A088567 -id:A088567 - cp's OEIS frontend

A089055 Solution to the non-squashing boxes problem (version 2).

2, 4, 8, 16, 28, 46, 72, 108, 156, 218, 298, 398, 524, 678, 868, 1096, 1372, 1698, 2086, 2538, 3070, 3684, 4398, 5214, 6156, 7226, 8450, 9830, 11400, 13162, 15152, 17372, 19868, 22642, 25742, 29170, 32986, 37192, 41850, 46962, 52606, 58784, 65576, 72984, 81106

Offset: 0

N. J. A. Sloane, Dec 04 2003

nonn

Given n+1 boxes labeled 0..n, such that box i weighs i grams and can support a total weight of i grams; a(n) = number of stacks of boxes that can be formed such that no box is squashed.

Amanda Folsom, Youkow Homma, Jun Hwan Ryu, and Benjamin Tong, On a general class of non-squashing partitions, Discrete Mathematics 339 (2016) 1482-1506.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.

Cf. A000123, A088567. Equals 2*A089054. Row sums of A089239.

See A089054 for g.f.

A110036 Constant terms of the partial quotients of the continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n), where each partial quotient has the form {x + a(n)} after the initial constant term of 1.

Original entry on oeis.org

1, -1, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, -2, 0, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, -2, 0, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, -2, 0

Offset: 0

Paul D. Hanna, Jul 08 2005

Suggested by Ralf Stephan.

For n>1, |a(n)| = 2*A090678(n) where A090678(n) = A088567(n) mod 2 and A088567(n) = number of "non-squashing" partitions of n into distinct parts.

			1 + 1/x + 1/x^2 + 1/x^4 + 1/x^8 + 1/x^16 + ... =
[1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, ...].

Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Cf. A090678, A088567.

PARI
```
contfrac(1+sum(n=0,10,1/x^(2^n)))
```

a(n)=polcoeff((1-x+3*x^2+x^3)/(1+x^2)- 2*sum(k=1,#binary(n),x^(3*2^(k-1))/prod(j=0,k,1+x^(2^j)+x*O(x^n))),n)

a(n)=subst(contfrac(1+sum(k=0,#binary(n+1),1/x^(2^k)))[n+1],x,0)

G.f. (1-x+3*x^2+x^3)/(1+x^2) - 2*Sum_{k>=1} x^(3*2^(k-1))/Product_{j=0..k} (1+x^(2^j)).

A279033 Irregular triangular array: T(n,i) = number of strict partitions of n having crossover index k; see Comments.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 1, 9, 1, 10, 2, 13, 2, 14, 4, 18, 4, 19, 8, 24, 8, 25, 13, 32, 14, 33, 21, 42, 22, 43, 33, 54, 35, 55, 49, 69, 53, 70, 72, 87, 78, 88, 103, 1, 109, 112, 1, 110, 145, 1, 136, 160, 137, 200, 3, 168, 220, 2, 169, 275, 4, 206, 303, 3

Offset: 1

Clark Kimberling, Dec 04 2016

Suppose that P = [p(1),p(2),...,p(k)] is a partition of n, where p(1) >= p(2) >= ... >= p(k). The crossover index of P is the least h such that p(1) + ... + p(h) > = n/2. Equivalently for k > 1, p(1) + ... + p(h) >= p(h+1) + ... + p(k). A strict partition is a partition into distinct parts. The n-th row sum is the number of strict partitions of n, A000009. Column 1 counts "non-squashing partitions", as in A088567.
First 32 rows (indexed by column 1):
..     1
..     1
..     2
..     2
..     3
..     4
..     5
..     6
..     7      1
..    9      1
..    10     2
..    13     2
..    14     4
..    18     4
..    19     8
..    24     8
..    25     13
..    32     14
..    33     21
..    42     22
..    43     33
..    54     35
..    55     49
..    69     53
..    70     72
..    87     78
..    88    103    1
..    109   112    1
..    110   145    1
..    136   160
..    137   200    3
..    168   220    3

Cf. A000009, A088567, A276468.

