A059268 -id:A059268 - cp's OEIS frontend

A130321 Triangle, (2^0, 2^1, 2^2, ...) in every column.

1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 16, 8, 4, 2, 1, 32, 16, 8, 4, 2, 1, 64, 32, 16, 8, 4, 2, 1, 128, 64, 32, 16, 8, 4, 2, 1, 256, 128, 64, 32, 16, 8, 4, 2, 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1

Offset: 0

Gary W. Adamson, May 24 2007

A130321^2 = A130322. Binomial transform of A130321 = triangle A027649. A007318^2 = A038207 = A007318(n,k) * A130321(n,k); i.e., the square of Pascal's triangle = dot product of Pascal's triangle rows and A130321 rows: A007318^2 = (1; 2,1; 4,4,1; 8,12,6,1;...), where row 3, (8,12,6,1) = (1,3,3,1) dot (8,4,2,1).
Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A130321 is the reverse reluctant sequence of sequence of power of 2 (A000079). - Boris Putievskiy, Dec 13 2012
From Wolfdieter Lang, Jan 10 2015: (Start)
This is the Riordan array (1/(1-2*x), x).
Row sums give A000225(n+1) = 2^(n+1) - 1.
Alternating row sums give A001045(n+1).
The inverse Riordan array is (1-2*x, x) = A251635. (End)

			The triangle T(n,m) begins:
  n\m     0   1   2   3  4  5  6  7  8  9 10 ...
  0:      1
  1:      2   1
  2:      4   2   1
  3:      8   4   2   1
  4:     16   8   4   2  1
  5:     32  16   8   4  2  1
  6:     64  32  16   8  4  2  1
  7:    128  64  32  16  8  4  2  1
  8:    256 128  64  32 16  8  4  2  1
  9:    512 256 128  64 32 16  8  4  2  1
 10:   1024 512 256 128 64 32 16  8  4  2  1
 ... Reformatted. - _Wolfdieter Lang_, Jan 10 2015

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A059268, A027649, A130322, A038207, A131816, A000225, A001045, A251635.

a130321 n k = a130321_tabl !! n !! k
a130321_row n = a130321_tabl !! n
a130321_tabl = iterate (\row -> (2 * head row) : row) [1]
-- Reinhard Zumkeller, Feb 27 2013

T[n_, m_] := 2^(n-m);
Table[T[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

Triangle, (1, 2, 4, 8, ...) in every column. Rows are reversals of A059268 terms.
a(n)=2^m, where m=(t*t + 3*t + 4)/2 - n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
From Wolfdieter Lang, Jan 10 2015: (Start)
T(n, m) = 2^(n-m) if n >= m >= 0 and 0 otherwise.
G.f. of row polynomials R(n,x) = Sum_{m=0..n} 2^(n-m)*x^m is 1/((1-2*z)*(1-x*z)) (Riordan property).
G.f. column m (with leading zeros) x^m/(1-2*x), m >= 0.
The diagonal sequences are D(k) = repeat(2^k), k >= 0. (End)

More terms from Philippe Deléham, Feb 08 2009

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Reinhard Zumkeller, Feb 28 2010

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

A140513 Repeat 2^n n times.

Original entry on oeis.org

2, 4, 4, 8, 8, 8, 16, 16, 16, 16, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024

Offset: 0

Paul Curtz, Jul 01 2008

Reinhard Zumkeller, Rows n = 0..127 of triangle, flattened
Sajed Haque, Chapter 2.6.2 of Discriminators of Integer Sequences, 2017, See p. 34.

