A130321 -id:A130321 - cp's OEIS frontend

A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).

1, 2, 2, 4, 3, 4, 8, 5, 5, 8, 16, 9, 7, 9, 16, 32, 17, 11, 11, 17, 32, 64, 33, 19, 15, 19, 33, 64, 128, 65, 35, 23, 23, 35, 65, 128, 256, 129, 67, 39, 31, 39, 67, 129, 256, 512, 257, 131, 71, 47, 47, 71, 131, 257, 512, 1024, 513, 259, 135, 79, 63, 79, 135, 259, 513, 1024

Offset: 0

Gary W. Adamson, Jul 18 2007

Row sums = A000295: (1, 4, 11, 26, 57, 120, ...).
If we regard the sequence as an infinite square array read by diagonals then it has the formula U(n,k) = (2^n + 2^k)/2 - 1. This appears to coincide with the number of n X k 0..1 arrays colored with only straight tiles, and new values 0..1 introduced in row major order, i.e., no equal adjacent values form a corner. (Fill the array with 0's and 1's. There must never be 3 adjacent identical values making a corner, only same values in a straight line.) Some solutions with n = k = 4 are:
  0 1 0 1     0 1 0 0     0 0 0 1     0 0 1 1     0 1 0 1
  1 0 1 0     1 0 1 1     1 1 1 0     1 1 0 0     1 0 1 0
  1 0 1 0     0 1 0 0     0 0 0 1     0 0 1 1     0 1 0 1
  0 1 0 1     1 0 1 1     1 1 1 0     1 1 0 0     0 1 0 1
(Observation from R. H. Hardin, cf. link.) - M. F. Hasler and N. J. A. Sloane, Feb 26 2013

			First few rows of the triangle:
    1;
    2,  2;
    4,  3,  4;
    8,  5,  5,  8;
   16,  9,  7,  9, 16;
   32, 17, 11, 11, 17, 32;
   64, 33, 19, 15, 19, 33, 64;
  128, 65, 35, 23, 23, 35, 65, 128;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
R. H. Hardin, Post to the SeqFan list, Feb 26 2013

Cf. A130321, A059268, A000012.

Row sums give A000295(n+2).

a131816 n k = a131816_tabl !! n !! k
a131816_row n = a131816_tabl !! n
a131816_tabl = map (map (subtract 1)) $
   zipWith (zipWith (+)) a130321_tabl a059268_tabl
-- Reinhard Zumkeller, Feb 27 2013

Table[Table[((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2, {m, 0, n}], {n, 0, 10}]; Flatten[%] (* Roger L. Bagula, Oct 16 2008 *)

T(n,m) = ((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2. - Roger L. Bagula, Oct 16 2008

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A130330 Triangle read by rows, the matrix product A130321 * A000012, both taken as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 7, 3, 1, 15, 7, 3, 1, 31, 15, 7, 3, 1, 63, 31, 15, 7, 3, 1, 127, 63, 31, 15, 7, 3, 1, 255, 127, 63, 31, 15, 7, 3, 1, 511, 255, 127, 63, 31, 15, 7, 3, 1, 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1, 2047, 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, May 24 2007

Row sums are A000295: (1, 4, 11, 26, 57, 120, 247, ...), the Eulerian numbers.
T(n,k) is the number of length n+1 binary words containing at least two 1's such that the first 1 is preceded by exactly (k-1) 0's. T(3,2) = 3 because we have: 0101, 0110, 0111. - Geoffrey Critzer, Dec 31 2013
Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A110441. - Peter Bala, Jul 22 2014
From Wolfdieter Lang, Oct 28 2019:(Start)
This triangle gives the solution of the following problem. Iterate the function f(x) = (x - 1)/2 to obtain f^{[k]}(x) = (x - (2^(k+1) - 1))/2^(k+1), for k >= 0. Find the positive integer x values for which the iterations stay integer and reach 1. Only odd integers x qualify, and the answer is x = x(n) = 2*T(n, 0) = 2*(2^(n+1) - 1), with the iterations T(n,0), ..., T(n,n) = 1.
This iteration is motivated by a problem posed by Johann Peter Hebel (1760 - 1826) in "Zweites Rechnungsexempel" from 1804, with the solution x = 31 corresponding to row n = 3 [15 7 3 1]. The egg selling woman started with 31 = T(4, 0) eggs and after four customers obtained, one after the other, always a number of eggs which was one half of the woman's remaining number of eggs plus 1/2 (selling only whole eggs, of course) she had one egg left. See the link and reference. [For Hebel's first problem see a comment in A000225.]
(End)

