A072197 -id:A072197 - cp's OEIS frontend

A080675 a(n) = (5*4^n - 8)/6.

2, 12, 52, 212, 852, 3412, 13652, 54612, 218452, 873812, 3495252, 13981012, 55924052, 223696212, 894784852, 3579139412, 14316557652, 57266230612, 229064922452, 916259689812, 3665038759252, 14660155037012, 58640620148052, 234562480592212, 938249922368852

Offset: 1

N. J. A. Sloane, Mar 02 2003

These numbers have a simple binary pattern: 10,1100,110100,11010100,1101010100, ... i.e., the n-th term has a binary expansion 1(10){n-1}0, that is, there are n-1 10's between the most significant 1 and the least significant 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..170
Andrei Asinowski, Cyril Banderier, Benjamin Hackl, On extremal cases of pop-stack sorting, Permutation Patterns (Zürich, Switzerland, 2019).
Index entries for linear recurrences with constant coefficients, signature (5,-4).

a(n) = A072197(n-1) - 1 = A014486(|A106191(n)|). a(n) = A079946(A020988(n-2)) for n>=2. Cf. also A122229.

[(5*4^n-8)/6: n in [1..40] ]; // Vincenzo Librandi, Apr 28 2011

(5*4^Range[30]-8)/6 (* or *) LinearRecurrence[{5,-4},{2,12},30] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=(5*4^n-8)/6 \\ Charles R Greathouse IV, Oct 07 2015

a(1)=2, a(2)=12, a(n)=5*a(n-1)-4*a(n-2). - Harvey P. Dale, Oct 16 2012

Further comments added by Antti Karttunen, Sep 14 2006

A206374 a(n) = (7*4^n - 1)/3.

Original entry on oeis.org

2, 9, 37, 149, 597, 2389, 9557, 38229, 152917, 611669, 2446677, 9786709, 39146837, 156587349, 626349397, 2505397589, 10021590357, 40086361429, 160345445717, 641381782869, 2565527131477, 10262108525909, 41048434103637, 164193736414549, 656774945658197

Offset: 0

Brad Clardy, Feb 07 2012

First bisection of A062092 and A081253, second bisection of A097163. - Bruno Berselli, Feb 12 2012

Except a(0)=2, this is the 3rd row of table A178415. - Michel Marcus, Apr 13 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A002450, A006666, A072197; A002042 (first differences), A178415, A347834.

Magma
```
[(7*4^n-1)/3 : n in [0..30]];
```

Table[(7(4^n) - 1)/3, {n, 0, 24}] (* Alonso del Arte, Feb 11 2012 *)
CoefficientList[Series[(2-x)/(1-5*x+4*x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-4},{2,9},30] (* Vincenzo Librandi, Mar 20 2012 *)

vector(20,n,(7*4^(n-1)-1)/3) \\ Derek Orr, Apr 12 2015

G.f.: (2-x)/(1-5*x+4*x^2). - Bruno Berselli, Feb 12 2012
a(n) = A083597(n)+1. - Bruno Berselli, Feb 12 2012
a(n) = 4*a(n-1)+1 for n>0, a(0)=2. - Bruno Berselli, Oct 22 2015
a(n) = 7*A002450(n) + 2. - Yosu Yurramendi, Jan 24 2017
A006666(a(n)) = 2*n+11 for n > 0. - Juan Miguel Barga Pérez, Jun 18 2020
a(n) = 5*a(n-1) - 4*a(n-2) for n >= 2. - Wesley Ivan Hurt, Jun 30 2020
a(n) = A178415(3, n) = A347834(4, n-1), arrays, for n >= 1.- Wolfdieter Lang, Nov 29 2021

A211016 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 4, 8, 12, 4, 40, 52, 12, 4, 168, 212, 52, 12, 4, 680, 852, 212, 52, 12, 4, 2728, 3412, 852, 212, 52, 12, 4, 10920, 13652, 3412, 852, 212, 52, 12, 4, 43688, 54612, 13652, 3412, 852, 212, 52, 12, 4, 174760, 218452, 54612, 13652, 3412, 852, 212, 52, 12, 4

Offset: 1

Omar E. Pol, Sep 18 2012

All internal regions in the toothpick structure are squares and rectangles.

