A010696 -id:A010696 - cp's OEIS frontend

A141498 a(n) = A010696(n-1) * A086892(n).

2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 46, 2730, 2, 6, 2, 510, 2, 798, 2, 1650, 2, 138, 94, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 46, 6, 142, 140100870, 2, 6, 2, 67650, 2, 12642, 862, 690, 2, 282, 2, 4501770, 2, 66, 2, 1590, 2, 798, 46, 870, 2, 354, 2, 283933650, 2, 6, 2, 510, 2

Offset: 1

Paul Curtz, Aug 10 2008

nonn

Cf. A010696, A086892.

A010696 := proc(n) op( (n mod 2)+1,[2,6]) ; end: A086892 := proc(n) gcd(2^n-1,3^n-1) ; end:
A141498 := proc(n) A010696(n-1)* A086892(n) ; end: seq(A141498(n),n=1..80) ; # R. J. Mathar, Sep 03 2009

Offset modified, extended by R. J. Mathar, Sep 03 2009

A010702 Period 2: repeat (3,4).

Original entry on oeis.org

3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3

Offset: 0

N. J. A. Sloane

Continued fraction expansion of A176102. - R. J. Mathar, Mar 08 2012

Also decimal expansion of 34/99. - Nicolas Bělohoubek, Nov 12 2021

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

Cf. A047355 (partial sums), A176102.

Cf. A010695, A010696.

a010702 = (+ 3) . (`mod` 2)
a010702_list = cycle [3,4]  -- Reinhard Zumkeller, Jul 05 2012

[3 + (n mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2014

&cat[[3,4]: n in [0..50]]; // Vincenzo Librandi, Aug 01 2015

A010702:=n->3+(n mod 2): seq(A010702(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2014

3 + Mod[Range[0, 100], 2] (* Wesley Ivan Hurt, Jul 24 2014 *)
PadRight[{}, 100, {3, 4}] (* Vincenzo Librandi, Aug 01 2015 *)

a(n)=3+n%2 \\ Charles R Greathouse IV, Dec 21 2011

def A010702(n): return 3 + (n & 1) # Chai Wah Wu, May 25 2022

G.f.: (3+4*x)/(1-x^2). - Jaume Oliver Lafont, Mar 20 2009
a(n) = floor((n+1)*7/2) - floor((n)*7/2). - Hailey R. Olafson, Jul 23 2014
a(n) = 3 + (n mod 2) = 4 - ((n+1) mod 2). - Wesley Ivan Hurt, Jul 24 2014
From Nicolas Bělohoubek, Nov 12 2021: (Start)
a(n) = 12/a(n-1). See also A010696.
a(n) = 7 - a(n-1). See also A010695. (End)
a(n) = (7-(-1)^n)/2. - Aaron J Grech, Jul 28 2024

A047463 Numbers that are congruent to {2, 4} mod 8.

Original entry on oeis.org

2, 4, 10, 12, 18, 20, 26, 28, 34, 36, 42, 44, 50, 52, 58, 60, 66, 68, 74, 76, 82, 84, 90, 92, 98, 100, 106, 108, 114, 116, 122, 124, 130, 132, 138, 140, 146, 148, 154, 156, 162, 164, 170, 172, 178, 180, 186, 188, 194, 196, 202, 204, 210, 212, 218, 220, 226, 228, 234

Offset: 1

N. J. A. Sloane

First differences in A010696.

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

Union of A017089 and A017113.

Cf. A014848.

[ n: n in [2..234 by 2] | n mod 8 in [2,4] ];  // Bruno Berselli, May 11 2011

Select[Range[250], MemberQ[{2, 4}, Mod[#, 8]] &] (* Amiram Eldar, Dec 18 2021 *)

a(n) = 8*n - a(n-1) - 10, with a(1)=2. - Vincenzo Librandi, Aug 06 2010
From Bruno Berselli, May 11 2011: (Start)
G.f.: 2*x*(1+x+2*x^2)/((1+x)*(1-x)^2).
a(n) = 4*n-(-1)^n-3.
Sum_{i=1..n} a(i) = 2*A014848(n).
a(n) = 2*A042963(n-1). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 + log(2)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A174114 Even central polygonal numbers (A193868) divided by 2.

