A130123 -id:A130123 - cp's OEIS frontend

A130125 Triangle defined by A128174 * A130123, read by rows.

1, 0, 2, 1, 0, 4, 0, 2, 0, 8, 1, 0, 4, 0, 16, 0, 2, 0, 8, 0, 32, 1, 0, 4, 0, 16, 0, 64, 0, 2, 0, 8, 0, 32, 0, 128, 1, 0, 4, 0, 16, 0, 64, 0, 256, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 1, 0, 4, 0, 16, 0, 64, 0, 256, 0, 1024, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 0, 2048

Offset: 0

Gary W. Adamson, May 11 2007

Row sums = A000975: (1, 2, 5, 10, 21, 42, ...).

			First few rows of the triangle are:
  1;
  0, 2;
  1, 0, 4;
  0, 2, 0, 8;
  1, 0, 4, 0, 16;
  0, 2, 0, 8,  0, 32; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000975, A128174, A130123.

Flat(List([0..10], n-> List([0..n], k-> 2^(k-1)*(1+(-1)^(n-k)) ))); # G. C. Greubel, Jun 05 2019

[[2^(k-1)*(1+(-1)^(n-k)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jun 05 2019

Table[2^(k-1)*(1+(-1)^(n-k)), {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = 2^(k-1)*(1+(-1)^(n-k))}; \\ G. C. Greubel, Jun 05 2019

[[2^(k-1)*(1+(-1)^(n-k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jun 05 2019

A128174 * A130123 as infinite lower triangular matrices. By columns, (2^k, 0, 2^k, 0, ...).

T(n,k) = 2^(k-1)*(1 + (-1)^(n-k)). - G. C. Greubel, Jun 05 2019

More terms added by G. C. Greubel, Jun 05 2019

A130124 Triangle defined by A130123 * A002260, read by rows.

Original entry on oeis.org

1, 2, 4, 4, 8, 12, 8, 16, 24, 32, 16, 32, 48, 64, 80, 32, 64, 96, 128, 160, 192, 64, 128, 192, 256, 320, 384, 448, 128, 256, 384, 512, 640, 768, 896, 1024, 256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096, 4608, 5120

Offset: 1

Gary W. Adamson, May 11 2007

Row sums = A001780, (1, 6, 24, 80, 240, ...).

			First few rows of the triangle are:
   1;
   2,  4;
   4,  8, 12;
   8, 16, 24,  32;
  16, 32, 48,  64,  80;
  32, 64, 96, 128, 160, 192; ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A001780, A130123, A002260.

Flat(List([1..12], n-> List([1..n], k-> 2^(n-1)*k ))); # G. C. Greubel, Jun 05 2019

[[2^(n-1)*k: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 05 2019

Table[2^(n-1)*k, {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = 2^(n-1)*k}; \\ G. C. Greubel, Jun 05 2019

[[2^(n-1)*k for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 05 2019

A130123 * A002260, where A130123 = the 2^n transform and A002260 = [1; 1, 2; 1, 2, 3; ...).

T(n, k) = 2^(n-1)*k. - G. C. Greubel, Jun 05 2019

More terms added by G. C. Greubel, Jun 05 2019

A134352 A130123 * A128174.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Oct 21 2007

Row sums = A134353.

			First few rows of the triangle:
   1;
   0,  2;
   4,  0,  4;
   0,  8,  0,  8;
  16,  0, 16,  0, 16;
   0, 32,  0, 32,  0, 32;
  ...

Cf. A130123, A128174, A134353.

A130123 * A128174 as infinite lower triangular matrices.

Triangle read by rows: even n-th row = n+1 terms of (2^n, 0, 2^n, ...); odd n-th row = n+1 terms of (0, 2^n, 0, 2^n, ...).

A143424 Triangle read by rows, A054525 * A130123, 1<=k<=n.

Original entry on oeis.org

1, -1, 2, -1, 0, 4, 0, -2, 0, 8, -1, 0, 0, 0, 16, 1, -2, -4, 0, 0, 32, -1, 0, 0, 0, 0, 0, 64, 0, 0, 0, -8, 0, 0, 0, 128, 0, 0, -4, 0, 0, 0, 0, 0, 256, 1, -2, 0, 0, -16, 0, 0, 0, 0, 512, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 2, 0, -8, 0, -32, 0, 0, 0, 0, 0, 2048

Offset: 1

Gary W. Adamson, Aug 14 2008

Row sums = A000740 starting with offset 1: (1, 1, 3, 6, 15, 27, 63,...).

Left border = mu(n), A008683.

			First few rows of the triangle =
1;
-1, 2;
-1, 0, 4;
0, -2, 0, 8;
-1, 0, 0, 0, 16;
1, -2, -4, 0, 0, 32;
-1, 0, 0, 0, 0, 0, 64;
0, 0, 0, -8, 0, 0, 0, 128;
...