			For n = 5 in the toothpick structure after 2^5 stages we have that:
T(5,1) = 168 is the number of squares of size 1 X 1.
T(5,2) = 212 is the number of rectangles of size 1 X 2.
T(5,3) = 52 is the total number of squares of size 2 X 2 and of rectangles of size 1 X 4.
T(5,4) = 12 is the number of rectangles of size 2 X 4.
T(5,5) = 4 is the number of rectangles of size 2 X 8.
Triangle begins:
       0;
       0,      4;
       8,     12,     4;
      40,     52,    12,     4;
     168,    212,    52,    12,    4;
     680,    852,   212,    52,   12,   4;
    2728,   3412,   852,   212,   52,  12,   4;
   10920,  13652,  3412,   852,  212,  52,  12,  4;
   43688,  54612, 13652,  3412,  852, 212,  52, 12,  4;
  174760, 218452, 54612, 13652, 3412, 852, 212, 52, 12, 4;

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Row sums give 0 together with A145655.

Cf. A020988, A072197, A080675, A139250, A160124, A211008, A211017, A211018, A211019.

T(n,k) = A211008(2^n,k) = 4*A211019(n,k).

T(n,1) = 4*A020988(n-2), n>=2.

A211019 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 4, n>=1, k>=1, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 1, 2, 3, 1, 10, 13, 3, 1, 42, 53, 13, 3, 1, 170, 213, 53, 13, 3, 1, 682, 853, 213, 53, 13, 3, 1, 2730, 3413, 853, 213, 53, 13, 3, 1, 10922, 13653, 3413, 853, 213, 53, 13, 3, 1, 43690, 54613, 13653, 3413, 853, 213, 53, 13, 3, 1, 174762, 218453

Offset: 1

Omar E. Pol, Sep 24 2012

All internal regions in the toothpick structure are squares and rectangles.

			Triangle begins:
0;
0,         1;
2,         3,     1;
10,       13,     3,    1;
42,       53,    13,    3,   1;
170,     213,    53,   13,   3,   1;
682,     853,   213,   53,  13,   3,  1;
2730,   3413,   853,  213,  53,  13,  3,  1;
10922, 13653,  3413,  853, 213,  53, 13,  3, 1;
43690, 54613, 13653, 3413, 853, 213, 53, 13, 3, 1;

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Row sums give 0 together with A014825.

Cf. A002450, A020988, A072197, A086462, A139250, A160124, A211016, A211017, A211018.

T(n,k) = A211016(n,k)/4.

T(n,1) = A020988(n-2), n>=2.

A356327 Replace 2^k in binary expansion of n with A039834(1+k).

Original entry on oeis.org

0, 1, -1, 0, 2, 3, 1, 2, -3, -2, -4, -3, -1, 0, -2, -1, 5, 6, 4, 5, 7, 8, 6, 7, 2, 3, 1, 2, 4, 5, 3, 4, -8, -7, -9, -8, -6, -5, -7, -6, -11, -10, -12, -11, -9, -8, -10, -9, -3, -2, -4, -3, -1, 0, -2, -1, -6, -5, -7, -6, -4, -3, -5, -4, 13, 14, 12, 13, 15, 16

Offset: 0

Rémy Sigrist, Aug 03 2022

This sequence has similarities with A022290, and is related to negaFibonacci representations.

			For n = 13:
- 13 = 2^3 + 2^2 + 2^0,
- so a(13) = A039834(4) + A039834(3) + A039834(1) = -3 + 2 + 1 = 0.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Wikipedia, NegaFibonacci coding
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000201, A001950, A003714, A004957, A020989, A022290, A026351, A039834, A060144, A072197, A189663, A215024, A215025, A309076, A356325, A356326.

