A048888 -id:A048888 - cp's OEIS frontend

A067401 Minimal non-uniquely factorizable OR-numbrals, i.e., numbrals that are not uniquely factorizable but for which all proper divisors are.

15, 85, 95, 111, 123, 125, 175, 191, 207, 223, 239, 243, 245, 247, 251, 253, 351, 367, 379, 381, 399, 415, 443, 445, 447, 463, 483, 487, 493, 499, 501, 507, 585, 603, 621, 631, 639, 685, 687, 701, 725, 729, 731, 735, 757, 763, 783, 799, 827, 831, 873, 877

Offset: 1

Jens Voß, Jan 24 2002

nonn

See A048888 for the definition of OR-numbral arithmetic.

			15 is in A067401 since [15] = [3] * [5] = [3]^3 all divisors of [15] are uniquely factorizable.

Cf. A003986, A007059, A048888, A067138, A067139, A067398, A067399, A067400.

A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 8, 7, 4, 2, 1, 1, 13, 13, 8, 4, 2, 1, 1, 21, 24, 15, 8, 4, 2, 1, 1, 34, 44, 29, 16, 8, 4, 2, 1, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1, 1, 144, 274, 208, 120, 63, 32, 16, 8, 4, 2, 1, 1, 233, 504, 401, 236, 125, 64, 32, 16, 8, 4, 2, 1

Offset: 2

Roger L. Bagula, Dec 03 2010

Antidiagonal sums are A048888. This is a transposed version of A048887, so the bivariate generating function is obtained by swapping the two arguments.

Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - N. J. A. Sloane, Oct 30 2018

			The array starts in row n=2 with columns k >= 1 as:
  1   1   1   1   1   1   1   1   1   1
  1   2   2   2   2   2   2   2   2   2
  1   3   4   4   4   4   4   4   4   4
  1   5   7   8   8   8   8   8   8   8
  1   8  13  15  16  16  16  16  16  16
  1  13  24  29  31  32  32  32  32  32
  1  21  44  56  61  63  64  64  64  64
  1  34  81 108 120 125 127 128 128 128
  1  55 149 208 236 248 253 255 256 256

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 125, 155.

Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014

Cf. A000045, A048887, A048004, A126198.

A092921 := proc(n,k) if k <= 0 or n <= 0 then 0; elif k = 1 or n = 1 then 1; else add( procname(n-i,k),i=1..k) ; end if; end proc:
A175331 := proc(n,k) A092921(n,k) ; end proc: # R. J. Mathar, Dec 17 2010

f[x_, n_] = (x - x^(m + 1))/(1 - 2*x + x^(m + 1))
a = Table[Table[SeriesCoefficient[
      Series[f[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];
Flatten[%]

T(n,k) = A092921(n,k), n >= 2.
T(n,2) = A000045(n).
T(n,3) = A000073(n+2).
T(n,4) = A000078(n+2).

A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 8, 11, 1, 12, 19, 26, 1, 19, 33, 45, 57, 1, 30, 58, 84, 102, 120, 1, 48, 101, 157, 197, 222, 247, 1, 77, 179, 292, 380, 436, 469, 502, 1, 124, 318, 546, 731, 855, 929, 971, 1013, 1, 200, 567, 1026, 1409, 1674, 1838, 1932, 1984, 2036

Offset: 1

Jonathan Vos Post, Nov 23 2006

Array of partial sums of diagonals of L(k,n) begins: 0.|.1...4..11...26...57..120..247..502.1013.2036.
|.1...8..19...45..102..222..469..971.1984.
|.1..12..33...84..197..436..929.1932.
|.1..19..58..157..380..855.1838.
|.1..30.101..292..731.1674.
|.1..48.179..546.1409.
|.1..77.318.1026.
|.1.124.567.
|.1.200.
|.1.

			Row 1 of the derived array is the partial sum of the diagonal above the main diagonal of array of k-step Lucas numbers as in A125127, hence the partial sums of: 1, 7, 11, 26, 57, 120, 247, 502, 103, ... are 1 = 1; 8 = 1 + 7; 19 = 1 + 7 + 11; 45 = 1 + 7 + 11 + 26; and so forth.