			First few rows of the triangle T(n, k):
n\k     0    1    2    3   4   5   6  7  8  9 10 11 12 ...
0:      1
1:      3    1
2:      7    3    1
3      15    7    3    1
4:     31   15    7    3   1
5:     63   31   15    7   3   1
6:    127   63   31   15   7   3   1
7:    255  127   63   31  15   7   3  1
8:    511  255  127   63  31  15   7  3  1
9:   1023  511  255  127  63  31  15  7  3  1
10:  2047 1023  511  255 127  63  31 15  7  3  1
11:  4095 2047 1023  511 255 127  63 31 15  7  3  1
12:  8191 4095 2047 1023 511 255 127 63 31 15  7  3  1
... reformatted and extended. - _Wolfdieter Lang_, Oct 28 2019

Johann Peter Hebel, Gesammelte Werke in sechs Bänden, Herausgeber: Jan Knopf, Franz Littmann und Hansgeorg Schmidt-Bergmann unter Mitarbeit von Ester Stern, Wallstein Verlag, 2019. Band 3, S. 36-37, Solution, S. 40-41. See also the link below.

Reinhard Zumkeller, Table of n, a(n) for n = 0..11324 [first 150 rows; offset shifted by _Georg Fischer_, Oct 29 2019]
Johann Peter Hebel, Zweites Rechnungsexempel., 1804; Solution: Auflösung des zweiten Rechnungsexempels. , 1805.

Cf. A130321, A000012, A000225. A110441.

a130330 n k = a130330_row n !! (k-1)
a130330_row n = a130330_tabl !! (n-1)
a130330_tabl = iterate (\xs -> (2 * head xs + 1) : xs) [1]
-- Reinhard Zumkeller, Mar 31 2012

nn=12;a=1/(1- x);b=1/(1-2x);Map[Select[#,#>0&]&,Drop[CoefficientList[Series[a x^2 b/(1-y x),{x,0,nn}],{x,y}],2]]//Grid  (* Geoffrey Critzer, Dec 31 2013 *)

A130321 * A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...) and A000012 = (1; 1,1; 1,1,1; ...).
In every column k with offset n = k: 2^(m+1) - 1 = A000225(m+1) = (1, 3, 7, 15, ...), for m >= 0.
G.f.: 1/((1-y*x)*(1-x)*(1-2x)). - Geoffrey Critzer, Dec 31 2013
T(n, k) = 2^((n - k) + 1) - 1, n >= 0, k = 0..n. - Wolfdieter Lang, Oct 28 2019

More terms from Geoffrey Critzer, Dec 31 2013

Edited by Wolfdieter Lang, Oct 28 2019

A130322 A130321^2.

Original entry on oeis.org

1, 4, 1, 12, 4, 1, 32, 12, 4, 1, 80, 32, 12, 4, 1, 192, 80, 32, 12, 4, 1, 448, 192, 80, 32, 12, 4, 1, 1024, 448, 192, 80, 32, 12, 4, 1, 2304, 1024, 448, 192, 80, 32, 12, 4, 1

Offset: 0

Gary W. Adamson, May 24 2007

Row sums = A000337: (1, 5, 17, 49, 129, 321, ...).

			First few rows of the triangle:
    1;
    4,  1;
   12,  4,  1;
   32, 12,  4,  1;
   80, 32, 12,  4,  1;
  192, 80, 32, 12,  4,  1;
  ...

Cf. A130321, A001787, A000337.

The square of triangle A130321 as an infinite lower triangular matrix.

A001787 in every column starting (1, 4, 12, 32, 80, 192, ...).

A130452 Triangle read by rows: A097806 * A130321 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 12, 6, 3, 1, 24, 12, 6, 3, 1, 48, 24, 12, 6, 3, 1, 96, 48, 24, 12, 6, 3, 1, 192, 96, 48, 24, 12, 6, 3, 1, 384, 192, 96, 48, 24, 12, 6, 3, 1, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1, 1536, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1, 3072, 1536, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, 190, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  3,  1;
  12,  6,  3,  1;
  24, 12,  6,  3,  1;
  48, 24, 12,  6,  3,  1;
  ...