Cf. A000079, A018900, A059268, A111650, A137688.

a140513 n k = a140513_tabl !! (n-1) !! (k-1)
a140513_row n = a140513_tabl !! (n-1)
a140513_tabl = iterate (\xs@(x:_) -> map (* 2) (x:xs)) [2]
a140513_list = concat a140513_tabl
-- Reinhard Zumkeller, Nov 14 2015

t={}; Do[r={}; Do[If[k==0||k==n, m=2^n, m=t[[n, k]] + t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t=Flatten[2 t] (* Vincenzo Librandi, Feb 17 2018 *)
Table[Table[2^n,n],{n,10}]//Flatten (* Harvey P. Dale, Dec 04 2018 *)

from math import isqrt
def A140513(n): return 1<<(m:=isqrt(k:=n+1<<1))+(k>m*(m+1)) # Chai Wah Wu, Nov 07 2024

a(n) = 2*A137688(n).
a(n) = A018900(n+1) - A059268(n). - Reinhard Zumkeller, Jun 24 2009
From Reinhard Zumkeller, Feb 28 2010: (Start)
Seen as a triangle read by rows: T(n,k)=2^n, 1 <= k <= n.
T(n,k) = A173786(n-1,k-1) + A173787(n-1,k-1), 1 <= k <= n. (End)
Sum_{n>=0} 1/a(n) = 2. - Amiram Eldar, Aug 16 2022

A130328 Triangle of differences between powers of 2, read by rows.

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 15, 14, 12, 8, 31, 30, 28, 24, 16, 63, 62, 60, 56, 48, 32, 127, 126, 124, 120, 112, 96, 64, 255, 254, 252, 248, 240, 224, 192, 128, 511, 510, 508, 504, 496, 480, 448, 384, 256

Offset: 0

Gary W. Adamson, May 24 2007

A130321 * A059268 as infinite lower triangular matrices.
Row sums = A000337: (1, 5, 17, 49, 129, 321, ...). A130329 = A059268 * A130321.
From Alonso del Arte, Mar 13 2008: (Start)
Column 0 contains the Mersenne numbers A000225.
Column 1 is A000918.
An even perfect number (A000396) is found in the triangle by reference to its matching exponent for the Mersenne prime p (A000043) thus: go to row 2p - 1 and then column p - 1 (remembering that the first position is column 0).
Likewise divisors of multiply perfect numbers, if not the multiply perfect numbers themselves, can also be found in this triangle. (End)

			First few rows of the triangle are;
   1;
   3,  2;
   7,  6,  4;
  15, 14, 12,  8;
  31, 30, 28, 24, 16;
  63, 62, 60, 56, 48, 32;
  ...
a(5, 2) = 28 because 2^5 = 32, 2^2 = 4 and 32 - 4 = 28.

Cf. A130321, A059268, A000337, A130329.

ColumnForm[Table[2^n - 2^k, {n, 15}, {k, 0, n - 1}], Center] (* Alonso del Arte, Mar 13 2008 *)

t(n, k) = 2^n - 2^k, where n is the row number and k is the column number, running from 0 to n - 1. (If k is allowed to reach n, then the triangle would have an extra diagonal filled with zeros) - Alonso del Arte, Mar 13 2008

Better definition from Alonso del Arte, Mar 13 2008

A077352 a(n) = (concatenation in ascending order of divisors of 2^n)/2^n.

Original entry on oeis.org

1, 6, 31, 156, 7801, 390051, 19502551, 9751275501, 4875637750501, 2437818875250501, 12189094376252505001, 60945471881262525005001, 304727359406312625025005001, 1523636797031563125125025005001, 76181839851578156256251250250050001, 3809091992578907812812562512502500050001

Offset: 0

Amarnath Murthy, Nov 05 2002

			a(6) = 1248163264/64 = 19502551.

Paolo Xausa, Table of n, a(n) for n = 0..80

Cf. A000079, A034887, A059268, A077351, A069872, A077353.

a:= n-> parse(cat(2^i$i=0..n))/2^n:
seq(a(n), n=0..15);  # Alois P. Heinz, May 16 2025

A077352[n_] := FromDigits[Flatten[IntegerDigits[Divisors[#]]]]/# & [2^n];
Array[A077352, 16, 0] (* or *)
FoldList[10^IntegerLength[2^#2]*#/2 + 1 &, 1, Range[15]] (* Paolo Xausa, May 19 2025 *)

For n>=1, a(n) = (a(n-1)*2^(n-1)*10^(floor(log_10(2^n))+1)+2^n)/2^n. - Sam Alexander, Feb 27 2005

Offset corrected by Sean A. Irvine, May 16 2025

A131436 Triangle read by rows, (n-1) zeros followed by 2^n - 1.

