A083420 -id:A083420 - cp's OEIS frontend

A065916 Denominator of sigma(8n^2)/sigma(4n^2).

7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 2047, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 8191, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 2047, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 32767, 7, 31, 7, 127, 7, 31, 7, 511, 7

Offset: 1

Labos Elemer, Nov 28 2001

The sequence is not periodic. The denominators are always of the form -1+2^s.

			sigma(72)/sigma(36) = 15/7, so a(3) = 7.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Index entries for sequences related to binary expansion of n.

Cf. A000203, A007814, A028982, A065915 (numerators), A083420, A220466.

nmax:=73: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2*4^(p+1) - 1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 12 2013

a[n_] := 2^(2*IntegerExponent[n, 2] + 3) - 1; Array[a, 100] (* Amiram Eldar, Jun 21 2024 *)

a(n) = denominator(sigma(8*n^2)/sigma(4*n^2)) \\ Harry J. Smith, Nov 04 2009

a(n)=2^(2*valuation(n,2)+3)-1 \\ Charles R Greathouse IV, Nov 18 2015

From Johannes W. Meijer, Feb 12 2013: (Start)
a((2*n-1)*2^p) = 2*4^(p+1) - 1 for p >= 0 and n >= 1. Observe that a(2^p) = A083420(p+1).
a(2^(p+3)*n + 2^(p+2) - 1) = a(2^(p+2)*n + 2^(p+1) - 1) for p >= 0. (End)
a(n) = 2^s-1, with s = 2*A007814(n) + 3. Recurrence: a(2n) = 4a(n)+3, a(2n+1) = 7. - Ralf Stephan, Aug 22 2013

A085903 Expansion of (1 + 2x^2)/((1 + x)(1 - 2x)(1 - 2*x^2)).

Original entry on oeis.org

1, 1, 7, 9, 31, 49, 127, 225, 511, 961, 2047, 3969, 8191, 16129, 32767, 65025, 131071, 261121, 524287, 1046529, 2097151, 4190209, 8388607, 16769025, 33554431, 67092481, 134217727, 268402689, 536870911, 1073676289, 2147483647

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003

Resultant of the polynomial x^n - 1 and the Chebyshev polynomial of the first kind T_2(x).

This sequence is the case P1 = 1, P2 = 0, Q = -2 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).

Cf. A083420, A028400, A062510, A088037, A107663.

Cf. A060867. A100047.

[Round((Sqrt(2)^n - 1)*(Sqrt(2)^n - (-1)^n)): n in [1..40]]; // Vincenzo Librandi, Apr 28 2014

seq(simplify((sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n)), n = 1..30); # Peter Bala, Apr 27 2014

CoefficientList[ Series[(1 + 2x^2)/(1 - x - 4x^2 + 2x^3 + 4x^4), {x, 0, 30}], x] (* Robert G. Wilson v, May 04 2013 *)
LinearRecurrence[{1,4,-2,-4},{1,1,7,9},40] (* Harvey P. Dale, Jul 25 2016 *)

a(n) = polresultant(x^n - 1, 2*x^2 - 1) \\ David Wasserman, Feb 10 2005

def A085903(n): return (1<>1))-1)**2 # Chai Wah Wu, Jun 19 2024

a(2*n) = 2*4^n - 1, a(2*n + 1) = (2^n - 1)^2; interlaces A083420 with A060867 (squares of Mersenne numbers A000225). - Creighton Dement, May 19 2005
A107663(2*n) = a(2*n) = A083420(n). - Creighton Dement, May 19 2005
From Peter Bala, Apr 27 2014: (Start)
a(n) = (sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n).
a(n) = Product_{k = 1..n} ( 2 - exp(4*k*Pi*i/n) ). (End)
E.g.f.: exp(-x) + exp(2*x) - 2*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Jun 16 2016

More terms from David Wasserman, Feb 10 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

Original entry on oeis.org

1, 91, 3751, 138811, 5028751, 181308931, 6529545751, 235085301451, 8463265086751, 304679288612371, 10968470088963751, 394865064451017691, 14215143591303768751, 511745180725868773411, 18422826609078989373751, 663221758853362301815531, 23875983327059668074930751

Offset: 1

Benoit Cloitre, Jun 18 2004

Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (50,-553,1800,-1296).

Cf. A083420, A096045, A096046, A096047, A096048, A096053.

a[n_] := 6^(2*n-1) * BernoulliB[2*n, 1/6] / BernoulliB[2*n]; Array[a, 15] (* Amiram Eldar, May 07 2025 *)

a(n)=(1/12)*36^n-(1/6)*9^n-(1/4)*4^n+1/2;

a(n) = (1/12)*(36^n - 2*9^n - 3*4^n+6).
From Colin Barker, May 30 2020: (Start)
G.f.: x*(1 - 6*x)*(1 + 47*x + 36*x^2) / ((1 - x)*(1 - 4*x)*(1 - 9*x)*(1 - 36*x)).
a(n) = 50*a(n-1) - 553*a(n-2) + 1800*a(n-3) - 1296*a(n-4) for n>4. (End)

A137410 a(n) = (5^n - 3)/2.

