] := s[t] = Sqrt[Sum[(t[[k]] - Mean[t])^2, {k, 1, Length[t]}]/Length[t]]
Table[Count[g[n], p_ /; s[p] < 1], {n, z}]   (* A239140 *)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (* A239141 *)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (* periodic 01 *)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (* A239142 *)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (* A239143 *)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
(* Peter J. C. Moses, Mar 03 2014 *)
Join[{1, 1, 2},LinearRecurrence[{-1, 0, 1, 1},{1, 2, 2, 2},83]] (* Ray Chandler, Aug 25 2015 *)

A083039(n) = (1+!(n%2)+!(n%3));
A239140(n) = if(n<=3,1+(3==n),A083039(n-3)); \\ Antti Karttunen, May 24 2021

a(n + 3) = A083039(n) for n >= 1 (periodic with period 6); a(n) + A239143(n) = A000009(n) for n >=1.

G.f.: -(x^6+x^5+x^4+2*x^3+3*x^2+2*x+1)*x / ((x-1)*(x+1)*(x^2+x+1)). - Alois P. Heinz, Mar 14 2014

A-number in the first formula corrected by Antti Karttunen, May 24 2021

A083040 Number of divisors of n that are <= 4.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4

Offset: 1

Daniele A. Gewurz (gewurz(AT)mat.uniroma1.it) and Francesca Merola (merola(AT)mat.uniroma1.it), May 06 2003

Periodic of period 12. Parker vector of the wreath product of S_4 and S, the symmetric group of a countable set.

D. A. Gewurz and F. Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., 6 (2003), 03.1.6
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A083039, A000005

G.f.: x/(1-x)+x^2/(1-x^2)+x^3/(1-x^3)+x^4/(1-x^4).
a(n) = a(n-12) = a(-n).
a(n) = 25/12 - (3/4)*( - 1)^n - 1/2*sin(Pi*n/2) - (1/3)*cos(2*Pi*n/3) - (1/3)*3^(1/2)*sin(2*Pi*n/3) [From Richard Choulet, Dec 12 2008]
a(n) = sum(k=1..1, cos(n*(k - 1)/1*2*Pi)/1) + sum(k=1..2, cos(n*(k - 1)/2*2*Pi)/2) + sum(k=1..3, cos(n*(k - 1)/3*2*Pi)/3) + sum(k=1..4, cos(n*(k - 1)/4*2*Pi)/4). - Mats Granvik, Sep 09 2012

A138553 Table read by rows: T(n,k) is the number of divisors of k that are <= n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 1, 3, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 5, 1, 2, 2, 3, 2, 4, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 1, 4, 1, 4, 2, 2, 1, 5

Offset: 1

Franklin T. Adams-Watters, Mar 24 2008

Suggested by a question from Eric Desbiaux.
The row lengths are the lengths before the pattern for n repeats.
Antidiagonal sums A070824. [From Eric Desbiaux, Dec 10 2009]

			The first few rows start:
1, [A000012]
1, 2, [A000034]
1, 2, 2, 2, 1, 3, [A083039]
1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, [A083040]

Row lengths A003418, row sums A025529, frequencies in rows A096180.

Cf. A243987

lista(nrows) = {for (n=1, nrows, for (k=1, lcm(vector(n, i, i)), print1(sumdiv(k, d, d <=n), ", ");); print(););} \\ Michel Marcus, Jun 19 2014

T(n,k) = sum_{i|k, i<=n} 1.

Definition corrected by Franklin T. Adams-Watters, Jun 19 2014

A360998 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling), 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 2, 3, 3, 4, 2, 4, 6, 5, 7, 5, 4

Offset: 1

Pontus von Brömssen, Feb 28 2023

It seems that each solution consists of n*k/(r*s) copies of an r X s piece (arranged in a simple grid, all pieces oriented in the same way), where r is a divisor of n, s is a divisor of k, and either r = s or r is not a divisor of k or s is not a divisor of n. If this is true, T(n,k) <= d(n)*d(k) - d(m)*(d(m)-1), where d = A000005 is the divisor count function and m = gcd(n,k). Equality does not always hold; for (n,k) = (3,2), for example, (r,s) = (1,2) satisfies the condition, but three 1 X 2 pieces can tile the 3 X 2 rectangle in more than one way.

