A173787 -id:A173787 - cp's OEIS frontend

A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048

Offset: 0

Reinhard Zumkeller, Feb 28 2010

Essentially the same as A048645. - T. D. Noe, Mar 28 2011

			Triangle begins as:
     2;
     3,    4;
     5,    6,    8;
     9,   10,   12,   16;
    17,   18,   20,   24,   32;
    33,   34,   36,   40,   48,   64;
    65,   66,   68,   72,   80,   96,  128;
   129,  130,  132,  136,  144,  160,  192,  256;
   257,  258,  260,  264,  272,  288,  320,  384,  512;
   513,  514,  516,  520,  528,  544,  576,  640,  768, 1024;
  1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;

T. D. Noe, Rows n = 0..100 of triangle, flattened
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).

Cf. A048645, A118413, A118416.

Cf. also A087112, A370121.

[2^n + 2^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 07 2021

Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* T. D. Noe, Jun 18 2013 *)

A173786(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)); }; \\ Antti Karttunen, Feb 29 2024, after David A. Corneth's PARI-program in A048645

from math import isqrt, comb
def A173786(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return (1<Chai Wah Wu, Jun 20 2025

flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 07 2021

1 <= A000120(T(n,k)) <= 2.
For n>0, 0<=kA048645(n+1,k+2) and T(n,n) = A048645(n+2,1).
Row sums give A006589(n).
Central terms give A161168(n).
T(2*n+1,n) = A007582(n+1).
T(2*n+1,n+1) = A028403(n+1).
T(n,k) = A140513(n,k) - A173787(n,k), 0<=k<=n.
T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0T(n,k) * A173787(n,k) = A173787(2*n,2*k), 0<=k<=n.
T(n,0) = A000051(n).
T(n,1) = A052548(n) for n>0.
T(n,2) = A140504(n) for n>1.
T(n,3) = A175161(n-3) for n>2.
T(n,4) = A175162(n-4) for n>3.
T(n,5) = A175163(n-5) for n>4.
T(n,n-4) = A110287(n-4) for n>3.
T(n,n-3) = A005010(n-3) for n>2.
T(n,n-2) = A020714(n-2) for n>1.
T(n,n-1) = A007283(n-1) for n>0.
T(n,n) = 2*A000079(n).

Typo in first comment line fixed by Reinhard Zumkeller, Mar 07 2010

A110286 a(n) = 15*2^n.

Original entry on oeis.org

15, 30, 60, 120, 240, 480, 960, 1920, 3840, 7680, 15360, 30720, 61440, 122880, 245760, 491520, 983040, 1966080, 3932160, 7864320, 15728640, 31457280, 62914560, 125829120, 251658240, 503316480, 1006632960, 2013265920, 4026531840, 8053063680, 16106127360

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A007283, A020714, A005009, A005010, A005015, A005029.

Cf. A000079, A051916, A173787.

[15*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

15*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,15,30] (* Harvey P. Dale, Oct 19 2014 *)

a(n)=15<Charles R Greathouse IV, Oct 07 2015

G.f.: 15/(1-2x). - Philippe Deléham, Nov 23 2008
a(n) = A000079(n)*15 = A007283(n)*5 = A020714(n)*3. - Omar E. Pol, Dec 17 2008
a(n) = A173787(n+4,n). - Reinhard Zumkeller, Feb 28 2010
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
a(n) = 2*a(n-1) (with a(0)=15). - Vincenzo Librandi, Dec 26 2010
E.g.f.: 15*exp(2*x). - Stefano Spezia, May 15 2021

Edited by Omar E. Pol, Dec 16 2008

A159741 a(n) = 8*(2^n - 1).

Original entry on oeis.org

8, 24, 56, 120, 248, 504, 1016, 2040, 4088, 8184, 16376, 32760, 65528, 131064, 262136, 524280, 1048568, 2097144, 4194296, 8388600, 16777208, 33554424, 67108856, 134217720, 268435448, 536870904, 1073741816, 2147483640, 4294967288, 8589934584, 17179869176, 34359738360

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

Fifth diagonal of the array which contains m-acci numbers in the m-th row.
The base array is constructed from m-acci numbers starting each with 1, 1, and 2 and filling one row of the table (see the examples).
The main and the upper diagonals of the table are the powers of 2, A000079.
The first subdiagonal is essentially A000225, followed by essentially A036563.
The next subdiagonal is this sequence here, followed by A159742, A159743, A159744, A159746, A159747, A159748.
a(n) written in base 2: 1000, 11000, 111000, 1111000, ..., i.e., n times 1 and 3 times 0 (A161770). - Jaroslav Krizek, Jun 18 2009
Also numbers for which n^8/(n+8) is an integer. - Vicente Izquierdo Gomez, Jan 03 2013

