A010694 -id:A010694 - cp's OEIS frontend

A194271 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194270.

0, 1, 4, 8, 16, 22, 24, 22, 40, 40, 32, 32, 56, 74, 96, 50, 88, 72, 32, 48, 72, 104, 128, 112, 144, 144, 152, 96, 152, 178, 240, 122, 184, 136, 32, 48, 72, 108, 144, 144, 184, 188, 200, 176, 272, 274, 416, 250, 288, 272, 216, 144, 208, 292, 384, 332, 376

Offset: 0

Omar E. Pol, Aug 23 2011

nonn

Essentially the first differences of A194270.

			Written as a triangle:
0,
1,
4,
8,
16,22,
24,22,40,40,
32,32,56,74,96,50,88,72,
32,48,72,104,128,112,144,144,152,96,152,178,240,122,184,136,
32,48,72,108,144,144,184,188,200,176,272,274,416,250,288,...

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A010694, A129370, A139250, A139251, A172311, A182839, A194270, A194441, A194443, A194445.

a(n) = n^2-(n-1)^2*(1-(-1)^n)/8, if 0 <= n <=4.
Let b(n) = A194441(n), let c(n) = A194443(n), let d(n) = A010694(n), then:
Conjecture: a(n) = 4*(b(n-1)-d(n)) + 2*(c(n)-d(n+1)) + 2*(c(n+2)-d(n+1)) + 8, if n >= 3.
Conjecture: a(2^k+2) = 32, if k >= 3.

More terms from Omar E. Pol, Sep 01 2011

A105397 Periodic with period 2: repeat [4,2].

Original entry on oeis.org

4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2

Offset: 0

Eric Angelini, May 01 2005

A simple "Fractal Jump Sequence" (FJS). An FJS is a sequence of digits containing an infinite number of copies of itself. Modus operandi: underline the first digit "a" of such a sequence then jump over the next "a" digits and underline the digit "b" on which you land. Jump from there over the next "b" digits and underline the digit "c" on which you land. Etc. The "abc...n..." succession of underlined digits is the sequence itself.

Simple continued fraction of 2+sqrt(6). - R. J. Mathar, Nov 21 2011

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010694 (period 2, repeat [2,4]).

First differences of A007310. - Fred Daniel Kline, Aug 17 2020

A105397:=n->3 + (-1)^n; seq(A105397(n), n=0..100); # Wesley Ivan Hurt, Mar 14 2014

Table[3 + (-1)^n, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 14 2014 *)
LinearRecurrence[{0, 1},{4, 2},75] (* Ray Chandler, Aug 25 2015 *)

contfrac(2+sqrt(6)) \\ Michel Marcus, Mar 18 2014

a(n) = 3 + (-1)^n = 4 - 2*(n mod 2) = 2 * 2^((n+1) mod 2). - Wesley Ivan Hurt, Mar 14 2014

Edited by N. J. A. Sloane, Jun 08 2010

A040003 Continued fraction for sqrt(6).

Original entry on oeis.org

2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4

Offset: 0

N. J. A. Sloane

			2.449489742783178098197284074... = 2 + 1/(2 + 1/(4 + 1/(2 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 143.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010464 (decimal expansion).
Equals twice A040001.
Essentially the same as A010694.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[6], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(6)); for (n=0, 20000, write("b040003.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

a(n-1) = gcd(2^n, 3^n+1) (empirical). - Michel Marcus, Sep 03 2020

G.f.: 2*(1 + x + x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A114697 Expansion of (1+x+x^2)/((1-x^2)(1-2x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 3, 9, 22, 55, 133, 323, 780, 1885, 4551, 10989, 26530, 64051, 154633, 373319, 901272, 2175865, 5253003, 12681873, 30616750, 73915375, 178447501, 430810379, 1040068260, 2510946901, 6061962063, 14634871029, 35331704122, 85298279275, 205928262673

Offset: 0

Creighton Dement, Feb 18 2006

Generating floretion: (- .5'j + .5'k - .5j' + .5k' + 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*('i + 'j + i').

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

Cf. A000129, A002203, A005409, A100828, A111954, A113224.

Cf. A114647, A114688, A114689, A114695, A116699.