Original entry on oeis.org

1, 2, 8, 11, 23, 28, 46, 53, 77, 86, 116, 127, 163, 176, 218, 233, 281, 298, 352, 371, 431, 452, 518, 541, 613, 638, 716, 743, 827, 856, 946, 977, 1073, 1106, 1208, 1243, 1351, 1388, 1502, 1541, 1661, 1702, 1828, 1871, 2003, 2048, 2186, 2233, 2377, 2426, 2576

Offset: 1

Reinhard Zumkeller, Mar 08 2010

Central terms of A170950, seen as a triangle of rows with an odd number of terms.
Equivalently, numbers of the form m*(4*m+3)+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - Bruno Berselli, Jan 05 2016
Conjecure: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n} 1 - q^k) = q + q^2 + q^8 + q^11 + q^23 + q^28 + .... Cf. A266883. - Peter Bala, May 10 2025

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

Cf. A002061, A002522, A010696, A170950, A193868, A266883.

Cf. A033951: numbers of the form m*(4*m+3)+1 for nonnegative m.

Select[Table[(n (n + 1)/2 + 1)/2, {n, 600}], IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)
(Select[PolygonalNumber@ Range@ 100, OddQ] + 1 )/2 (* Version 10.4, or *)
Rest@ CoefficientList[Series[-x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)

a(n)=(2*n-1)*(2*n-1-(-1)^n)\4+1 \\ Charles R Greathouse IV, Jun 11 2015

a(n+3) - a(n+2) - a(n+1) + a(n) = A010696(n+1).
a(n) = A170950(A002061(n)).
a(n) = A193868(n)/2. - Omar E. Pol, Aug 16 2011
G.f.: -x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 18 2011
E.g.f.: ((2 + x + 2*x^2)*cosh(x) + (1 - x + 2*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Nov 16 2024
Sum_{n>=1} 1/a(n) = 4*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - Amiram Eldar, May 12 2025

New name from Omar E. Pol, Aug 16 2011

A040008 Continued fraction for sqrt(12).

Original entry on oeis.org

3, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6

Offset: 0

N. J. A. Sloane

Eventual period is (2,6). - Zak Seidov, Mar 05 2011

Decimal expansion of 323/990. - R. J. Mathar, Aug 22 2025

			3.464101615137754587054892683... = 3 + 1/(2 + 1/(6 + 1/(2 + 1/(6 + ...)))). - _Harry J. Smith_, Jun 02 2009

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. A010469 Decimal expansion, A010696.

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

ContinuedFraction[Sqrt[12],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)

{ allocatemem(932245000); default(realprecision, 24000); x=contfrac(sqrt(12)); for (n=0, 20000, write("b040008.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

G.f.: (3 + 2*x + 3*x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A176394 Decimal expansion of 3+2*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of 3+2*sqrt(3) is A010696 preceded by 6.
a(n) = A010469(n) for n > 1.
Largest radius of three circles tangent to a circle of radius 1. - Charles R Greathouse IV, Jan 14 2013
For a spinning black hole the phase transition to positive specific heat happens at a point governed by 2*sqrt(3)-3 (according to a discussion on John Baez's blog), and not at the golden ratio as claimed by Paul Davis. - Peter Luschny, Mar 02 2013
In particular: a black hole with J > (2*sqrt(3)-3) Gm^2/c has positive specific heat, and negative specific heat if J is less, where J is its angular momentum, m is its mass, G is the gravitational constant, and c is the speed of light. For a solar mass black hole, this is 4.08 * 10^41 joule-seconds or a rotation every 1.61 days with the sun's inertia. - Charles R Greathouse IV, Sep 20 2013

			3+2*sqrt(3) = 6.46410161513775458705...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
John Baez, Black Holes and The Golden Ratio, Mar 01 2013.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (decimal expansion of sqrt(3)), A010469 (decimal expansion of sqrt(12)), A010696 (repeat 2, 6).

Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}], {Blue, Circle[{0, 0}, 1]}]]]; Circs[3] (* Charles R Greathouse IV, Jan 14 2013 *)

3+2*sqrt(3) \\ Charles R Greathouse IV, Jan 14 2013

Equals Sum_{n>=1} (sqrt(3)/2)^n = (sqrt(3)/2)/(1 - (sqrt(3)/2)). - Fred Daniel Kline, Mar 03 2014

A176053 Decimal expansion of (3+2*sqrt(3))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (3+2*sqrt(3))/3 is A010696.
a(n) = A020832(n-1) for n > 1; a(1) = 2.
This equals the ratio of the radius of the outer Soddy circle and the common radius of the three kissing circles. See A343235, also for links. - Wolfdieter Lang, Apr 19 2021
Previous comment is, together with A246724, the answer to the 1st problem proposed during the 4th Canadian Mathematical Olympiad in 1972. - Bernard Schott, Mar 20 2022

			2.15470053837925152901...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993 - Canadian Mathematical Society & Société Mathématique du Canada, Problem 1, 1972, page 37, 1993.