			From _R. J. Mathar_, Apr 22 2009: (Start)
The base table is
.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
.2..2....2....2....2....2....2....2....2....2....2....2....2....2
.0..2....3....4....4....4....4....4....4....4....4....4....4....4
.0..2....5....7....8....8....8....8....8....8....8....8....8....8
.0..2....8...13...15...16...16...16...16...16...16...16...16...16
.0..2...13...24...29...31...32...32...32...32...32...32...32...32
.0..2...21...44...56...61...63...64...64...64...64...64...64...64
.0..2...34...81..108..120..125..127..128..128..128..128..128..128
.0..2...55..149..208..236..248..253..255..256..256..256..256..256
.0..2...89..274..401..464..492..504..509..511..512..512..512..512
.0..2..144..504..773..912..976.1004.1016.1021.1023.1024.1024.1024
.0..2..233..927.1490.1793.1936.2000.2028.2040.2045.2047.2048.2048
.0..2..377.1705.2872.3525.3840.3984.4048.4076.4088.4093.4095.4096
Columns: A000045, A000073, A000078, A001591, A001592 etc. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000079, A000225, A036563, A159742, A159743, A159744, A159746, A159747, A159748.

Cf. A000918, A028399, A161770, A173787, A175161, A175166.

[8*(2^n -1): n in [1..50]]; // G. C. Greubel, May 22 2018

T := proc(n,m) option remember ; if n < 0 then 0; elif n <= 1 then 1; elif n = 2 then 2; else add(procname(n-i,m),i=1..m) ; fi: end: A159741 := proc(n) T(n+4,n+1) ; end: seq(A159741(n),n=1..40) ; # R. J. Mathar, Apr 22 2009

Table[8(2^n-1),{n,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{3,-2},{8,24},30] (* Harvey P. Dale, Jan 01 2019 *)

a(n)=8*(2^n-1) \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Apr 22 2009: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = 8*(2^n-1).
G.f.: 8*x/((2*x-1)*(x-1)). (End)
From Jaroslav Krizek, Jun 18 2009: (Start)
a(n) = Sum_{i=3..(n+2)} 2^i.
a(n) = Sum_{i=1..n} 2^(i+2).
a(n) = a(n-1) + 2^(n+2) for n >= 2. (End)
a(n) = A173787(n+3,3) = A175166(2*n)/A175161(n). - Reinhard Zumkeller, Feb 28 2010
From Elmo R. Oliveira, Jun 15 2025: (Start)
E.g.f.: 8*exp(x)*(exp(x) - 1).
a(n) = 8*A000225(n) = 4*A000918(n+1) = 2*A028399(n+2). (End)

More terms from R. J. Mathar, Apr 22 2009
Edited by Al Hakanson (hawkuu(AT)gmail.com), May 11 2009
Comments claiming negative entries deleted by R. J. Mathar, Aug 24 2009

A059153 a(n) = 2^(n+2)*(2^(n+1)-1).

Original entry on oeis.org

4, 24, 112, 480, 1984, 8064, 32512, 130560, 523264, 2095104, 8384512, 33546240, 134201344, 536838144, 2147418112, 8589803520, 34359476224, 137438429184, 549754765312, 2199021158400, 8796088827904, 35184363700224, 140737471578112, 562949919866880

Offset: 0

Jonas Wallgren, Feb 02 2001

A hierarchical sequence (S(W'2{2}c) - see A059126).
a(n) written in base 2: 100, 11000, 1110000, ..., i.e., (n+1) times 1 and (n+2) times 0 (see A163663). - Jaroslav Krizek, Aug 12 2009
Also, the number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 513", based on the 5-celled von Neumann neighborhood. - Robert Price, May 04 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Harry J. Smith, Table of n, a(n) for n = 0..200
J. Wallgren, Hierarchical sequences
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A006516, A020522, A059126, A163663, A173787, A272702.

Table[2^(n + 2)*(2^(n + 1) - 1), {n, 0, 23}] (* and *) LinearRecurrence[{6, -8}, {4, 24}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n) = { 2^(n + 2)*(2^(n + 1) - 1) } \\ Harry J. Smith, Jun 25 2009

a(n) = A173787(2*n+3,n+2) = 4*A006516(n+1). - Reinhard Zumkeller, Feb 28 2010
From Colin Barker, Apr 28 2013: (Start)
a(n) = 6*a(n-1) - 8*a(n-2).
G.f.: 4 / ((2*x-1)*(4*x-1)). (End)
a(n) = 2*A020522(n+1). - Hussam al-Homsi, Jun 06 2021
E.g.f.: 4*exp(2*x)*(2*exp(2*x) - 1). - Elmo R. Oliveira, Dec 10 2023

Revised by Henry Bottomley, Jun 27 2005

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

A175164 a(n) = 16*(2^n - 1).

