A092440 -id:A092440 - cp's OEIS frontend

A276918 a(2n) = A060867(n+1), a(2n+1) = A092440(n+1).

1, 5, 9, 25, 49, 113, 225, 481, 961, 1985, 3969, 8065, 16129, 32513, 65025, 130561, 261121, 523265, 1046529, 2095105, 4190209, 8384513, 16769025, 33546241, 67092481, 134201345, 268402689, 536838145, 1073676289, 2147418113, 4294836225, 8589803521, 17179607041

Offset: 0

Daniel Poveda Parrilla, Jan 26 2017

nonn

In binary there is a pattern in how the zeros and ones appear:
a(0)  =         01
a(1)  =        101
a(2)  =        1001
a(3)  =       11001
a(4)  =       110001
a(5)  =      1110001
a(6)  =      11100001
a(7)  =     111100001
a(8)  =     1111000001
a(9)  =    11111000001
a(10) =    111110000001
a(11) =   1111110000001
a(12) =   11111100000001
a(13) =  111111100000001
a(14) =  1111111000000001
a(15) = 11111111000000001
Graphically, each term can be obtained by successively and alternately forming squares and centered squares as shown in the illustration.

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..1000
Daniel Poveda Parrilla, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Cf. A000225, A060867, A092440, A224195.

Table[1+2^(n+2)-2^(1+n/2)+(-1)^(n+1) 2^(1+n/2)-2^((n+1)/2)+(-1)^(n+2) 2^((n+1)/2), {n,0,28}] (*or*)
CoefficientList[Series[(-1 - 2 x + 6 x^2 - 4 x^3)/(-1 + 3 x - 6 x^3 + 4 x^4), {x,0,28}], x] (*or*)
LinearRecurrence[{3, 0, -6, 4}, {1, 5, 9, 25}, 29]

Vec((-1-2*x+6*x^2-4*x^3) / (-1+3*x-6*x^3+4*x^4) + O(x^29))

a(n) = 1 + 2^(n+2) - 2^(1 + n/2) + (-1)^(n+1)*2^(1 + n/2) - 2^((n+1)/2) + (-1)^(n+2)*2^((n+1)/2).
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4) for n>3.
G.f.: (-1-2*x+6*x^2-4*x^3)/(-1+3*x-6*x^3+4*x^4).

A169699 Total number of ON cells at stage n of two-dimensional 5-neighbor outer totalistic cellular automaton defined by "Rule 510".

Original entry on oeis.org

1, 5, 12, 25, 28, 56, 56, 113, 60, 120, 120, 240, 120, 240, 240, 481, 124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985, 252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016

Offset: 0

N. J. A. Sloane, Apr 17 2010

We work on the square grid. Each cell has 4 neighbors, N, S, E, W. If none of your 4 neighbors are ON, your state does not change. If all 4 of your neighbors are ON, your state flips. In all other cases you turn ON. We start with one ON cell.

As observed by Packard and Wolfram (see Fig. 2), a slice along the E-W line shows the successive states of the 1-D CA Rule 126 (see A071035, A071051).

			When arranged into blocks of sizes 1,1,2,4,8,16,...:
1,
5,
12, 25,
28, 56, 56, 113,
60, 120, 120, 240, 120, 240, 240, 481,
124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985,
252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032,
..., the initial terms in the rows (after the initial rows) have the form 2^m-4 and the final terms are given by A092440. The row beginning with 2^m-4 is divisible by 2^(m-2)-1 (see formula).

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

Robert Price, Table of n, a(n) for n = 0..128
N. H. Packard and S. Wolfram, Two-Dimensional Cellular Automata, Journal of Statistical Physics, 38 (1985), 901-946.
N. J. A. Sloane, Illustration of first 28 generations
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
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

Cf. A000120, A071035, A071051, A008574, A092440, A169700.

See A253089 for 9-celled neighborhood version.