Paolo Xausa, Table of n, a(n) for n = 1..10000
The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
Michael Penn, An inscribed tower of squares, YouTube video, 2020.
Bernard Schott, Illustration of the Soddy circles.
Index to sequences related to Olympiads.

Cf. A002194 (sqrt(3)), A020832 (1/sqrt(75)), A010696 (repeat 2, 6).

Cf. A246724, A343235.

RealDigits[1+2/3Sqrt[3],10,100][[1]] (* Paolo Xausa, Aug 10 2023 *)

Equals 2 + A246724.

A086970 Fix 1, then exchange the subsequent odd numbers in pairs.

Original entry on oeis.org

1, 5, 3, 9, 7, 13, 11, 17, 15, 21, 19, 25, 23, 29, 27, 33, 31, 37, 35, 41, 39, 45, 43, 49, 47, 53, 51, 57, 55, 61, 59, 65, 63, 69, 67, 73, 71, 77, 75, 81, 79, 85, 83, 89, 87, 93, 91, 97, 95, 101, 99, 105, 103, 109, 107, 113, 111, 117, 115, 121, 119

Offset: 0

Paul Barry, Jul 26 2003

Partial sums are A086955.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004442, A014681, A065190.

[1] cat [2*n+1-2*(-1)^n: n in [1..70]]; // Vincenzo Librandi, Jun 21 2017

Join[{1}, LinearRecurrence[{1, 1, -1}, {5, 3, 9}, 60]] (* Vincenzo Librandi, Jun 21 2017 *)

Vec((1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2) + O(x^100)) \\ Michel Marcus, Jun 21 2017

G.f.: (1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2).
a(n) = n + abs(2 - (n + 1)*(-1)^n). - Lechoslaw Ratajczak, Dec 09 2016
a(n) = 2*A065190(n+1)-1 and a(n) = 2*A014681(n)+1. - Michel Marcus, Dec 10 2016
From Guenther Schrack, Jun 09 2017: (Start)
a(n) = 2*n + 1 - 2*(-1)^n for n > 0.
a(n) = 2*n + 1 - 2*cos(n*Pi) for n > 0.
a(n) = 4*n - a(n-1) for n > 1.
Linear recurrence: a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
First differences: 2 - 4*(-1)^n for n > 1; -(-1)^n*A010696(n) for n > 1.
a(n) = A065164(n+1) + n for n > 0.
a(A014681(n)) = A005408(n) for n >= 0.
a(A005408(A014681(n)) for n >= 0.
a(n) = A005408(A103889(n)) for n >= 0.
A103889(a(n)) = 2*A065190(n+1) for n >= 0.
a(2*n-1) = A004766(n) for n > 0.
a(2*n+2) = A004767(n) for n >= 0. (End)

A141517 A141498(n)/A027760(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 47, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 71, 73, 1, 1, 1, 5, 1, 7, 431, 1, 1, 1, 1, 97, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 1, 1, 1, 601, 1, 23, 1, 1, 5, 1, 1, 167, 7, 1, 431, 1, 1, 1, 1, 1, 1, 1, 1, 191, 193, 1

Offset: 1

Paul Curtz, Aug 11 2008

nonn

Composite entries are a(100)=25, a(120)=1205, a(144)=55969, a(156)=4069 etc.

A010696 := proc(n) op( (n mod 2)+1,[2,6]) ; end: A086892 := proc(n) gcd(2^n-1,3^n-1) ; end:
A141498 := proc(n) A010696(n-1)* A086892(n) ; end: A027760 := proc(n) local s, p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end:
A141517 := proc(n) A141498(n)/A027760(n) ; end: seq(A141517(n),n=1..120) ; # R. J. Mathar, Sep 03 2009

Offset modified, extended by R. J. Mathar, Sep 03 2009

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

A141498 a(n) = A010696(n-1) * A086892(n).

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A010702 Period 2: repeat (3,4).

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A047463 Numbers that are congruent to {2, 4} mod 8.

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Magma

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A174114 Even central polygonal numbers (A193868) divided by 2.

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Mathematica

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A040008 Continued fraction for sqrt(12).

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A176394 Decimal expansion of 3+2*sqrt(3).

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Mathematica

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A176053 Decimal expansion of (3+2*sqrt(3))/3.

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