A140529 -id:A140529 - cp's OEIS frontend

A140660 a(n) = 3*4^n + 1.

4, 13, 49, 193, 769, 3073, 12289, 49153, 196609, 786433, 3145729, 12582913, 50331649, 201326593, 805306369, 3221225473, 12884901889, 51539607553, 206158430209, 824633720833, 3298534883329, 13194139533313, 52776558133249

Offset: 0

Paul Curtz, Jul 10 2008

nonn

An Engel expansion of 4/3 to the base 4 as defined in A181565, with the associated series expansion 4/3 = 4/4 + 4^2/(4*13) + 4^3/(4*13*49) + 4^4/(4*13*49*193) + .... Cf. A199115. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A181565, A199115.

[3*4^n+1: n in [0..30] ]; // Vincenzo Librandi, May 23 2011

LinearRecurrence[{5,-4}, {4,13}, 50] (* or *) CoefficientList[Series[ (7*x-4)/((1-x)*(4*x-1)), {x,0,50}], x] (* G. C. Greubel, Sep 15 2017 *)

x='x+O('x^50); Vec((7*x-4)/((1-x)*(4*x-1))) \\ G. C. Greubel, Sep 15 2017

a(n) = A002001(n+1) + 1.
a(n) = 4*a(n-1) - 3.
First differences: a(n+1) - a(n) = A002063(n).
a(n+k) - a(n) = 3*(4^k - 1)*A000302(n) = 9*A002450(k)*A000302(n).
a(n) = A140529(n) - A096045(n).
O.g.f.: (7*x - 4)/((1 - x)*(4*x - 1)). - R. J. Mathar, Jul 14 2008
From G. C. Greubel, Sep 15 2017: (Start)
E.g.f.: 3*exp(4*x) + exp(x).
a(n) = 5*a(n-1) - 4*a(n-2). (End)

Edited and extended R. J. Mathar, Jul 14 2008

A156760 5*4^n-1.

Original entry on oeis.org

4, 19, 79, 319, 1279, 5119, 20479, 81919, 327679, 1310719, 5242879, 20971519, 83886079, 335544319, 1342177279, 5368709119, 21474836479, 85899345919, 343597383679, 1374389534719, 5497558138879, 21990232555519, 87960930222079, 351843720888319

Offset: 0

Paul Curtz, Feb 15 2009

Second column of the array A132207, or, if this array is flattened, a(n)=A132207(A007583(n)).

			Binary.......................................Decimal
100................................................4
10011.............................................19
1001111...........................................79
100111111........................................319
10011111111.....................................1279
1001111111111...................................5119
100111111111111................................20479
10011111111111111..............................81919
1001111111111111111...........................327679
100111111111111111111........................1310719
10011111111111111111111......................5242879
1001111111111111111111111...................20971519
100111111111111111111111111.................83886079
10011111111111111111111111111..............335544319
1001111111111111111111111111111...........1342177279
... - _Philippe Deléham_, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500

[5*4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Table[5*4^n - 1, {n, 0, 18}] (* Michael De Vlieger, Apr 20 2015 *)

a(n) mod 9 = A070403(n+2).
a(n+1) = 10*A083420(n)+9 .
a(n) = 5*A000302(n)-1.
a(n) = ( A024036(n+1)+A140529(n) )/2.
a(n) = 4a(n-1)+3, a(0)=4.
a(n) = A003947(n+1)-1 = 5*a(n-1)-4*a(n-2). G.f.: (4-x)/((1-x)(1-4x)). - R. J. Mathar, Feb 23 2009
a(n) = A198693(n) + 2^(2n+1). - Bob Selcoe, Apr 20 2015

Edited and extended by R. J. Mathar, Feb 23 2009

A140683 a(n) = 3(-1)^(n+1)2^n - 1.

Original entry on oeis.org

-4, 5, -13, 23, -49, 95, -193, 383, -769, 1535, -3073, 6143, -12289, 24575, -49153, 98303, -196609, 393215, -786433, 1572863, -3145729, 6291455, -12582913, 25165823, -50331649, 100663295, -201326593, 402653183, -805306369, 1610612735, -3221225473

Offset: 0

Paul Curtz, Jul 11 2008

sign

Alternated reading of negative of A140660 and A140529.
The binomial transform yields -4 followed by the negative of A140657.
The inverse binomial transform yields essentially a signed version of A000244. - R. J. Mathar, Aug 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 2).