Table[Reverse[#].Fibonacci[-Range[Length[#]]] &@ IntegerDigits[n, 2], {n, 0, 69}] (* Rémy Sigrist, Aug 05 2022 *)

a(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=fibonacci(-1-k)); return (v) }

from sympy import fibonacci
def A356327(n): return sum(fibonacci(-a)*int(b) for a, b in enumerate(bin(n)[:1:-1],start=1)) # Chai Wah Wu, Aug 31 2022

a(n) = Sum_{k>=0} A030308(n,k)*A039834(1+k).
a(A215024(n)) = n.
a(A215025(n)) = -n.
a(A003714(n)) = A309076(n).
Empirically:
- a(n) = 0 iff n = 0 or n belongs to A072197,
- a(n) = 1 iff n belongs to A020989,
- a(2*A215024(n)) = -A000201(n) for n > 0,
- a(3*A215024(n)) = -A060143(n),
- a(floor(A215024(n)/2)) = -A060143(n),
- a(4*A215024(n)) = A001950(n) for n > 0,
- a(floor(A215024(n)/4)) = A189663(n) for n > 0,
- a(2*A215025(n)) = A026351(n),
- a(3*A215025(n)) = A019446(n) for n > 0,
- a(floor(A215025(n)/2)) = A019446(n) for n > 0,
- a(4*A215025(n)) = -A004957(n),
- a(floor(A215025(n)/4)) = -A060144(n+1) for n >= 0.

A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

Original entry on oeis.org

1, 3, 9, 7, 17, 11, 25, 15, 33, 19, 41, 23, 49, 27, 57, 31, 65, 35, 73, 39, 81, 43, 89, 47, 97, 51, 105, 55, 113, 59, 121, 63, 129, 67, 137, 71, 145, 75, 153, 79, 161, 83, 169, 87, 177, 91, 185, 95, 193, 99, 201, 103, 209, 107, 217, 111, 225, 115, 233, 119, 241, 123, 249

Offset: 1

T. D. Noe, May 28 2010

The odd non-multiples of 3 are 1, 5, 7, 11,... (A007310). The odd multiples of 3 have no odd numbers their Collatz pre-image. The next odd number in the Collatz iteration of a(2n) is 6n-1. The next odd number in the Collatz iteration of a(2n+1) is 6n+1. For each non-multiple of 3, there are an infinite number of odd numbers in its Collatz pre-image. For example:
Odd pre-images of 1: 1, 5, 21, 85, 341,... (A002450)
Odd pre-images of 5: 3, 13, 53, 213, 853,... (A072197)
Odd pre-images of 7: 9, 37, 149, 597, 2389,...
Odd pre-images of 11: 7, 29, 117, 469, 1877,...(A072261)
In each case, the pre-image sequence is t(k+1) = 4*t(k) + 1 with t(0)=a(n). The array of pre-images is in A178415.
a(n) = A047529(P(n)), with the permutation P(n) = A006368(n-1) + 1, for n >= 1. This shows that this sequence gives the numbers {1, 3, 7} (mod 8) uniquely. - Wolfdieter Lang, Sep 21 2021

Matthew House, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A002450, A006368, A007310, A022998, A047529, A072197, A072261, A114752.

Riffle[1+8*Range[0,50], 3+4*Range[0,50]]

a(n) = (n - 1)*(3 - (-1)^n) + 1. [Bogart B. Strauss, Sep 20 2013, adapted to the offset by Matthew House, Feb 14 2017]
From Matthew House, Feb 14 2017: (Start)
G.f.: x*(1 + 3*x + 7*x^2 + x^3)/((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4). (End)
From Philippe Deléham, Nov 06 2023: (Start)
a(2*n) = 4*n-1, a(2*n+1) = 8*n+1.
a(n) = 2*A022998(n-1)+1.
a(n) = 2*A114752(n)-1. (End)

A211018 Triangle read by rows: T(n,k) = total area of all squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 8, n>=1, k>=1, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 5, 13, 6, 4, 21, 53, 26, 12, 8, 85, 213, 106, 52, 24, 16, 341, 853, 426, 212, 104, 48, 32, 1365, 3413, 1706, 852, 424, 208, 96, 64, 5461, 13653, 6826, 3412, 1704, 848, 416, 192, 128, 21845, 54613, 27306, 13652, 6824, 3408, 1696, 832, 384, 256

Offset: 1

Omar E. Pol, Sep 24 2012

All internal regions in the toothpick structure are squares and rectangles. The area of every internal region is a power of 2.