A000120 := proc(n) add(i,i=convert(n,base,2)) end:
ht:=n->floor(log[2](n));
f:=proc(n) local a,t1;
if n=0 then 1 else
a:=(2^(ht(n)+1)-1)*2^(1+A000120(n));
if 2^log[2](n)=n then a:=a+1; fi; a; fi; end;
[seq(f(n),n=0..65)]; # A169699

Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 510, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]]
ArrayPlot /@ CellularAutomaton[{510, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 28]

For n>0, it is easy to show that if 2^k <= n < 2^(k+1) then a(n) =

(2^(k+1)-1)*2^(1+wt(n)), where wt is the binary weight A000120, except that if n is a power of 2 we must add 1 to the result.

Entry revised with more precise definition, formula and additional information, N. J. A. Sloane, Aug 24 2014

A220978 a(n) = 3^(2n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6n+3) + 1.

Original entry on oeis.org

1, 19, 217, 2107, 19441, 176419, 1592137, 14342347, 129120481, 1162202419, 10460176057, 94142647387, 847287015121, 7625592702019, 68630363015977, 617673353237227, 5559060437415361, 50031544711579219, 450283904728735897, 4052555149532191867

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A198410(n+2): 3^(6*n+3) + 1 = (3^(2*n+1) + 1) * a(n) * A198410(n+2).

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (13, -39, 27).

Cf. A092440, A085601, A198410, A220979-A220990.

Table[3^(2n+1) - 3^(n+1) + 1, {n, 0, 30}]
LinearRecurrence[{13,-39,27},{1,19,217},30] (* Harvey P. Dale, Mar 17 2013 *)

Vec((1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)) + O(x^30)) \\ Michel Marcus, Feb 12 2015

a(n) = 13*a(n-1) - 39*a(n-2) + 27*a(n-3).

G.f.: (1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)).

A220979 a(n) = 5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1: the left Aurifeuillian factor of 5^(10n+5) - 1.

Original entry on oeis.org

11, 12851, 9384251, 6054921251, 3808599606251, 2383422998031251, 1490020755615156251, 931310653778075781251, 582075119020843503906251, 363797694444713592519531251, 227373652160169124603222656251, 142108544241637027263641113281251

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220980.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (781,-101530,2538250,-12203125,9765625).

Cf. A092440, A085601, A220978, A198410, A220980-A220990.

Table[5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1, {n, 0, 30}]

a(n)=5^(4*n+2)-5^(3*n+2)+3*5^(2*n+1)-5^(n+1)+1 \\ Charles R Greathouse IV, Sep 28 2015

Aurifeuillian factorization: 5^(10n+5) - 1 = (5^(2n+1) - 1) * a(n) * A220980(n).

G.f.: -(4296875*x^4+2662500*x^3+464450*x^2+4260*x+11) / ((x-1)*(5*x-1)*(25*x-1)*(125*x-1)*(625*x-1)). - Colin Barker, Jan 03 2013

A220990 a(n) = 12^(2n+1) + 6 * 12^n + 1: the right Aurifeuillian factor of 12^(6n+3) + 1.

Original entry on oeis.org

19, 1801, 249697, 35842177, 5159904769, 743009863681, 106993223294977, 15407021789577217, 2218611109320327169, 319479999401581608961, 46005119909741205651457, 6624737266953695061344257, 953962166440743626203987969

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220989.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (157,-1884,1728).

Cf. A092440, A085601, A220978, A198410, A220979-A220989.

Table[12^(2n+1) + 6 * 12^n + 1, {n, 0, 10}]
LinearRecurrence[{157,-1884,1728},{19,1801,249697},20] (* Harvey P. Dale, Mar 26 2022 *)

a(n)=12^(2*n+1)+6*12^n+1 \\ Charles R Greathouse IV, Sep 28 2015

Aurifeuillian factorization: 12^(6n+3) + 1 = (12^(2n+1) + 1) * A220989(n) * a(n).

G.f.: -(2736*x^2-1182*x+19) / ((x-1)*(12*x-1)*(144*x-1)). - Colin Barker, Jan 03 2013

A092443 Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.