[3*(-1)^(n+1)*2^n-1: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

Table[3(-1)^(n+1)2^n-1,{n,0,40}] (* or *) LinearRecurrence[{-1,2},{-4,5},40] (* Harvey P. Dale, May 26 2011 *)

a(2n) = -A140660(n). a(2n+1) = A140529(n).
a(n+1) - a(n) = (-1)^n*A005010(n). a(2n) + a(2n+1) = A096045(n).
a(n) = A140590(n+1) - 2*A140590(n).
O.g.f: (4-x)/((x-1)(2x+1)). - R. J. Mathar, Aug 02 2008
a(n) = -a(n-1) + 2*a(n-2); a(0)=-4, a(1)=5. - Harvey P. Dale, May 26 2011

Edited and extended by R. J. Mathar, Aug 02 2008

A199116 a(n) = 6*4^n + 1.

Original entry on oeis.org

7, 25, 97, 385, 1537, 6145, 24577, 98305, 393217, 1572865, 6291457, 25165825, 100663297, 402653185, 1610612737, 6442450945, 25769803777, 103079215105, 412316860417, 1649267441665, 6597069766657, 26388279066625, 105553116266497, 422212465065985, 1688849860263937

Offset: 0

Vincenzo Librandi, Nov 04 2011

Bisection (odd part) of A181565 and A201630. - Bruno Berselli, Dec 04 2011

First differences of A221130, a(n) = A221130(n+2) - A221130(n+1). - Jaroslav Krizek, Jan 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A140529, A181565, A201630, A221130.

Magma
```
[6*4^n+1: n in [0..30]];
```

6*4^Range[0,30]+1 (* or *) LinearRecurrence[{5,-4},{7,25},30] (* Harvey P. Dale, Apr 18 2024 *)

a(n) = 4*a(n-1) - 3.
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: (7-10*x)/((1-x)*(1-4*x)). - Bruno Berselli, Nov 04 2011
From Elmo R. Oliveira, May 08 2025: (Start)
E.g.f.: exp(x)*(6*exp(3*x) + 1).
a(n) = A140529(n) + 2. (End)

A140589 Triangle A(k,n) = (-2)^k+2^n read by rows.

Original entry on oeis.org

2, -1, 0, 5, 6, 8, -7, -6, -4, 0, 17, 18, 20, 24, 32, -31, -30, -28, -24, -16, 0, 65, 66, 68, 72, 80, 96, 128, -127, -126, -124, -120, -112, -96, -64, 0, 257, 258, 260, 264, 272, 288, 320, 384, 512, -511, -510, -508, -504, -496, -480, -448, -384, -256, 0, 1025, 1026, 1028, 1032

Offset: 0

Paul Curtz, Jul 06 2008

The flattened sequence a(A000217(k)+j)=A(k,j) obeys a(n+1)-2a(n)= -5, 2, 5, -4, -4, -23, 8, 8, 8, 17, -16, -16, -16, -16, -95, ..., which is a dispersion of 2, -4, -4, 8, 8, 8, ... (a signed version of A140513) with -5, 5, -23, 17, -95, 65,... The latter sequence is A(k,0)-2*A(k-1,k-1), an alternation of the negative of A140529 with each second element of A000051.

			Rows starting at k=0: (2), (-1,0); (5, 6, 8); (-7,-6,-4,0); (17,18,20,24,32);...

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

A(k,n) = A000079(n)+A122803(k).

Edited by R. J. Mathar, Jul 08 2008

A267622 Binary representation of the n-th iteration of the "Rule 187" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 101, 10111, 1011111, 101111111, 10111111111, 1011111111111, 101111111111111, 10111111111111111, 1011111111111111111, 101111111111111111111, 10111111111111111111111, 1011111111111111111111111, 101111111111111111111111111, 10111111111111111111111111111

Offset: 0

Robert Price, Jan 18 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
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 Elementary Cellular Automata

Cf. A267621, A140529.

rule=187; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}]   (* Binary Representation of Rows *)

Conjectures from Colin Barker, Jan 19 2016 and Apr 20 2019: (Start)
a(n) = 101*a(n-1)-100*a(n-2) for n>2.
G.f.: (1+10*x^2) / ((1-x)*(1-100*x)).
(End)

cp's OEIS Frontend

A140660 a(n) = 3*4^n + 1.

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A156760 5*4^n-1.

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A140683 a(n) = 3*(-1)^(n+1)*2^n - 1.

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A199116 a(n) = 6*4^n + 1.

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A140589 Triangle A(k,n) = (-2)^k+2^n read by rows.

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A267622 Binary representation of the n-th iteration of the "Rule 187" elementary cellular automaton starting with a single ON (black) cell.

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A140683 a(n) = 3(-1)^(n+1)2^n - 1.