Original entry on oeis.org

1, 0, 3, 0, 0, 7, 0, 0, 0, 15, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 63, 0, 0, 0, 0, 0, 0, 127, 0, 0, 0, 0, 0, 0, 0, 255, 0, 0, 0, 0, 0, 0, 0, 0, 511, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4095, 0, 0

Offset: 1

Gary W. Adamson, Jul 11 2007

			First few rows of the triangle are:
1;
0, 3;
0, 0, 7;
0, 0, 0, 15;
0, 0, 0, 0, 31;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A131437, A131438, A131439.

Flatten[Table[{PadRight[{},n-1,0],2^n-1},{n,20}]] (* Harvey P. Dale, May 23 2012 *)

2*A059268(n)-A059268(n+1). - Paul Curtz, Jul 03 2008

More terms from Harvey P. Dale, May 23 2012

A220280 The reluctant sequence for A002260.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Dec 12 2012

The reluctant sequence B for a sequence A is a triangular array in which row k (>= 1) consists of the first k terms of A.

Here A002260 is the reluctant sequence for the sequence 1,2,3,... of positive numbers (A000027).

			A002260 begins
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, ...
so the first few rows of the new triangle are
   1,
   1, 1,
   1, 1, 2,
   1, 1, 2, 1,
   1, 1, 2, 1, 2,
   1, 1, 2, 1, 2, 3,
   ...
                                                                               ~

Boris Putievskiy, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002260, A002262, A037126, A059268, A215026.

t=int((math.sqrt(8*n-7) - 1)/ 2)
n1=n-t*(t+1)/2
t1=int((math.sqrt(8*n1-7) - 1)/ 2)
a=n1-t1*(t1+1)/2

a(n) = n1 - t1(t1+1)/2, where n1 = n - t(t+1)/2, t1 = floor[(-1+sqrt(8*n1-7))/2], t=floor[(-1+sqrt(8*n-7))/2]. For example, a(6)=2 since t=2, t1=1, n1=3.

Edited by N. J. A. Sloane, Jun 07 2024

A140531 Concatenate subsequences 0, 1, 2, 4, ..., 2^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 4, 0, 1, 2, 4, 8, 0, 1, 2, 4, 8, 16, 0, 1, 2, 4, 8, 16, 32, 0, 1, 2, 4, 8, 16, 32, 64, 0, 1, 2, 4, 8, 16, 32, 64, 128, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

Offset: 0

Paul Curtz, Jul 03 2008

A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A140531 is the reluctant sequence of A131577. - Boris Putievskiy, Dec 12 2012

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000079, A059268.

a(n) = floor(2^(n-2-t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2), n>=1. - Boris Putievskiy, Dec 13 2012

A164281 Triangle read by rows, a Petoukhov sequence (cf. A164279) generated from (1,2).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2, 4, 8, 16, 8, 16, 32, 16, 8, 4, 8, 16, 8, 4, 8, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2, 4, 8, 16, 8

Offset: 0

Gary W. Adamson, Aug 12 2009

Row sums = powers of 3: (1, 3, 9, 27, 81, ...). A164279 = a Petoukhov sequence generated through analogous principles from (3,2), with row sums = powers of 5.
Essentially, A164281 converts the terms (1,2,4,8,...) into rows with a binomial distribution as to frequency of terms. For example, row 3 has one 1, three 2's, three 4's, and one 8. This property arises due to the origin of the system of codes in A164056 (derived from the Gray code).
A Gray code origin also preserves the "one bit" (in this case, a "one product operation") since in each row, the next term is either twice current term or (1/2) current term.
Rows tend to A166242. - Gary W. Adamson, Oct 10 2009

			First few rows of the triangle =
  1;
  1, 2;
  1, 2, 4, 2;
  1, 2, 4, 2, 4, 8, 4, 2;
  1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2;
  ...
Example: row 3 of A164056 = (0, 1, 1, 0, 1, 1, 0, 0), so beginning with "1" at left, row 3 of A164281 = (1, 2, 4, 2, 4, 8, 4, 2).