Original entry on oeis.org

3, 12, 50, 210, 882, 3696, 15444, 64350, 267410, 1108536, 4585308, 18929092, 78004500, 320932800, 1318498920, 5409723510, 22169259090, 90751353000, 371125269900, 1516311817020, 6189965556060, 25249187564640, 102917884095000, 419218847880300, 1706543186909652

Offset: 1

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

The sequence 1, 3, 12, 50, ... is ((n+2)/2)*C(2n,n) with g.f. F(1/2,3;2;4x). - Paul Barry, Sep 18 2008

			a(3) = 5!/2!2! + 6!/3!3! = 50.

James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).

Michael De Vlieger, Table of n, a(n) for n = 1..1659
Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, arXiv:1910.14048 [cond-mat.str-el], 2019.
James Propp, Publications and Preprints.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics, Cambridge University Press, Cambridge, 1999, pp. 255-291.
Eric Rowland and Jason Wu, The entries of the Sinkhorn limit of an m X n matrix, arXiv:2409.02789 [math.NT], 2024. See p. 11.

Cf. A000984, A001622, A001700, A002457.

Cf. A092437, A092438, A092439, A092440, A092441, A092442.

Array[Binomial[2 # + 1, # + 1] &[# - 1]*(# + 2) &, 22] (* Michael De Vlieger, Dec 17 2017 *)

combinat::catalan(n) *binomial(n+2,2) $ n = 1..22 // Zerinvary Lajos, Feb 15 2007

a(n) = (n+2)*binomial(2*n-1, n); \\ Altug Alkan, Dec 17 2017

a(n) = (2*n-1)!/((n-1)!)^2+(2*n)!/(n!)^2 = A002457(n-1) + A000984(n).
a(n) = (n+2)*A001700(n-1). - Vladeta Jovovic, Jul 12 2004
n*a(n) + (-7*n+4)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
From Amiram Eldar, Jan 27 2024: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi*(11*sqrt(3)-3*Pi)/9 - 13.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)*(13*sqrt(5)-30*log(phi))/5 - 11, where phi is the golden ratio (A001622). (End)
From Peter Bala, Aug 02 2024: (Start)
a(n) = 1/(n + 1)^2 * Sum_{k = 1..n+1} (k^3)*binomial(n+1, k)^2 = hypergeom([2, -n, -n], [1, 1], 1).
a(n) = 2*(n + 2)*(2*n - 1)/(n*(n + 1)) * a(n-1) with a(1) = 3. (End)

A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

113, 34925927, 4651514210561, 556919483179733591, 65684998500756890925713, 7730533744900130305342957127, 909535949164303794596648514307361, 107006774488854204226839526889653524791, 12589253114717671385404089651370543317211313

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220984.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (137257, -2354918349, 5670354183893, -1944931485075299, 95029449572634843, -651636050170246351, 558545864083284007).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1, {n, 0, 20}]

a(n) = 7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * a(n) * A220984(n).
G.f.: -(184010736563880737*x^6 +268740854387875086*x^5 +14564007567924591*x^4 +73553506117028*x^3 +123792021759*x^2 +19415886*x +113) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A220984 The right Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

911, 46489241, 4845303761663, 560176314330212777, 65739735996793498937711, 7731453717973685046293120441, 909551411151743369070229385367263, 107007034358477098527617255914118283977, 12589257482346423369016062830670344414194511

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220983.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (137257, -2354918349, 5670354183893, -1944931485075299, 95029449572634843, -651636050170246351, 558545864083284007).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1, {n, 0, 20}]

a(n) = 7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * A220983(n) * a(n).
G.f.: -(1483484787696419039*x^6 -1087259214306211086*x^5 +71725962948861585*x^4 -562870083909028*x^3 +609660625665*x^2 -78551886*x +911) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 15, 17, 25, 29, 31, 33, 49, 57, 61, 63, 65, 97, 113, 121, 125, 127, 129, 193, 225, 241, 249, 253, 255, 257, 385, 449, 481, 497, 505, 509, 511, 513, 769, 897, 961, 993, 1009, 1017, 1021, 1023, 1025, 1537, 1793, 1921, 1985, 2017, 2033, 2041, 2045, 2047