p[n_] := p[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
t[n_, k_] := t[n, k] = p[n][[k]];
q[n_, k_] := q[n, k] = Select[Range[50], Sum[t[n, k][[i]], {i, 1, #}] >= n/2 &, 1];
u[n_] := u[n] = Flatten[Table[q[n, k], {k, 1, Length[p[n]]}]];
c1[n_, k_] := c1[n, k] = Count[u[n], k];
m[n_] := -1 + Min[Flatten[Position[Table[c1[n, k], {k, 1, n + 1}], 0]]]
u = Table[c1[n, k], {n, 1, 50}, {k, 1, m[n]}]
TableForm[u] (* A279033 array *)
Flatten[u]   (* A279033 sequence *)

A242634 G.f. A(x) satisfies A(x) = A(x^2) / (1 - x) + x / (1 - x^2).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19, 24, 25, 31, 32, 40, 41, 50, 51, 63, 64, 77, 78, 95, 96, 114, 115, 138, 139, 163, 164, 194, 195, 226, 227, 266, 267, 307, 308, 357, 358, 408, 409, 471, 472, 535, 536, 612, 613, 690, 691, 785, 786, 881, 882

Offset: 0

Michael Somos, May 19 2014

nonn

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 7*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A088567, A088585.

a[ n_] := If[ n < 0, 0, Module[{A = 0}, Do[A = (x + (1 + x) (A /. x -> x^2)) / (1 - x^2), {IntegerLength[ n, 2]}]; SeriesCoefficient[ A, {x, 0, n}]]];

{a(n) = my(A = O(x)); if( n<0, 0, for(k=1, #binary(n), A = (x + (1 + x) * subst(A, x, x^2)) / (1 - x^2)); polcoeff(A, n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=0, #binary(n\3), x^(2^k*3 \ 2) / prod(j=0, k, 1 - x^2^j), x * O(x^n)), n))};

G.f.: x / (1 - x) + Sum_{k>0} x^(3*2^(k-1)) / Product_{j=0..k} (1 - x^(2^j)).
a(n) = a(n-2) + a(floor(n/2)) unless n=1.
a(n) = A088585(n) - A088585(n-1) if n>=1.
a(n) = A088567(n) if n>0.
a(2*n + 1) = a(2*n) + 1 = A088585(n) if n>=0.

A089198 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of non-squashing partitions of n into distinct parts of which the greatest is k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 3, 3, 2, 2, 1, 1, 1

Offset: 0

N. J. A. Sloane, Dec 10 2003

			Triangle begins:
1
0 1
0 0 1
0 0 1 1
0 0 0 1 1
0 0 0 1 1 1
0 0 0 1 1 1 1
0 0 0 0 2 1 1 1
0 0 0 0 1 2 1 1 1

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Row sums = A088567. Rows read from right to left also give (essentially) A088567.

T[n_, m_] := T[n, m] = Which[n==m, 1, mn, 0, True, Sum[T[n-m, i], {i, 0, m-1}]];
Table[T[n, m], {n, 0, 12}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 13 2019 *)

The nonzero values of T(n, m) lie within a certain cone: T(n, m) = 0 if m < n/2 or if m > n. For m <= n <= 2m, T(n, m) = sum_{i=0}^{m-1} T(n-m, i).

For m <= n <= 2m, T(n, m) = b(n-m) if n < 2m, = b(n-m) - 1 if n = 2m, where b = A088567.

A121241 Change 0 to -1 in A090678.

Original entry on oeis.org

1, 1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1

Offset: 0

Philippe Deléham, Aug 22 2006

sign

a(n) = (-1)^(1+A088567(n)).

cp's OEIS Frontend

A089055 Solution to the non-squashing boxes problem (version 2).

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A110036 Constant terms of the partial quotients of the continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n), where each partial quotient has the form {x + a(n)} after the initial constant term of 1.

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A279033 Irregular triangular array: T(n,i) = number of strict partitions of n having crossover index k; see Comments.

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A242634 G.f. A(x) satisfies A(x) = A(x^2) / (1 - x) + x / (1 - x^2).

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A089198 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of non-squashing partitions of n into distinct parts of which the greatest is k.

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A121241 Change 0 to -1 in A090678.

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