Sergei Petoukhov & Matthew He, "Symmetrical Analysis Techniques for Genetic Systems and Bioinformatics - Advanced Patterns and Applications", IGI Global, 978-1-60566-127-9, October 2009, Chapters 2, 4, and 6.

Jon Maiga, Table of n, a(n) for n = 0..1022 (Rows 0..9)

Cf. A088696, A164279, A164056.

Cf. A166242.

A088696[n_]:=A088696[n]=Flatten[NestList[Join[#,Reverse[#]+1]&,{1},15]][[n]];
A164281[0]=1;
A164281[n_]:=If[IntegerQ[Log2[n+1]], 1, If[A088696[n+1]>A088696[n], 2*A164281[n-1], A164281[n-1]/2]]
Array[A164281,100,0] (* Jon Maiga, Oct 04 2019 *)

Given row terms of triangle A059268: (1; 1,2; 1,2,4; 1,2,4,8;...) and the digital codes in A164056: (0; 0,1; 0,1,1,0; 0,1,1,0,1,1,0,0;...); beginning with "1" in each row, multiply by 2 to obtain the next term to the right, if the corresponding positional term in A164056 = "1". Divide by 2 if the corresponding A164056 term = 0.

A(n, k) = 2^(A088696(n+1, k)-1). - Andrey Zabolotskiy, Feb 18 2025

Corrected and more terms from Jon Maiga, Oct 04 2019

A126064 Triangle read by rows, obtained by multiplying columns of triangle in A094587 by 1,2,4,8,16,... respectively.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 12, 12, 8, 24, 48, 48, 32, 16, 120, 240, 240, 160, 80, 32, 720, 1440, 1440, 960, 480, 192, 64, 5040, 10080, 10080, 6720, 3360, 1344, 448, 128, 40320, 80640, 80640, 53760, 26880, 10752, 3584, 1024, 256, 362880, 725760, 725760, 483840, 241920, 96768, 32256, 9216, 2304, 512

Offset: 0

N. J. A. Sloane, Feb 28 2007

Also obtained by multiplying the n-th rows of A094587 by the first (n+1) powers of 2: T(n,k) = A094587(n,k) * A059268(n,k), 0 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

			1
1, 2
2, 4, 4
6, 12, 12, 8
24, 48, 48, 32, 16
120, 240, 240, 160, 80, 32
720, 1440, 1440, 960, 480, 192, 64
5040, 10080, 10080, 6720, 3360, 1344, 448, 128

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Peter Luschny, Variants of Variations.

a126064 n k = a126064_tabl !! n !! k
a126064_row n = a126064_tabl !! n
a126064_tabl =  zipWith (zipWith (*)) a094587_tabl a059268_tabl
-- Reinhard Zumkeller, Jul 05 2012

A126064 := proc(n,k) binomial(n,k)*(n-k)!*2^k ; end: for n from 0 to 13 do for k from 0 to n do printf("%d,",A126064(n,k)) ; od: od: # R. J. Mathar, Nov 02 2007

m = 9;
T = Transpose[2^Range[0, m] Table[n!/k!, {k, 0, m}, {n, 0, m}]];
Table[T[[n+1, k+1]], {n, 0, m}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 04 2020 *)

More terms from R. J. Mathar, Nov 02 2007

cp's OEIS Frontend

A130321 Triangle, (2^0, 2^1, 2^2, ...) in every column.

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A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

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A140513 Repeat 2^n n times.

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A130328 Triangle of differences between powers of 2, read by rows.

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A077352 a(n) = (concatenation in ascending order of divisors of 2^n)/2^n.

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A131436 Triangle read by rows, (n-1) zeros followed by 2^n - 1.

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A220280 The reluctant sequence for A002260.

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