Original entry on oeis.org

-1, 1, 11, 61, 311, 1561, 7811, 39061, 195311, 976561, 4882811, 24414061, 122070311, 610351561, 3051757811, 15258789061, 76293945311, 381469726561, 1907348632811, 9536743164061, 47683715820311, 238418579101561, 1192092895507811, 5960464477539061, 29802322387695311, 149011611938476561

Offset: 0

Ctibor O. Zizka, Apr 15 2008

Sequence is a(n) = a(n;5,3,1) where a(n;A,B,r) = (A^n - B^r)/(A - B) for arbitrary integers A, B, r with A != B.
Primes of this form are sometimes of interest, examples:
A=2, B=1, r=1 gives A000225 and subsequence of primes: A001348,
A=3, B=1, r=1 gives A003462 and subsequence of primes: A028491,
A=3, B=2, r=1 gives A058481 and subsequence of primes: A014224,
A=4, B=1, r=1 gives A002450,
A=4, B=2, r=1 gives A083420,
A=4, B=2, r=2 gives A002446,
A=5, B=1, r=1 gives A003463 and subsequence of primes: A004061,
A=5, B=2, r=1 gives A037577.
Sum of n-th row of triangle of powers of 5: 1; 5 1 5; 25 5 1 5 25; 125 25 5 1 5 25 125; ... (cf. Examples). - Philippe Deléham, Feb 24 2014
Integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n (see Campbell and Zujev). - Michel Marcus, Mar 02 2016

			From _Philippe Deléham_, Feb 24 2014: (Start)
a(1) = 1;
a(2) = 5 + 1 + 5 = 11;
a(3) = 25 + 5 + 1 + 5 + 25 = 61;
a(4) = 125 + 25 + 5 + 1 + 5 + 25 + 125 = 311;
etc. (End)

M. F. Hasler, Table of n, a(n) for n = 0..100
Geoffrey B Campbell and Aleksander Zujev, On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, arXiv:1603.00080 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000040, A000225, A001348, A002446, A002450, A003462, A003463, A004061, A014224, A024049, A028491, A037577, A058481, A083420, A125831.

[(5^n-3)/2: n in [0..25]]; // Vincenzo Librandi, Mar 02 2016

LinearRecurrence[{6, -5}, {1, 11}, 25] (* Vincenzo Librandi, Mar 02 2016 *)
(5^Range[30]-3)/2 (* Harvey P. Dale, Feb 24 2017 *)

a(n) = (5^n - 3) / 2; \\ Michel Marcus, Mar 02 2016

a(n) = (5^n - 3)/2.
From Colin Barker, May 01 2012: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: (-1+7*x)/((1-x)*(1-5*x)). (End)
a(n) = 5*a(n-1) + 6, a(1) = 1. - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Dec 11 2023: (Start)
a(n) = A024049(n)/2 - 1 = A125831(n) - 1.
E.g.f.: (1/2)*(exp(5*x) - 3*exp(x)). (End)

More terms from Michel Marcus, Mar 02 2016

Edited and missing term a(0) inserted by M. F. Hasler, Jul 10 2018

A180938 Smallest k such that k*n has an even number of 1's in its base-2 expansion.

Original entry on oeis.org

3, 3, 1, 3, 1, 1, 9, 3, 1, 1, 3, 1, 3, 9, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 9, 1, 1, 33, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 9, 1, 1, 3, 1, 3, 33, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 5, 3, 1, 1, 5, 1, 11, 3, 1, 1, 3, 3, 1, 3, 1, 1, 9, 3, 1

Offset: 1

Jeffrey Shallit, Sep 26 2010

From Robert G. Wilson v, Sep 29 2010: (Start)
k must always be odd.
First occurrence of odd k: 3, 1, 87, 109, 7, 93, 457, 1143, 5501, 7921, 889, 12775, 11753, 635, 111209, 6093, 31, 33823, 7665, ..., .
(End)

			For n = 7, a(n) = 9, since the smallest multiple of 7 with an even number of 1's in its base-2 expansion is 9*7 = 63.

Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A001969, A178757.

Cf. A083420 (where records occur). - Alois P. Heinz, Oct 16 2011

a[n_] := Block[{k = 1}, While[OddQ@ DigitCount[k*n, 2, 1], k++ ]; k]; Array[a, 100] (* Robert G. Wilson v, Sep 29 2010 *)

A180938(n) = my(k=1); while(hammingweight(k*n)%2, k += 2); k; \\ Antti Karttunen, Jul 09 2017

def a(n):
    k=1
    while True:
        if not bin(k*n)[2:].count('1')%2: return k
        k+=1
print([a(n) for n in range(1, 61)]) # Indranil Ghosh, Jul 11 2017

More terms from Robert G. Wilson v, Sep 29 2010

A267890 Decimal representation of the n-th iteration of the "Rule 239" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 6, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Robert Price, Jan 21 2016

Seems to differ from A267888 and A083420 only at a(1). - R. J. Mathar, Jun 24 2025

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. A267871.

rule=239; 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 22 2016 and Apr 17 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>3.
G.f.: (1+x+5*x^2-4*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 2^(2*n+1) - 1 for n>1. - Colin Barker, Nov 25 2016

A285475 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 4, 15, 16, 63, 64, 255, 256, 1023, 1024, 4095, 4096, 16383, 16384, 65535, 65536, 262143, 262144, 1048575, 1048576, 4194303, 4194304, 16777215, 16777216, 67108863, 67108864, 268435455, 268435456, 1073741823, 1073741824, 4294967295, 4294967296

Offset: 0

Robert Price, Apr 19 2017

Initialized with a single black (ON) cell at stage zero.