Table[(3*LucasL[n, 2] +10*Fibonacci[n, 2] -3 +(-1)^n)/4, {n,0,30}] (* G. C. Greubel, May 24 2021 *)

Vec((1+x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^40)) \\ Colin Barker, Jun 24 2015

[(4*lucas_number1(n+2,2,-1) -2*lucas_number1(n+1,2,-1) -3 +(-1)^n)/4 for n in (0..30)] # G. C. Greubel, May 24 2021

a(n+2) - 2*a(n+1) + a(n) = A111955(n+2).
G.f.: (1+x+x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
From Raphie Frank, Oct 01 2012: (Start)
a(2*n) = A216134(2*n+1).
a(2*n+1) = A006452(2*n+3)-1.
Lim_{n->infinity} a(n+1)/a(n) = A014176. (End)
a(n) = (2*A078343(n+2) - A010694(n))/4. - R. J. Mathar, Oct 02 2012
From Colin Barker, May 26 2016: (Start)
a(n) = ( 2*(-3 +(-1)^n) + (6-5*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(6+5*sqrt(2)) )/8.
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3. (End)
a(n) = (3*A002203(n) + 10*A000129(n) - 3 + (-1)^n)/4. - G. C. Greubel, May 24 2021

A274913 Square array read by antidiagonals upwards in which each new term is the least positive integer distinct from its neighbors.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 2, 3, 2, 3, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Omar E. Pol, Jul 11 2016

This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the square array we have that:
Antidiagonal sums give the positive terms of A008851.
Odd-indexed rows give A010684.
Even-indexed rows give A010694.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed antidiagonals give the initial terms of A010685.
Even-indexed antidiagonals give the initial terms of A010693.
Main diagonal gives A010685.
This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the triangle we have that:
Row sums give the positive terms of A008851.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed diagonals give A010684.
Even-indexed diagonals give A010694.
Odd-indexed rows give the initial terms of A010685.
Even-indexed rows give the initial terms of A010693.
Odd-indexed antidiagonals give the initial terms of A010684.
Even-indexed antidiagonals give the initial terms of A010694.

			The corner of the square array begins:
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, ...
1, 3, 1, 3, ...
2, 4, 2, ...
1, 3, ...
2, ...
...
The sequence written as a triangle begins:
1;
2, 3;
1, 4, 1;
2, 3, 2, 3;
1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3, 2, 3;
...

Cf. A000034, A008851, A010684, A010685, A010693, A010694, A010702, A274912, A274921.

Table[1 + Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

a(n) = A274912(n) + 1.

A382713 Simple continued fraction expansion of sqrt(3/2).

Original entry on oeis.org

1, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2

Offset: 0

N. J. A. Sloane, Apr 08 2025

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A105397, A115754, A142238, A142239.

Essentially the same as A106469, A040003, A010694.

with(numtheory); cfrac (sqrt(3/2, 70, 'quotients');

PadRight[{1}, 100, {2, 4}] (* Paolo Xausa, Apr 14 2025 *)

def A382713(n): return 1<<1+(n&1) if n else 1 # Chai Wah Wu, Apr 09 2025

A176051 Decimal expansion of (2+sqrt(6))/2.

Original entry on oeis.org

2, 2, 2, 4, 7, 4, 4, 8, 7, 1, 3, 9, 1, 5, 8, 9, 0, 4, 9, 0, 9, 8, 6, 4, 2, 0, 3, 7, 3, 5, 2, 9, 4, 5, 6, 9, 5, 9, 8, 2, 9, 7, 3, 7, 4, 0, 3, 2, 8, 3, 3, 5, 0, 6, 4, 2, 1, 6, 3, 4, 6, 2, 8, 3, 6, 2, 5, 4, 8, 0, 1, 8, 8, 7, 2, 8, 6, 5, 7, 5, 1, 3, 2, 6, 9, 9, 2, 9, 7, 1, 6, 5, 5, 2, 3, 2, 0, 1, 1, 7, 4, 0, 9, 2, 9

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (2+sqrt(6))/2 is A010694.

a(n) = A115754(n) for n > 1; a(1) = 2.

			(2+sqrt(6))/2 = 2.22474487139158904909...
Note also that (1+sqrt(6))/2 = 1.724744871391589049098642..., the mis-typed golden ratio. - _N. J. A. Sloane_, Jan 19 2025

Cf. A010464 (decimal expansion of sqrt(6)), A115754 (decimal expansion of sqrt(3/2)), A010694 (repeat 2, 4).

A176213 Decimal expansion of 2+sqrt(6).