Offset: 1

Brad Clardy, Apr 01 2013

The table is constructed so that row labels are 2^n - 1, and column labels are 2^n. The body of the table is the row*col + 1. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner.
All of these numbers have the following property:
   let m be a member of A(n),
   if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then
   the differences between successive members of B(n) is a repeating series
   of 1's with the last difference in the pattern m. The number of ones in
   the pattern is 2^j - 1, where j is the column index.
As an example consider A(4) which is 9,
   the sequence B(n) where i XOR 8 = i - 8 starts as:
   8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)
   with successive differences of:
   1, 1, 1, 1, 1, 1, 1, 9.
The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).
The formula for diagonals above the main diagonal
  2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal
  2^(2*n) - 2^(n  + (b/2)) + 1    n>=(b/2)+1 b=even number above diagonal
The formulas for diagonals below the main diagonal
  2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal
  2^(2*n) - 2^(n - (b/2)) + 1       n>=(b/2)+1 b=even number below diagonal
Primes of this sequence are in A152449.

			Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
  +1  |    2    4     8    16    32     64    128    256     512    1024 ...
  ----|-----------------------------------------------------------------
  1   |    3    5     9    17    33     65    129    257     513    1025
  3   |    7   13    25    49    97    193    385    769    1537    3073
  7   |   15   29    57   113   225    449    897   1793    3585    7169
  15  |   31   61   121   241   481    961   1921   3841    7681   15361
  31  |   63  125   249   497   993   1985   3969   7937   15873   31745
  63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
  127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
  255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
  511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
  1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
  ...

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).
Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).
Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).
Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).

//program generates values in a table form
for i:=1 to 10 do
    m:=2^i - 1;
    m,[ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(i-j) + 1;
       if IsPrime(k) then k,"*";
          else k;
       end if;;
    end for;
end for;

Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* Paolo Xausa, Apr 02 2024 *)

a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by Jason Yuen, Feb 22 2025]

a(n) = A081118(n)+2; a(n)=(2^i-1)*2^j+1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

A229767 Largest prime factor of 2^(2*n+1)-2^(n+1)+1.

Original entry on oeis.org

5, 5, 113, 37, 397, 1613, 61, 953, 457, 14449, 30269, 8101, 246241, 107367629, 384773, 312709, 47392381, 184481113, 1249, 12112549, 1759217765581, 54001, 140737471578113, 4981857697937, 26317, 1801439824104653, 415878438361, 525313, 174877, 368140581013

Offset: 1

Colin Barker, Sep 29 2013

nonn

2^(2*n+1)-2^(n+1)+1 is a factor of 4^(2*n+1)+1.

			For n=5, 2^(2*n+1)-2^(n+1)+1 = 1985 = 5*397, so a(5)=397.

Daniel Suteu, Table of n, a(n) for n = 1..547

Cf. A092440, A207262, A229747, A229768.

a(n) = {f=factor(2^(2*n+1)-2^(n+1)+1); f[matsize(f)[1],1]}

cp's OEIS Frontend

A276918 a(2n) = A060867(n+1), a(2n+1) = A092440(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A169699 Total number of ON cells at stage n of two-dimensional 5-neighbor outer totalistic cellular automaton defined by "Rule 510".

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A220978 a(n) = 3^(2*n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6*n+3) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A220979 a(n) = 5^(4n+2) - 5^(3n+2) + 3 * 5^(2n+1) - 5^(n+1) + 1: the left Aurifeuillian factor of 5^(10n+5) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A220990 a(n) = 12^(2n+1) + 6 * 12^n + 1: the right Aurifeuillian factor of 12^(6n+3) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A092443 Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

MuPAD

PARI

Formula

A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

A220978 a(n) = 3^(2n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6n+3) + 1.