Cf. A003945, A033484, A097806, A130321.

A097806 * A130321 as infinite lower triangular matrices. A097806 = the pairwise operator, A130321 = [1; 2,1; 4,2,1; ...]. Triangle, A003945 (1, 3, 6, 12, 24, 48, ...) in every column.

a(28) = 1 inserted and more terms from Georg Fischer, May 29 2023

A251635 Riordan array (1-2x,x), inverse of Riordan array (1/(1-2x), x) = A130321.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Jan 10 2015

This is a simple Riordan array, an infinite lower triangular matrix. It is the inverse matrix of A130321 (with zeros above the diagonal).
Row sums have o.g.f. (1-2*x)/(1-x) and give 1, repeat(-1), i.e., A153881(n+1), n >= 0.
Alternate row sums have o.g.f. (1-2*x)/(1+x) and give 1, repeat(-3,3), i.e., (-1)^n*A122553(n).

			The triangle T(n, k) begins:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
0:   1
1:  -2  1
2:   0 -2  1
3:   0  0 -2  1
4:   0  0  0 -2  1
5:   0  0  0  0 -2  1
6:   0  0  0  0  0 -2 1
7:   0  0  0  0  0  0 -2  1
8:   0  0  0  0  0  0  0 -2  1
9:   0  0  0  0  0  0  0  0 -2  1
10:  0  0  0  0  0  0  0  0  0 -2  1
...

Cf. A130321, A153881, A122553.

a251635 n k = a251635_tabl !! n !! k
a251635_row n = a251635_tabl !! n
a251635_tabl = [1] : iterate (0 :) [-2, 1]
-- Reinhard Zumkeller, Jan 11 2015

T(n, k) = 0 if n < k and k = 0..(n-2) for n >= 2, and T(n, n) = 1 and T(n, n-1) = -2.
G.f. for row polynomials P(n, x) = -2^x^(n-1) + x^n is (1-2*z)/(1-x*z).
G.f. for k-th column: (1-2*x)*x^k, k >= 0.

A130329 A059268 * A130321.

Original entry on oeis.org

1, 5, 2, 21, 10, 4, 85, 42, 20, 8, 341, 170, 84, 40, 16, 1365, 682, 340, 168, 80, 32, 5461, 2730, 1364, 680, 336, 160, 64, 21845, 10922, 5460, 2728, 1360, 672, 320, 128, 87381, 43690, 21844, 10920, 5456, 2720, 1344, 640, 256

Offset: 0

Gary W. Adamson, May 24 2007

Row sums = A006095: (1, 7, 35, 155, 651, 2667, ...).
Left border = A002450: (1, 5, 21, 85, 341, 1365, ...).
A130328 = A130321 * A059268.

			First few rows of the triangle:
    1;
    5,   2;
   21,  10,  4;
   85,  42, 20,  8;
  341, 170, 84, 40, 16;
  ...

Cf. A059268, A130321, A006095, A002450, A130328.

A059268 * A130321 as infinite lower triangular matrices.

A131817 a(n) = A051340(n) + A130321(n) - A000012(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 3, 8, 4, 2, 4, 16, 8, 4, 2, 5, 32, 16, 8, 4, 2, 6, 64, 32, 16, 8, 4, 2, 7, 128, 64, 32, 16, 8, 4, 2, 8, 256, 128, 64, 32, 16, 8, 4, 2, 9, 512, 256, 128, 64, 32, 16, 8, 4, 2, 10, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 11, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 12

Offset: 0

Gary W. Adamson, Jul 18 2007

			First few rows of the triangle:
   1;
   2,  2;
   4,  2,  3;
   8,  4,  2,  4;
  16,  8,  4,  2,  5;
  32, 16,  8,  4,  2,  6;
  64, 32, 16,  8,  4,  2,  7;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)

Cf. A051340, A083706 (row sums), A130321.

Array[Append[ConstantArray[1, # - 1], #] + 2^Range[# - 1, 0, -1] - 1 &, 12] // Flatten (* Michael De Vlieger, Aug 19 2021 *)

A051340 + A130321 - A000012 as infinite lower triangular matrices.