Cf. A000740, A130123, A054525, A008683.

Mobius transform of A130123, where A130123 = an infinite lower triangular matrix with (1, 2, 4, 8,...) in the main diagonal and the rest zeros.

A143425 Triangle read by rows A051731 * A130123, 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 0, 4, 1, 2, 0, 8, 1, 0, 0, 0, 16, 1, 2, 4, 0, 0, 32, 1, 0, 0, 0, 0, 0, 64, 1, 2, 0, 8, 0, 0, 0, 128, 1, 0, 4, 0, 0, 0, 0, 0, 256, 1, 2, 0, 0, 16, 0, 0, 0, 0, 512, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 1, 2, 4, 8, 0, 32, 0, 0, 0, 0, 0, 2048

Offset: 1

Gary W. Adamson, Aug 14 2008

Row sums = A034729: (1, 3, 5, 11, 17, ...).

			First few rows of the triangle are:
  1;
  1, 2;
  1, 0, 4;
  1, 2, 0, 8;
  1, 0, 0, 0, 16;
  1, 2, 4, 0,  0, 32;
  ...

Cf. A034729, A051731, A130123.

tabl(nn) = {ma = matrix(nn, nn, n, k, !(n%k)); mb = matrix(nn, nn, n, k, n--; k--; if (n==k, 2^n, 0)); m = ma*mb; for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Jun 30 2017

Triangle read by rows A051731 * A130123, 1<=k<=n, where A130123 = an infinite lower triangular matrix with (1, 2, 4, 8, ...) in the main diagonal and the rest zeros. A051731 = the inverse Mobius transform.

Typo in data corrected by Michel Marcus, Jun 30 2017

A023758 Numbers of the form 2^i - 2^j with i >= j.

Original entry on oeis.org

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

Offset: 1

Olivier Gérard

Numbers whose digits in base 2 are in nonincreasing order.
Might be called "nialpdromes".
Subset of A077436. Proof: Since a(n) is of the form (2^i-1)*2^j, i,j >= 0, a(n)^2 = (2^(2i) - 2^(i+1))*2^(2j) + 2^(2j) where the first sum term has i-1 one bits and its 2j-th bit is zero, while the second sum term switches the 2j-th bit to one, giving i one bits, as in a(n). - Ralf Stephan, Mar 08 2004
Numbers whose binary representation contains no "01". - Benoit Cloitre, May 23 2004
Every polynomial with coefficients equal to 1 for the leading terms and 0 after that, evaluated at 2. For instance a(13) = x^4 + x^3 + x^2 at 2, a(14) = x^4 + x^3 + x^2 + x at 2. - Ben Paul Thurston, Jan 11 2008
From Gary W. Adamson, Jul 18 2008: (Start)
As a triangle by rows starting:
   1;
   2,  3;
   4,  6,  7;
   8, 12, 14, 15;
  16, 24, 28, 30, 31;
  ...,
equals A000012 * A130123 * A000012, where A130123 = (1, 0,2; 0,0,4; 0,0,0,8; ...). Row sums of this triangle = A000337 starting (1, 5, 17, 49, 129, ...). (End)
First differences are A057728 = 1; 1; 1; 1; 2,1; 1; 4,2,1; 1; 8,4,2,1; 1; ... i.e., decreasing powers of 2, separated by another "1". - M. F. Hasler, May 06 2009
Apart from first term, numbers that are powers of 2 or the sum of some consecutive powers of 2. - Omar E. Pol, Feb 14 2013
From Andres Cicuttin, Apr 29 2016: (Start)
Numbers that can be digitally generated with twisted ring (Johnson) counters. This is, the binary digits of a(n) correspond to those stored in a shift register where the input bit of the first bit storage element is the inverted output of the last storage element. After starting with all 0’s, each new state is obtained by rotating the stored bits but inverting at each state transition the last bit that goes to the first position (see link).
Examples: for a(n) represented by three bits
             Binary
a(5)= 4   ->  100   last bit = 0
a(6)= 6   ->  110   first bit = 1 (inverted last bit of previous number)
a(7)= 7   ->  111
and for a(n) represented by four bits
             Binary
a(8) = 8   -> 1000
a(9) = 12  -> 1100  last bit = 0
a(10)= 14  -> 1110  first bit = 1 (inverted last bit of previous number)
a(11)= 15  -> 1111
(End)
Powers of 2 represented in bases which are terms of this sequence must always contain at least one digit which is also a power of 2. This is because 2^i mod (2^i - 2^j) = 2^j, which means the last digit always cycles through powers of 2 (or if i=j+1 then the first digit is a power of 2 and the rest are trailing zeros). The only known non-member of this sequence with this property is 5. - Ely Golden, Sep 05 2017
Numbers k such that k = 2^(1 + A000523(k)) - 2^A007814(k). - Daniel Starodubtsev, Aug 05 2021
A002260(n) = v(a(n)/2^v(a(n))+1) and A002024(n) = A002260(n) + v(a(n)) where v is the dyadic valuation (i.e., A007814). - Lorenzo Sauras Altuzarra, Feb 01 2023