			0;
0,        1;
1,        3,    2;
5,       13,    6,    4;
21,      53,   26,   12,    8;
85,     213,  106,   52,   24,  16;
341,    853,  426,  212,  104,  48,  32;
1365,  3413, 1706,  852,  424, 208,  96,  64;
5461, 13653, 6826, 3412, 1704, 848, 416, 192, 128;

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Rows sums give A006516. Right border gives A131577.

Cf. A002450, A072197, A139250, A160124, A211012, A211016, A211017, A211019

T(n,k) = A211017(n,k)/8.

T(n,1) = A002450(n-2), n>=2.

A324036 Modified reduced Collatz map fs acting on positive odd integers.

Original entry on oeis.org

1, 5, 1, 11, 7, 17, 3, 23, 13, 29, 5, 35, 19, 41, 7, 47, 25, 53, 9, 59, 31, 65, 11, 71, 37, 77, 13, 83, 43, 89, 15, 95, 49, 101, 17, 107, 55, 113, 19, 119, 61, 125, 21, 131, 67, 137, 23, 143, 73, 149, 25, 155, 79, 161, 27, 167, 85, 173, 29, 179, 91, 185, 31, 191, 97, 197, 33, 203, 103, 209

Offset: 0

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 08 2019

This is a modification of the reduced Collatz map given in A075677.
The Collatz conjecture is that iteration of the map fs leads to 1 for all positive odd integers.
In the Vaillant-Delarue (V-D) reference the present map fs: Odd -> Odd, 2*n+1 -> a(n) = fs(2*n+1), for n >= 0, is called f_{s}. The differences from b(n) = A075677(n+1) = fCr(2*n+1) (called f_{cr} in V-D) occur for the positions n = 2 + 4*k, for k >= 1: b(2 + 4*k) = b(k) = A075677(k+1) but a(2 + 4*k) = 1 + 2*k, which differs.
The advantage of the map fs (or a) over fCr (or b) is an explicit formula over a recurrence.
Additional steps are introduced in the iteration of fs versus fCr. This leads to an incomplete binary tree, called CfsTree, given in A324038. No such tree is available for fCr.
Such additional steps in fs can only occur after odd numbers congruent to 5 modulo 8: fs(5 + 8*k) = a(2 + 4*k) = 1 + 2*k  and fs(1 + 2*k) = a(k). On the other hand, fCr(5 + 8*k) = b(2 + 4*k) = b(k).
The appearance of exactly N consecutive steps in fs versus fCr, for N >= 2, can be shown recursively to start with the odd numbers O(N;k) = 1 + 4*O(N-1;k), for N >= 3, with input O(2;k) = 53 + (4^3)*k. These are the numbers  O(N;k) = A072197(N) + A000302(N+1)*k, for N >= 2. Therefore only one additional step follows directly after an odd number 5 (mod 8) if it is not of the O(N;k) type for N >= 2.
The minimal number of iterations of function fs acting on 2*n + 1  (or a acting on n), for n >= 0, to reach 1 is given in A324037 (if for very large n the number 1 should not be reached A324037(n) is set to -1).

			Iteration of fs on 11: 11, 17, 13, 3, 5, 1, whereas for fCr: 11, 17, 13 , 5, 1. The additional step (N = 1) occurs for 13 == 5 (mod 8), and 13 does not belong to the O(N;k) sets for N >= 2.
The first additional N = 2 steps occur for 53 = a(26): 53, 13, 3, 5, 1, versus iteration of fCr: 53, 5, 1. Such N = 2 steps occur precisely after 53 + 64*k as 13 + 16*k and 3 + 4*k.
The first additional N = 3 steps occur for 213 = a(106): 213, 53, 13, 3, 5, 1 versus 213, 5, 1 for fCr.
The first additional N = 4 steps occur for 853 = a(426): 853, 213, 53, 13, 3, 5, 1 versus 853, 5, 1 for fCr.