More terms from Jinyuan Wang, Aug 19 2021

A131897 A130321 + A131821 - A000012.

Original entry on oeis.org

1, 3, 2, 6, 2, 3, 11, 4, 2, 4, 20, 8, 4, 2, 5, 37, 16, 8, 4, 2, 6, 70, 32, 16, 8, 4, 2, 7, 135, 64, 32, 16, 8, 4, 2, 8, 264, 128, 64, 32, 16, 8, 4, 2, 9, 521, 256, 128, 64, 32, 16, 8, 4, 2, 10

Offset: 0

Gary W. Adamson, Jul 25 2007

Left column = A006127: (1, 3, 6, 11, 20, 37, ...).

Row sums = A131898: (1, 5, 11, 21, 39, 73, ...).

			First few rows of the triangle:
   1;
   3,  2;
   6,  2,  3;
  11,  4,  2,  4;
  20,  8,  4,  2,  5;
  37, 16,  8,  4,  2,  6;
  70, 32, 16,  8,  4,  2,  7;
  ...

Cf. A130321, A131821, A006127, A131898.

A131321 + A131821 - A000012 as infinite lower triangular matrices.

A131953 A130321 + A059268 - A000012(signed).

Original entry on oeis.org

1, 4, 2, 4, 5, 4, 10, 5, 7, 8, 16, 11, 7, 11, 16, 34, 17, 13, 11, 19, 32, 64, 35, 19, 17, 19, 35, 64, 130, 65, 37, 23, 25, 35, 67, 128, 256, 131, 67, 41, 31, 41, 67, 131, 256, 514, 257, 133, 71, 49, 47, 73, 131, 259, 512

Offset: 0

Gary W. Adamson, Jul 30 2007

Row sums = A101622 starting (1, 6, 13, 30, 61, 126, ...).

			First few rows of the triangle:
   1;
   4,  2;
   4,  5,  4;
  10,  5,  7,  8;
  16, 11,  7, 11, 16;
  34, 17, 13, 11, 19, 32;
  64, 35, 19, 17, 19, 35, 64;
  ...

Cf. A130321, A059268, A101622.

A130321 + A059268 - A000012 (signed + - + -, ... by columns) as infinite lower triangular matrices.

A135086 Triangle, antidiagonals of an array formed by A000012 * A130321(transform).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 3, 6, 8, 1, 3, 7, 12, 16, 1, 3, 7, 14, 24, 32, 1, 3, 7, 15, 28, 48, 64, 1, 3, 7, 15, 30, 56, 96, 128, 1, 3, 7, 15, 31, 60, 112, 192, 256, 1, 3, 7, 15, 31, 62, 120, 224, 384, 512

Offset: 1

Gary W. Adamson, Nov 18 2007

Row sums = A011377: (1, 3, 8, 18, 39, 81, 166, ...).

			First few rows of the array:
  1, 2, 4,  8, 16, 32, ...
  1, 3, 6, 12, 24, 48, ...
  1, 3, 7, 14, 28, 56, ...
  1, 3, 7, 15, 30, 60, ...
  1, 3, 7, 15, 31, 62, ...
  ...
First few rows of the triangle:
  1;
  1,  2;
  1,  3,  4;
  1,  3,  6,  8;
  1,  3,  7, 12, 16;
  1,  3,  7, 14, 24, 32;
  1,  3,  7, 15, 28, 48, 64;
  ...

Cf. A130321, A000012, A011377.

Triangle, antidiagonals of the array formed by A000012 * A130321.

cp's OEIS Frontend

A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).

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A130330 Triangle read by rows, the matrix product A130321 * A000012, both taken as infinite lower triangular matrices.

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A130322 A130321^2.

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A130452 Triangle read by rows: A097806 * A130321 as infinite lower triangular matrices.

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A251635 Riordan array (1-2*x,x), inverse of Riordan array (1/(1-2*x), x) = A130321.

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A130329 A059268 * A130321.

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A131817 a(n) = A051340(n) + A130321(n) - A000012(n).

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A131897 A130321 + A131821 - A000012.

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A251635 Riordan array (1-2x,x), inverse of Riordan array (1/(1-2x), x) = A130321.