			a(22) = 64 = 32 + 32 = 2^5 + a(16) = 2^A003056(20) + a(22-5-1).
a(23) = 96 = 64 + 32 = 2^6 + a(16) = 2^A003056(21) + a(23-6-1).
a(24) = 112 = 64 + 48 = 2^6 + a(17) = 2^A003056(22) + a(24-6-1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5051 terms from T. D. Noe)
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
S. M. Shabab Hossain, Md. Mahmudur Rahman and M. Sohel Rahman, Solving a Generalized Version of the Exact Cover Problem with a Light-Based Device, Optical Supercomputing, Lecture Notes in Computer Science, 2011, Volume 6748/2011, 23-31, DOI: 10.1007/978-3-642-22494-2_4.
Eric Weisstein's World of Mathematics, Digit.
Wikipedia, Ring counter.
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A000337(r) = sum of row T(r, c) with 0 <= c < r. See also A002024, A003056, A140129, A140130, A221975.
Cf. A007088, A130123, A101082 (complement), A340375 (characteristic function).
This is the base-2 version of A064222. First differences are A057728.
Subsequence of A077436, of A129523, of A277704, and of A333762.
Subsequences: A043569 (nonzero even terms, or equally, nonzero terms doubled), A175332, A272615, A335431, A000396 (its even terms only), A324200.
Positions of zeros in A049502, A265397, A277899, A284264.
Positions of ones in A283983, A283989.
Positions of nonzero terms in A341509 (apart from the initial zero).
Positions of squarefree terms in A260443.
Fixed points of A264977, A277711, A283165, A334666.
Distinct terms in A340632.
Cf. also A002024, A002260, A007814, A133797, A152449, A181666, A211705, A277699, A306441.
Cf. also A309758, A309759, A309761 (for analogous sequences).

import Data.Set (singleton, deleteFindMin, insert)
a023758 n = a023758_list !! (n-1)
a023758_list = 0 : f (singleton 1) where
f s = x : f (if even x then insert z s' else insert z $ insert (z+1) s')
where z = 2*x; (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 24 2014, Dec 19 2012

a:=proc(n) local n2,d: n2:=convert(n,base,2): d:={seq(n2[j]-n2[j-1],j=2..nops(n2))}: if n=0 then 0 elif n=1 then 1 elif d={0,1} or d={0} or d={1} then n else fi end: seq(a(n),n=0..2100); # Emeric Deutsch, Apr 22 2006

Union[Flatten[Table[2^i - 2^j, {i, 0, 100}, {j, 0, i}]]] (* T. D. Noe, Mar 15 2011 *)
Select[Range[0, 2^10], NoneTrue[Differences@ IntegerDigits[#, 2], # > 0 &] &] (* Michael De Vlieger, Sep 05 2017 *)

for(n=0,2500,if(prod(k=1,length(binary(n))-1,component(binary(n),k)+1-component(binary(n),k+1))>0,print1(n,",")))

A023758(n)= my(r=round(sqrt(2*n--))); (1<<(n-r*(r-1)/2)-1)<<(r*(r+1)/2-n)
/* or, to illustrate the "decreasing digit" property and analogy to A064222: */
A023758(n,show=0)={ my(a=0); while(n--, show & print1(a","); a=vecsort(binary(a+1)); a*=vector(#a,j,2^(j-1))~); a} \\ M. F. Hasler, May 06 2009

is(n)=if(n<5,1,n>>=valuation(n,2);n++;n>>valuation(n,2)==1) \\ Charles R Greathouse IV, Jan 04 2016

list(lim)=my(v=List([0]),t); for(i=1,logint(lim\1+1,2), t=2^i-1; while(t<=lim, listput(v,t); t*=2)); Set(v) \\ Charles R Greathouse IV, May 03 2016

def a_next(a_n): return (a_n | (a_n >> 1)) + (a_n & 1)
a_n = 1; a = [0]
for i in range(55): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022

from math import isqrt
def A023758(n): return (1<<(m:=isqrt(n-1<<3)+1>>1))-(1<<(m*(m+1)-(n-1<<1)>>1)) # Chai Wah Wu, Feb 23 2025