Is d(n)*d(k) - T(n,k) eventually periodic in n for each k?

			Triangle begins:
  n\k|  1  2  3  4  5  6
  ---+------------------
  1  |  1
  2  |  2  2
  3  |  2  3  2
  4  |  3  4  4  3
  5  |  2  3  3  4  2
  6  |  4  6  5  7  5  4
The T(4,3) = 4 nonrearrangeable tilings of the 4 X 3 rectangle are:
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +           +   +           +   +   +   +   +   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +           +   +---+---+---+   +   +   +   +   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +           +   +           +   +   +   +   +   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+

Columns: A000005 (k = 1), A360999 (k = 2), A361000 (k = 3).

Cf. A083039, A360629, A361001.

T(n,1) = d(n) = A000005(n).
T(n,2) = A360999(n) = 2*d(n) - 1 - [n even] for n >= 2.
T(n,3) = A361000(n) = 2*d(n) - A083039(n) for n >= 3.
It appears that T(n,4) = 3*d(n) - 2 - 2*[n even] - [n divisible by 3] - 2*[n divisible by 4] for n >= 4.
It appears that T(n,n) = d(n). (It is easy to see that T(n,n) >= d(n).)

A361000 Number of tilings of an n X 3 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Original entry on oeis.org

2, 3, 2, 4, 3, 5, 3, 6, 4, 6, 3, 9, 3, 6, 6, 8, 3, 9, 3, 10, 6, 6, 3, 13, 5, 6, 6, 10, 3, 13, 3, 10, 6, 6, 7, 15, 3, 6, 6, 14, 3, 13, 3, 10, 10, 6, 3, 17, 5, 10, 6, 10, 3, 13, 7, 14, 6, 6, 3, 21, 3, 6, 10, 12, 7, 13, 3, 10, 6, 14, 3, 21, 3, 6, 10, 10, 7, 13, 3

Offset: 1

Pontus von Brömssen, Feb 28 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Third column of A360998.

Cf. A000005, A083039, A360632, A361005.

A083039(n) = ([3, 1, 2, 2, 2, 1][n%6+1]);
A361000(n) = if(n<3, 1+n, 2*numdiv(n) - A083039(n)); \\ Antti Karttunen, Jul 19 2024

a(n) = 2*A000005(n) - A083039(n) for n >= 3.
Proof: (Start)
Note that a nonrearrangeable tiling must have full symmetry, otherwise it could be rotated or reflected to another tiling.
Identical pieces of size m X 1 tile the n X 3 rectangle in a unique way exactly when m is a divisor of n, except m = 2 and m = 3. Identical pieces of size m X 3 tile the rectangle in a unique way exactly when m is a divisor of n, except m = 1. The only other pieces that fit in the rectangle are of size m X 2, where m = 2 or m >= 4, but it is easily seen that the rectangle cannot be tiled with identical pieces of those sizes, so there are 2*A000005(n) - 1 - [n even] - [n divisible by 3] = 2*A000005(n) - A083039(n) nonrearrangeable tilings of the rectangle with identical pieces.
We now prove that there is no nonrearrangeble tiling of the rectangle that contains pieces of different sizes. Assume that we have such a tiling of the rectangle of height n and width 3, for the smallest possible value of n.
A fault of a tiling is a line through the interior of the rectangle that does not pass through the interior of any piece. If the tiling has any horizontal (or vertical) faults, all the resulting horizontal (or vertical) strips between the faults must have the same size and be tiled in the same way, otherwise the strips could be permuted to another tiling.
If the tiling has any vertical faults, there must be three identically tiled vertical n X 1 strips. This implies that a single strip must contain pieces of different lengths, which is clearly impossible because those pieces could be permuted to another tiling.
If there are no vertical faults, there must be a piece in the tiling with a horizontal side of length at least 2, and by symmetry one of its horizontal edges must be part of a horizontal fault, otherwise another tiling could be obtained by reflection. The resulting strips (of width 3) must all be tiled in the same way (and thus contain pieces of different sizes), and the tiling of one strip must be nonrearrangeble. We have thus obtained a nonrearrangeable tiling with pieces of different sizes for a smaller height, a contradiction.
(End)