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

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
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 (0,5,0,-4).

Cf. A285473, A285474, A080924.
Cf. A083420.
Cf. A000302 (even bisection), A024036 (odd bisection).

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 3; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

From Colin Barker, Apr 19 2017: (Start)
G.f.: (1 + 3*x - x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = (-1 - (-2)^n + (-1)^n + 3*2^n)/2.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
a(2*n-1) + a(2*n) = A083420(n). - Paul Curtz, Dec 16 2024

A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 35, 57, 65, 85, 93, 119, 127, 147, 155, 201, 217, 257, 273, 325, 341, 381, 397, 455, 471, 511, 527, 579, 595, 635, 651, 745, 777, 857, 889, 993, 1025, 1105, 1137, 1253, 1285, 1365, 1397, 1501, 1533, 1613, 1645, 1767, 1799, 1879, 1911, 2015

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			5 is a term since the negabinary representation of -5 is 1111 which is palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002113, A005352, A006995, A014190, A039724, A094202, A212529, A331191, A331891.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[2000], PalindromeQ @ negabin[-#] &]

A132207 Triangle read by rows: row n lists first 24^n terms of an array read by rows, in which row k gives 24^n + 3k - 1; 54^n + 3*k - 1, with k>=0 in each array.

Original entry on oeis.org

1, 4, 7, 19, 10, 22, 13, 25, 16, 28, 31, 79, 34, 82, 37, 85, 40, 88, 43, 91, 46, 94, 49, 97, 52, 100, 55, 103, 58, 106, 61, 109, 64, 112, 67, 115, 70, 118, 73, 121, 76, 124, 127, 319, 130, 322, 133, 325, 136, 328, 139, 331, 142, 334, 145, 337, 148, 340, 151, 343, 154

Offset: 0

Paul Curtz, Nov 06 2007

			Contribution from _Omar E. Pol_, Jan 06 2009: (Start)
Triangle begins:
1,4;
7,19,10,22,13,25,16,28;
31,79,34,82,37,85,40,88,43,91,......,76,124;
127,319,130,322,133,325,136,328,139,331,142,334,145,337,......,316,508;
...
Array, in row 0, is
1, 4
...
Array, in row 1, is
7, 19
10, 22
13, 25
16, 28
...
Array, in row 2, begins:
31, 79
34, 82
37, 85
40, 88
And so on.
(End)

Cf. A000302, A016777, A083420. [From Omar E. Pol, Jan 06 2009]

Edited with better definition by Omar E. Pol, Jan 04 2009

A140252 Inverse binomial transform of A140420.

Original entry on oeis.org

0, 1, 1, 7, 7, 31, 31, 127, 127, 511, 511, 2047, 2047, 8191, 8191, 32767, 32767, 131071, 131071, 524287, 524287, 2097151, 2097151, 8388607, 8388607, 33554431, 33554431, 134217727, 134217727, 536870911, 536870911

Offset: 0

Paul Curtz, Jun 23 2008

nonn

Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 23 2017

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

Harvey P. Dale, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
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 (1, 4, -4).

Join[{0},LinearRecurrence[{1,4,-4},{1,1,7},30]] (* Harvey P. Dale, May 28 2012 *)

a(2n+1) = a(2n+2)= A083420(n).
a(n+1)-2a(n) = (-1)^n*A014551(n), n>0.
a(n+1)-2a(n)-1 = 2*(-1)^n*A131577(n).
O.g.f.: x(1+2x^2)/((2x-1)(1+2x)(x-1)). - R. J. Mathar, Aug 02 2008
a(n) = a(n-1)+4*a(n-2)-4*a(n-3), a(0)=0, a(1)=1, a(2)=1, a(3)=7. - Harvey P. Dale, May 28 2012

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

cp's OEIS Frontend

A065916 Denominator of sigma(8*n^2)/sigma(4*n^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A085903 Expansion of (1 + 2*x^2)/((1 + x)*(1 - 2*x)*(1 - 2*x^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A137410 a(n) = (5^n - 3)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A180938 Smallest k such that k*n has an even number of 1's in its base-2 expansion.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A267890 Decimal representation of the n-th iteration of the "Rule 239" elementary cellular automaton starting with a single ON (black) cell.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A285475 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood.

Views

A065916 Denominator of sigma(8n^2)/sigma(4n^2).

A085903 Expansion of (1 + 2x^2)/((1 + x)(1 - 2x)(1 - 2*x^2)).

A132207 Triangle read by rows: row n lists first 24^n terms of an array read by rows, in which row k gives 24^n + 3k - 1; 54^n + 3*k - 1, with k>=0 in each array.