A052901 -id:A052901 - cp's OEIS frontend

A208131 Partial products of A052901.

1, 3, 6, 12, 36, 72, 144, 432, 864, 1728, 5184, 10368, 20736, 62208, 124416, 248832, 746496, 1492992, 2985984, 8957952, 17915904, 35831808, 107495424, 214990848, 429981696, 1289945088, 2579890176, 5159780352, 15479341056, 30958682112, 61917364224, 185752092672

Offset: 0

Reinhard Zumkeller, Apr 04 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,12).

Cf. A001222, A052901, A026549, A026532.

a208131 n = a208131_list !! n
a208131_list = scanl (*) 1 $ a052901_list
-- Reinhard Zumkeller, Mar 29 2012

FoldList[Times,1,PadRight[{},30,{3,2,2}]] (* Harvey P. Dale, Mar 19 2013 *)

a(n+1) = a(n) * A052901(n).
A001222(a(n)) = n.
a(n) = 12^floor(n/3)*(r+1)*(r+2)/2 with r = n mod 3. G.f.: -(6*x^2+3*x+1) / (12*x^3-1). - Alois P. Heinz, Apr 05 2012
Sum_{n>=0} 1/a(n) = 18/11. - Amiram Eldar, Feb 13 2023

A069705 a(n) = 2^n mod 7.

Original entry on oeis.org

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

Offset: 0

Jon Perry, Jan 14 2003

Periodic sequence with period [1,2,4]. - Philippe Deléham, Sep 25 2006
From Klaus Brockhaus, May 23 2010: (Start)
Continued fraction expansion of (11 + sqrt(229))/18.
Decimal expansion of 124/999. (End)

			a(4)=16 mod 7=2, a(5)=32 mod 7=4, a(6)=64 mod 7=1.

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

Cf. A000079, A050985.
Cf. A178233 (decimal expansion of (11+sqrt(229))/18).
Cf. A052901, A144437.

List([0..83],n->PowerMod(2,n,7)); # Muniru A Asiru, Jan 31 2019

[Modexp(2, n, 7): n in [0..100]]; // Vincenzo Librandi, Mar 25 2016

A069705 := proc(n) op((n mod 3)+1,[1,2,4]) ; end proc: # R. J. Mathar, Feb 05 2011

PowerMod[2,Range[0,110],7] (* or *) PadRight[{},110,{1,2,4}] (* Harvey P. Dale, Mar 28 2015 *)

a(n)=2^(n%3)%7 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = lift(Mod(2, 7)^n); \\ Altug Alkan, Mar 25 2016

[power_mod(2,n,7) for n in range(0, 105)] # Zerinvary Lajos, Jun 07 2009

n=0 mod 3 -> a(n)=1 n=1 mod 3 -> a(n)=2 n=2 mod 3 -> a(n)=4.
a(n) = 2^(n mod 3). - Paul Barry, Oct 06 2003
a(n) = cubefree part of 2^n = A000079(A050985(n)). - Artur Jasinski, Oct 15 2008
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-3).
G.f.: (1+2*x+4*x^2)/((1-x) * (1+x+x^2)). (End)
a(n) = (7+5*cos(2*(n+1)*Pi/3)-sqrt(3)*sin(2*(n+1)*Pi/3))/3. - Wesley Ivan Hurt, Oct 01 2017
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 8/(a(n-2)*a(n-1)).
a(n) = 7 - a(n-2) - a(n-1). See also A052901 or A144437. (End)
a(n) = n + 1 + floor((n+1)/3) - 4*floor(n/3). - Ridouane Oudra, Sep 25 2024
E.g.f.: (7*exp(x) - 2*exp(-x/2)*(2*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Sep 27 2024

A144437 Period 3: repeat [3, 3, 1].