a(n) = 2^s(n) - 2^((s(n)^2 + s(n) - 2n)/2) where s(n) = ceiling((-1 + sqrt(1+8n))/2). - Sam Alexander, Jan 08 2005
a(n) = 2^k + a(n-k-1) for 1 < n and k = A003056(n-2). The rows of T(r, c) = 2^r-2^c for 0 <= c < r read from right to left produce this sequence: 1; 2, 3; 4, 6, 7; 8, 12, 14, 15; ... - Frank Ellermann, Dec 06 2001
For n > 0, a(n) mod 2 = A010054(n). - Benoit Cloitre, May 23 2004
A140130(a(n)) = 1 and for n > 1: A140129(a(n)) = A002262(n-2). - Reinhard Zumkeller, May 14 2008
a(n+1) = (2^(n - r(r-1)/2) - 1) 2^(r(r+1)/2 - n), where r=round(sqrt(2n)). - M. F. Hasler, May 06 2009
Start with A000225. If k is in the sequence, then so is 2k. - Ralf Stephan, Aug 16 2013
G.f.: (x^2/((2-x)*(1-x)))*(1 + Sum_{k>=0} x^((k^2+k)/2)*(1 + x*(2^k-1))). The sum is related to Jacobi theta functions. - Robert Israel, Feb 24 2015
A049502(a(n)) = 0. - Reinhard Zumkeller, Jun 17 2015
a(n) = a(n-1) + a(n-d)/a(d*(d+1)/2 + 2) if n > 1, d > 0, where d = A002262(n-2). - Yuchun Ji, May 11 2020
A277699(a(n)) = a(n)^2, A306441(a(n)) = a(n+1). - Antti Karttunen, Feb 15 2021 (the latter identity from A306441)
Sum_{n>=2} 1/a(n) = A211705. - Amiram Eldar, Feb 20 2022

Definition changed by N. J. A. Sloane, Jan 05 2008

A130128 Triangle read by rows: T(n,k) = (n - k + 1)*2^(k-1).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 6, 8, 8, 5, 8, 12, 16, 16, 6, 10, 16, 24, 32, 32, 7, 12, 20, 32, 48, 64, 64, 8, 14, 24, 40, 64, 96, 128, 128, 9, 16, 28, 48, 80, 128, 192, 256, 256, 10, 18, 32, 56, 96, 160, 256, 384, 512, 512, 11, 20, 36, 64, 112, 192, 320, 512, 768, 1024, 1024

Offset: 1

Gary W. Adamson, May 11 2007

T(n,k) is the number of paths from node 0 to odd k in a directed graph with 2n+1 vertices labeled 0, 1, ..., 2n+1 and edges leading from i to i+1 for all i, from i to i+2 for even i, and from i to i-2 for odd i. - Grace Work, Mar 01 2020

			First few rows of the triangle are:
  1;
  2,  2;
  3,  4,  4;
  4,  6,  8,  8;
  5,  8, 12, 16, 16;
  6, 10, 16, 24, 32, 32;
  7, 12, 20, 32, 48, 64, 64;
  ...
From _Peter Munn_, Sep 22 2022: (Start)
As a square array, showing top left:
    1,   2,   3,    4,    5,    6,    7, ...
    2,   4,   6,    8,   10,   12,   14, ...
    4,   8,  12,   16,   20,   24,   28, ...
    8,  16,  24,   32,   40,   48,   56, ...
   16,  32,  48,   64,   80,   96,  112, ...
   32,  64,  96,  128,  160,  192,  224, ...
  ...
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. Krom and M. M. Roughan, Path Counting and Eulerian Numbers, Girls' Angle Bulletin, Vol. 13, No. 3 (2020), 8-10.

Row sums are A000295.

Cf. A004736, A054582 (subtable of square array), A130123.

Table[(n - k + 1)*2^(k - 1), {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Mar 23 2020 *)

T(n,k)={(n - k + 1)*2^(k-1)} \\ Andrew Howroyd, Mar 01 2020

Equals A004736 * A130123 as infinite lower triangular matrices.
As a square array, n >= 0, k >= 1, read by descending antidiagonals, A(n,k) = k * 2^n. - Peter Munn, Sep 22 2022
G.f.: x*y/( (1-x)^2 * (1-2*x*y) ). - Kevin Ryde, Sep 24 2022

Name clarified by Grace Work, Mar 01 2020

Terms a(56) and beyond from Andrew Howroyd, Mar 01 2020

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A130125 Triangle defined by A128174 * A130123, read by rows.

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A130124 Triangle defined by A130123 * A002260, read by rows.

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A134352 A130123 * A128174.

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A143424 Triangle read by rows, A054525 * A130123, 1<=k<=n.

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A143425 Triangle read by rows A051731 * A130123, 1<=k<=n.

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A023758 Numbers of the form 2^i - 2^j with i >= j.

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