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125127, A000295.

Row 0 = SUM[i=1..n]L(i,i) = A127128 = partial sum of main diagonal of array of A125127. Row 1 = SUM[i=1..n]L(i,i+1) = partial sum of diagonal above main diagonal of array of A125127. Row 2 = SUM[i=1..n]L(i,i+2) = partial sum of diagonal 2 above main diagonal of array of A125127. .. Row m = SUM[i=1..n]L(i,i+m) = partial sum of diagonal 2 above main diagonal of array of A125127.

A067150 Number of integers i=1,2,...,n such that (n,i) has Property F3, i.e., i and n are consecutive terms of a sequence b(k) satisfying b(1)=1, b(n) = (b(n-1) OR 2b(n-1)) + b(n-2), where the OR is taken bitwise.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 3, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 1, 1, 0, 1, 2, 0, 3, 5, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

John W. Layman, Jan 05 2002

nonn

Surprisingly, for k > 0, we find that a(2^k) = F(k-1), where {F(n)} is the sequence of Fibonacci numbers (A000045). Also, except for n = 2^3 = 8, these values are exactly those where new records in a(n) are made.
The definition can be restated as follows: a(n) is the number of integers i, 0 < i < n such that i and n are consecutive terms of some sequence b(k) satisfying b(1)=1 and b(k) = 3#b(k-1) + b(k-2), where # denotes OR-numbral multiplication (see A048888 for the definition).
If the OR-numbral multiplier 3 in the definition is replaced by 7, the resulting sequence has as record values the tribonacci numbers in A000073.

A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.

Cf. A000045, A000073, A067148.

A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 52, 101, 200, 396, 787, 1564, 3117, 6214, 12398, 24749, 49427, 98740, 197303, 394323, 788201, 1575695, 3150265, 6298732, 12594595, 25184598, 50361842, 100711888, 201404839, 402779246, 805509560, 1610940381, 3221753990

Offset: 0

Joerg Arndt, Apr 21 2013

nonn

			The a(5) = 14 such compositions of 5 are
01:  [ 1 1 1 1 1 ]
02:  [ 1 1 1 2 ]
03:  [ 1 1 2 1 ]
04:  [ 1 1 3 ]
05:  [ 1 2 1 1 ]
06:  [ 1 2 2 ]
07:  [ 1 3 1 ]
08:  [ 1 4 ]
09:  [ 2 1 1 1 ]
10:  [ 2 1 2 ]
11:  [ 2 2 1 ]
12:  [ 2 3 ]
13:  [ 3 2 ]
14:  [ 5 ]
(the two forbidden compositions are [ 3 1 1 ] and [ 4 1 ]).

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

Cf. A171682 (compositions such that p(j) >= p(1)).
Cf. A079501 (compositions such that p(j) > p(1)).
Cf. A048888 (compositions such that p(j) <= p(1) + 1).
Cf. A007059 (compositions such that p(j) < p(1)).
Cf. A079500 (compositions such that p(j) <= p(1)).

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j,
      `if`(i=0, max(1, j-1), i)), j=`if`(i=0, 1, i)..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, If[i == 0, Max[1, j - 1], i]], {j, If[i == 0, 1, i], n}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ 3 * 2^(n-3). - Vaclav Kotesovec, May 01 2014

cp's OEIS Frontend

A067401 Minimal non-uniquely factorizable OR-numbrals, i.e., numbrals that are not uniquely factorizable but for which all proper divisors are.

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A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.

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A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

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A067150 Number of integers i=1,2,...,n such that (n,i) has Property F3*, i.e., i and n are consecutive terms of a sequence b(k) satisfying b(1)=1, b(n) = (b(n-1) OR 2*b(n-1)) + b(n-2), where the OR is taken bitwise.

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A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

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A067150 Number of integers i=1,2,...,n such that (n,i) has Property F3, i.e., i and n are consecutive terms of a sequence b(k) satisfying b(1)=1, b(n) = (b(n-1) OR 2b(n-1)) + b(n-2), where the OR is taken bitwise.