Original entry on oeis.org

3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3

Offset: 1

Paul Curtz, Oct 05 2008

The sequence is generated from numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...
Conjecture: a(n) is the separatix. See A045944.
Also the decimal expansion of the constant 3310/999. - R. J. Mathar, May 21 2009
Continued fraction expansion of A171417.
Greatest common divisor of (n+1)^2-1 and (n+1)^2+2. - Bruno Berselli, Mar 08 2017

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

Cf. A000217, A045944, A109007, A164306, A167192, A169609, A256095.

Numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...

&cat [[3, 3, 1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

seq(op([3, 3, 1]), n=1..50); # Wesley Ivan Hurt, Jul 02 2016

A144437[n_]:=Denominator[n/3]; Array[A144437,100] (* Enrique Pérez Herrero, Oct 05 2011 *)
CoefficientList[Series[(3 + 3 x + x^2)/(1 - x^3), {x, 0, 120}], x] (* Michael De Vlieger, Jul 02 2016 *)
Table[Mod[2*n^2 + 1, 3,1], {n,1,50}] (* G. C. Greubel, Aug 24 2017 *)

a(n)=if(n%3,3,1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = (7-4*cos(2*Pi*n/3))/3. - Jaume Oliver Lafont, Nov 23 2008
G.f.: x*(3 + 3*x + x^2)/((1 - x)*(1 + x + x^2)). - R. J. Mathar, May 21 2009
a(n) = 3/gcd(n,3). - Reinhard Zumkeller, Oct 30 2009
a(n) = denominator(n^k/3), where k>0 is an integer. - Enrique Pérez Herrero, Oct 05 2011
a(n) = gcd(T(n+1), T(2)) = A256095(n+1, 2), with the triangular numbers T = A000217, for n >= 1. - Wolfdieter Lang, Mar 17 2015
a(n) = a(n-3) for n>3; a(n) = A169609(n) for n>0. - Wesley Ivan Hurt, Jul 02 2016
E.g.f.: (1/3)*(7*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2) - 3). - G. C. Greubel, Aug 24 2017
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 9/(a(n-2)*a(n-1)).
a(n) = 7 - a(n-2) - a(n-1). See also A052901 or A069705. (End)

Edited by R. J. Mathar, May 21 2009

A082204 Begin with a 1, then place the smallest (as far as possible distinct) digits, such that, beginning from the n-th term, n terms form a palindrome.

Original entry on oeis.org

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

Offset: 0

Amarnath Murthy, Apr 10 2003

			The first six palindromes are 1, 22, 232, 3223, 22322, 232232.

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

Cf. A082205, A082206.

Essentially the same as A052901.

Join[{1},LinearRecurrence[{0, 0, 1},{2, 2, 3},104]] (* Ray Chandler, Aug 25 2015 *)

a(1) = 1; for k > 0, a(3k-1) = a(3k) = 2; a(3k+1) = 3. - David Wasserman, Aug 19 2004

More terms from David Wasserman, Aug 19 2004

A176979 Decimal expansion of (15+sqrt(365))/10.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2010

Continued fraction expansion of (15+sqrt(365))/10 is A052901.

			(15+sqrt(365))/10 = 3.41049731745428001791...

Cf. A176980 (decimal expansion of sqrt(365)), A052901 (repeat 3, 2, 2).

A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, 17, 14, 20, 15, 22, 17, 24, 19, 27, 20, 29, 22, 31, 24, 34, 25, 36, 27, 38, 29, 41, 30, 43, 32, 45, 34, 48, 35, 50, 37, 52, 39, 55, 40, 57, 42, 59, 44, 62, 45, 64, 47, 66, 49, 69, 50, 71, 52, 73, 54, 76, 55, 78

Offset: 0

Paul Curtz, Nov 22 2015

A260708 difference table rows have the same nine-step recurrence:
0, 1, 3,  6, 10, 16, 21, 29, 36, 46, 55, 65, 78,  93, ...
1, 2, 3,  4,  6,  5,  8,  7, 10,  9, 13, 10, 15,  12, ...     = a(n)
1, 1, 1,  2, -1,  3, -1,  3, -1,  4, -3,  5, -3,   5, ...     = b(n)
0, 0, 1, -3,  4, -4,  4, -4,  5, -7,  8, -8,  8,  -8, ... (see A042965(n)).
(b(2n) + b(2n+1) = A052901(n+2).)