Original entry on oeis.org

0, 16, 48, 112, 240, 496, 1008, 2032, 4080, 8176, 16368, 32752, 65520, 131056, 262128, 524272, 1048560, 2097136, 4194288, 8388592, 16777200, 33554416, 67108848, 134217712, 268435440

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), this sequence (m=16), A175165 (m=32), A175166 (m=64).

Cf. A140504, A173787.

I:=[0,16]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

16*(2^Range[0,40] - 1) (* G. C. Greubel, Jul 08 2021 *)

def A175164(n): return (1<Chai Wah Wu, Jun 27 2023

[16*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+4) - 16.
a(n) = A173787(n+4, 4).
a(2*n) = A140504(n+2)*A028399(n).
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=16. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 16*x/((1-x)*(1-2*x)).
E.g.f.: 16*(exp(2*x) - exp(x)). (End)

A175166 a(n) = 64*(2^n - 1).

Original entry on oeis.org

0, 64, 192, 448, 960, 1984, 4032, 8128, 16320, 32704, 65472, 131008, 262080, 524224, 1048512, 2097088, 4194240, 8388544, 16777152, 33554368, 67108800, 134217664, 268435392, 536870848, 1073741760

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), A175165 (m=32), this sequence (m=64).

Cf. A173787, A175161.

I:=[0,64]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

LinearRecurrence[{3,-2},{0,64},30] (* Harvey P. Dale, Apr 08 2015 *)

def A175166(n): return (1<Chai Wah Wu, Jun 27 2023

[64*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+6) - 64.
a(n) = A173787(n+6, 6).
a(2*n) = A175161(n)*A159741(n) for n > 0.
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=64. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 64*x/((1-x)*(1-2*x)).
E.g.f.: 64*(exp(2*x) - exp(x)). (End)

A175165 a(n) = 32*(2^n - 1).

Original entry on oeis.org

0, 32, 96, 224, 480, 992, 2016, 4064, 8160, 16352, 32736, 65504, 131040, 262112, 524256, 1048544, 2097120, 4194272, 8388576, 16777184, 33554400, 67108832, 134217696, 268435424, 536870880

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), this sequence (m=32), A175166 (m=64).

Cf. A173787.

I:=[0,32]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

32(2^Range[0,30] -1) (* or *) LinearRecurrence[{3,-2},{0,32},30] (* Harvey P. Dale, Mar 23 2015 *)

def A175165(n): return (1<Chai Wah Wu, Jun 27 2023

[32*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+5) - 32.
a(n) = A173787(n+5, 5).
a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0, a(1)=32. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 32*x/((1-x)*(1-2*x)).
E.g.f.: 32*(exp(2*x) - exp(x)). (End)

A204983 a(n) = 2^(k-1)-2^(j-1), where (2^(k-1),2^(j-1)) is the least pair of distinct positive powers of 2 for which n divides 2^(k-1)-2^(j-1).

Original entry on oeis.org

1, 2, 3, 4, 15, 6, 7, 8, 63, 30, 1023, 12, 4095, 14, 15, 16, 255, 126, 262143, 60, 63, 2046, 2047, 24, 1048575, 8190, 262143, 28, 268435455, 30, 31, 32, 1023, 510, 4095, 252, 68719476735, 524286, 4095, 120, 1048575, 126, 16383, 4092, 4095

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

(Conjecture) Equivalently, the solution set of 2^p * (2^q - 1) = x * y, OR 2^q - 1 = 2^p * x * y, for at most one of the naturals x and y being given; unknown p and q in the integers; then a(n) = 2^p * (2^q - 1) where p and q are directly related to n (see formula). - Andrew T. Porter, Dec 20 2022

Cf. A204979, A204892, A204984.
Cf. A000079, A173787.
Cf. A007733, A007814.

Mathematica
```
(See the program at A204979.)
```

a(n) = for (k=1, oo, for (j=1, k-1, my(d=2^(k-1)-2^(j-1)); if (!(d % n), return(d)););); \\ Michel Marcus, Sep 16 2023

Conjecture: a(n) = 2^A007814(n) * (2^A007733(n) - 1). - Andrew T. Porter, Dec 20 2022

Previous Showing 11-19 of 19 results.

cp's OEIS Frontend

A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

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A110286 a(n) = 15*2^n.

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A159741 a(n) = 8*(2^n - 1).

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A059153 a(n) = 2^(n+2)*(2^(n+1)-1).

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A140513 Repeat 2^n n times.

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A175164 a(n) = 16*(2^n - 1).

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