Wolfdieter Lang, Collatz Trees from Vaillant-Delarue Maps
Nicolas Vaillant and Philippe Delarue, The hidden face of the 3x+1 problem. Part I: Intrinsic algorithm, April 26 2019.

Cf. A000302, A072197, A075677, A324037, A324038.

a(n) = my(m=Mod(n,4)); if (m==0, (2 + 3*n)/2, if (m==2, n/2, 2 + 3*n)); \\ Michel Marcus, Aug 10 2023

a(n) = fs(1 + 2*n) = (2 + 3*n)/2 if n == 0 (mod 4), a(n) = 2 + 3*n, for n == 1 or 3 (mod 4), and a(n) = n/2 if n == 2 (mod 4). This corresponds to fs(1 + 8*k) = 1 + 6*k, fs(3 + 8*k) = 5 + 12*k, fs(5 + 8*k) = 1 + 2*k, and fs(7 + 8*k) = 11 + 12*k, for k >= 0.
Conjectures from Colin Barker, Oct 14 2019: (Start)
G.f.: (1 + 5*x + x^2 + 11*x^3 + 5*x^4 + 7*x^5 + x^6 + x^7) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = 2*a(n-4) - a(n-8) for n>7.
(End)

More terms from Michel Marcus, Aug 10 2023

A086462 Expansion of (1+x)(1+4x)/((1-x)(1-4x)).

Original entry on oeis.org

1, 10, 50, 210, 850, 3410, 13650, 54610, 218450, 873810, 3495250, 13981010, 55924050, 223696210, 894784850, 3579139410, 14316557650, 57266230610, 229064922450, 916259689810, 3665038759250, 14660155037010, 58640620148050

Offset: 0

Paul Barry, Jul 21 2003

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

Cf. A001045, A002450, A072197.

CoefficientList[Series[(1+x)(1+4x)/((1-x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[{5,-4},{1,10,50},30] (* Harvey P. Dale, May 27 2023 *)

a(n) = 0^n + 10*(4^n-1)/3; \\ Ruud H.G. van Tol, Oct 11 2023

a(0) = 1, a(1) = 10, a(2) = 50, a(n) = 5*a(n-1) - 4*a(n-2).

a(n) = 10*(4^n-1)/3 + 0^n = 10*A002450(n) + 0^n = 10*A001045(2*n) + 0^n.

A266753 Decimal representation of the n-th iteration of the "Rule 163" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 18, 74, 298, 1194, 4778, 19114, 76458, 305834, 1223338, 4893354, 19573418, 78293674, 313174698, 1252698794, 5010795178, 20043180714, 80172722858, 320690891434, 1282763565738, 5131054262954, 20524217051818, 82096868207274, 328387472829098

Offset: 0

Robert Price, Jan 17 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. A263919, A266752.

rule=163; 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]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 20 2016 and Apr 20 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>2.
G.f.: (1-x+2*x^2) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = (7*4^n - 4)/6 for n>1. - Colin Barker, Nov 25 2016 and Apr 20 2019
a(n) = 4*a(n-1) + 2, n>1, conjectured. - Yosu Yurramendi, Jan 22 2017
a(n) = 2*A020988(n) - A020988(n-1) = A020988(n) + 2^(2n-1) for n > 0, conjectured. - Yosu Yurramendi, Jan 24 2017 [n range correction - Karl V. Keller, Jr., May 07 2022]
a(n) = A072197(n-1) + A002450(n), n > 0, conjectured. - Yosu Yurramendi, Mar 03 2017

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - N. J. A. Sloane, Jun 13 2022

cp's OEIS Frontend

A080675 a(n) = (5*4^n - 8)/6.

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A206374 a(n) = (7*4^n - 1)/3.

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A211016 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

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A211019 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 4, n>=1, k>=1, assuming the toothpicks have length 2.

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A356327 Replace 2^k in binary expansion of n with A039834(1+k).

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A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

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A211018 Triangle read by rows: T(n,k) = total area of all squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 8, n>=1, k>=1, assuming the toothpicks have length 2.

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A324036 Modified reduced Collatz map fs acting on positive odd integers.

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