G. C. Greubel, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1,0,-1).

Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.

I:=[1,2,3,4,6,5,8,7];[n le 8 select I[n] else Self(n-2) + Self(n-6) - Self(n-8): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015

RecurrenceTable[{a[n] == a[n-2] + a[n-6] - a[n-8], a[0]=1, a[1]=2, a[2]=3, a[3]=4, a[4]=6, a[5]=5, a[6]=8, a[7]=7}, a, {n,0,100}] (* G. C. Greubel, Nov 23 2015 *)

Vec((x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 22 2015

vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ Altug Alkan, Nov 24 2015

a(2n) = A047282(n). a(2n+1) = A047212(n+1).
a(n) = A260708(n+1) - A260708(n).
a(n+6) = a(n) + period of length 2: repeat 7, 5.
a(2n) + a(2n+1) = 3 + 4*n.
a(n) = n + 1 + (-1)^n*A152467(n+2).
From Colin Barker, Nov 22 2015: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) for n>7.
G.f.: (x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)).
(End)

A267068 a(n) = (n+1) / A189733(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 2, 11, 1, 13, 2, 3, 2, 17, 1, 19, 2, 3, 2, 23, 1, 25, 2, 3, 2, 29, 1, 31, 2, 3, 2, 35, 1, 37, 2, 3, 2, 41, 1, 43, 2, 3, 2, 47, 1, 49, 2, 3, 2, 53, 1, 55, 2, 3, 2, 59, 1, 61, 2, 3, 2, 65, 1, 67, 2, 3, 2

Offset: 0

Paul Curtz, Jan 10 2016

nonn

A189733(n) is the denominator of an autosequence of the first kind (the main diagonal is A000004).

Cf. A000004, A047238, A047245, A052901, A130196, A189733.

CoefficientList[Series[(1 + 2 x + 3 x^2 + 2 x^3 + 5 x^4 + x^5 + 5 x^6 - 2 x^7 - 3 x^8 - 2 x^9 + x^10 - x^11)/((1 - x)^2 (1 + x)^2 (1 - x + x^2)^2 (1 + x + x^2)^2), {x, 0, 69}], x] (* or *)
b[m_, n_] := b[m, n] = Which[m == n, 0, n == m + 1, (-1)^(n + 1)/n, n > m, b[m, n - 1] + b[m + 1, n - 1], n < m, b[m - 1, n + 1] - b[m - 1, n]]; Table[(n + 1)/Denominator@ b[0, n], {n, 0, 69}] (* Michael De Vlieger, Jan 15 2016, Jean-François Alcover at A189733 *)

a(2n+1) = A130196(n+1).
A052901(n+2) = period 3: 2, 3, 2  is at rank A047245(n+1) = 1, 2, 3, 7, 8, 9, ... .
Conjectures from Colin Barker, Jan 10 2016: (Start)
a(n) = 2*a(n-6) - a(n-12) for n>11.
G.f.: (1+2*x+3*x^2+2*x^3+5*x^4+x^5+5*x^6-2*x^7-3*x^8-2*x^9+x^10-x^11) / ((1-x)^2*(1+x)^2*(1-x+x^2)^2*(1+x+x^2)^2).
(End)
a(3n) + a(3n+1) + a(3n+2) = A047238(n+3).

cp's OEIS Frontend

A208131 Partial products of A052901.

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A069705 a(n) = 2^n mod 7.

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A144437 Period 3: repeat [3, 3, 1].

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A082204 Begin with a 1, then place the smallest (as far as possible distinct) digits, such that, beginning from the n-th term, n terms form a palindrome.

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A176979 Decimal expansion of (15+sqrt(365))/10.

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A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

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A267068 a(n) = (n+1) / A189733(n).

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