A117198 Generalized Riordan array (1,x)+(x,x^2)+(x^2,x^3).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Mar 02 2006

Row sums are A083039. Diagonal sums are A117199. Inverse is A117200.

			Triangle begins
1,
1, 1,
1, 0, 1,
0, 1, 0, 1,
0, 0, 0, 0, 1,
0, 1, 1, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 1, 0, 0, 0, 1,
0, 0, 1, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Number triangle T(n,k)=if(n=k,1,0) OR if(n=2k+1,1,0) OR if(n=3k+2,1,0).

A106253 First difference of A106252.

Original entry on oeis.org

2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2

Offset: 1

John W. Layman, Apr 27 2005

nonn

It appears that this sequence is periodic with period 6.

Index entries for linear recurrences with constant coefficients, signature (-1, 0, 1, 1).

Cf. A106252.

Cf. A083039. - R. J. Mathar, Aug 24 2008

LinearRecurrence[{-1, 0, 1, 1},{2, 2, 2, 1},105] (* Ray Chandler, Aug 26 2015 *)

A143981 The number of unigraphical partitions of 2m; that is, the number of partitions of 2m which are realizable as the degree sequence of one and only one graph (where loops are not allowed but multiple edges are allowed).

Original entry on oeis.org

1, 3, 6, 9, 15, 19, 26, 36, 46, 59, 80, 100, 128, 167, 211, 267, 341, 429, 541, 682, 850, 1063, 1327, 1647, 2035, 2520, 3100, 3810, 4669, 5708, 6955, 8468, 10267, 12441, 15026, 18120, 21788, 26175, 31355, 37510, 44769, 53362, 63460, 75384, 89348

Offset: 1

Michael David Hirschhorn and James Sellers, Sep 06 2008

nonn

			For m = 4, the number of unigraphical partitions is A000041(4) + A001399(1) + A000005(5) + A083039(2) + 4 - 5 = 5 + 1 + 2 + 2 + 4 - 5 = 9.

S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Indust. Appl. Math., vol. 10 (1962), 496-506.
S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. II. Uniqueness, J. Soc. Indust. Appl. Math., vol. 11 (1963), 135-147.

Cf. A000041, A001399, A000005, A083039.

with(combinat): with(numtheory): a:=proc(m) it:=round(m^2/12)+numbpart(m)+tau(m+1)+m-5: if m mod 6 = 0 then it:=it+2 fi: if m mod 6 = 1 then it:=it+1 fi: if m mod 6 = 2 then it:=it+3 fi: if m mod 6 = 3 then it:=it+1 fi: if m mod 6 = 4 then it:=it+2 fi: if m mod 6 = 5 then it:=it+2 fi: RETURN(it): end:

For m >= 3, a(2m) = A000041(m) + A001399(m-3) + A000005(m+1) + A083039(m-2) + m - 5.

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A239140 Number of strict partitions of n having standard deviation σ < 1.

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A083040 Number of divisors of n that are <= 4.

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A138553 Table read by rows: T(n,k) is the number of divisors of k that are <= n.

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A360998 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling), 1 <= k <= n.

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A361000 Number of tilings of an n X 3 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

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A117198 Generalized Riordan array (1,x)+(x,x^2)+(x^2,x^3).

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A106253 First difference of A106252.

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A143981 The number of unigraphical partitions of 2m; that is, the number of partitions of 2m which are realizable as the degree sequence of one and only one graph (where loops are not allowed but multiple edges are allowed).

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