Original entry on oeis.org

4, 4, 4, 9, 4, 8, 9, 7, 4, 2, 7, 8, 3, 1, 7, 8, 0, 9, 8, 1, 9, 7, 2, 8, 4, 0, 7, 4, 7, 0, 5, 8, 9, 1, 3, 9, 1, 9, 6, 5, 9, 4, 7, 4, 8, 0, 6, 5, 6, 6, 7, 0, 1, 2, 8, 4, 3, 2, 6, 9, 2, 5, 6, 7, 2, 5, 0, 9, 6, 0, 3, 7, 7, 4, 5, 7, 3, 1, 5, 0, 2, 6, 5, 3, 9, 8, 5, 9, 4, 3, 3, 1, 0, 4, 6, 4, 0, 2, 3, 4, 8, 1, 8, 5, 9

Offset: 1

Klaus Brockhaus, Apr 12 2010

Continued fraction expansion of 2+sqrt(6) is A010694.

a(n) = A010464(n) = A086180(n) for n > 1, a(1) = 4.

			2+sqrt(6) = 4.44948974278317809819...

Cf. A010464 (decimal expansion of sqrt(6)), A086180 (decimal expansion of 1+sqrt(6)), A010694 (repeat 4, 2).

RealDigits[2+Sqrt[6],10,120][[1]] (* Harvey P. Dale, Aug 19 2018 *)

A099517 A transform of (1-x)/(1-2x).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 27, 44, 71, 116, 188, 305, 493, 798, 1291, 2090, 3382, 5473, 8855, 14328, 23183, 37512, 60696, 98209, 158905, 257114, 416019, 673134, 1089154, 1762289, 2851443, 4613732, 7465175, 12078908, 19544084, 31622993, 51167077

Offset: 0

Paul Barry, Oct 20 2004

A transform of A011782 under the mapping g(x)->(1/(1+x^3))g(x/(1+x^3))

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,0,-1).

Cf. A099505, A099508, A099516, A000071.

G.f.: (1-x+x^3)/((1+x^3)*(1-2*x+x^3)).
a(n) = 2*a(n-1)-2*a(n-3)+2*a(n-4)-a(n-6).
a(n) = sum{k=0..floor(n/3), binomial(n-2*k, k)*(-1)^k*(2^(n-3*k)+0^(n-3*k))/2}.
a(n) = A057079(n+1)/6 +A000045(n+3)/2 -A010694(n)/6. - R. J. Mathar, Sep 21 2012

A155158 Period 4: repeat [1, 5, 7, 3].

Original entry on oeis.org

1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5, 7, 3, 1, 5

Offset: 0

Paul Curtz, Jan 21 2009

Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A010694, A010696, A048473, A105398.

&cat [[1, 5, 7, 3]^^30]; // Wesley Ivan Hurt, Jul 08 2016

seq(op([1, 5, 7, 3]), n=0..50); # Wesley Ivan Hurt, Jul 08 2016

PadRight[{}, 300, {1, 5, 7, 3}] (* Wesley Ivan Hurt, Jul 08 2016 *)

a(n) = A048473(n) mod 10.
First differences: a(n+1)-a(n) = (-1)^floor(n/2)*A010694(n+1).
Second differences: a(n+2)-2*a(n+1)+a(n) = (-1)^floor(1+n/2)*A010696(n).
Third differences: a(n+3)-3*a(n+2)+3*a(n+1)-a(n) = (-1)^floor((n+3)/2)*A105398(n).
G.f.: (1+4*x+3*x^2)/(1-x+x^2-x^3). - Colin Barker, Feb 28 2012
a(n) = a(n-1) - a(n-2) + a(n-3) for n>2. - Wesley Ivan Hurt, Jul 08 2016

cp's OEIS Frontend

A194271 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194270.

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A105397 Periodic with period 2: repeat [4,2].

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A040003 Continued fraction for sqrt(6).

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A114697 Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

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A274913 Square array read by antidiagonals upwards in which each new term is the least positive integer distinct from its neighbors.

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Mathematica

Formula

A382713 Simple continued fraction expansion of sqrt(3/2).

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Maple

Mathematica

Python

A176051 Decimal expansion of (2+sqrt(6))/2.

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Mathematica

A176213 Decimal expansion of 2+sqrt(6).

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A114697 Expansion of (1+x+x^2)/((1-x^2)(1-2x-x^2)); a Pellian-related sequence.