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Search: a052901 -id:a052901
Displaying 1-6 of 6 results found. page 1
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A208131 Partial products of A052901. +20
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,12).
FORMULA
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
MATHEMATICA
FoldList[Times, 1, PadRight[{}, 30, {3, 2, 2}]] (* Harvey P. Dale, Mar 19 2013 *)
PROG
(Haskell)
a208131 n = a208131_list !! n
a208131_list = scanl (*) 1 $ a052901_list
-- Reinhard Zumkeller, Mar 29 2012
CROSSREFS
Cf. A001222, A052901, A026549, A026532.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 04 2012
STATUS
approved
A069705 a(n) = 2^n mod 7. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Periodic sequence with period [1,2,4]. - Philippe Deléham, Sep 25 2006
a(n) = cubefree part of 2^n. - Artur Jasinski, Oct 15 2008
From Klaus Brockhaus, May 23 2010: (Start)
Continued fraction expansion of (11 + sqrt(229))/18.
Decimal expansion of 124/999. (End)
LINKS
Table of n, a(n) for n=0..104.
Index entries for linear recurrences with constant coefficients, signature (0,0,1).
FORMULA
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
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)
EXAMPLE
a(4)=16 mod 7=2, a(5)=32 mod 7=4, a(6)=64 mod 7=1.
MAPLE
A069705 := proc(n) op((n mod 3)+1, [1, 2, 4]) ; end proc: # R. J. Mathar, Feb 05 2011
MATHEMATICA
PowerMod[2, Range[0, 110], 7] (* or *) PadRight[{}, 110, {1, 2, 4}] (* Harvey P. Dale, Mar 28 2015 *)
PROG
(Sage) [power_mod(2, n, 7) for n in range(0, 105)] # Zerinvary Lajos, Jun 07 2009
(PARI) a(n)=2^(n%3)%7 \\ Charles R Greathouse IV, Jun 11 2015
(PARI) a(n) = lift(Mod(2, 7)^n); \\ Altug Alkan, Mar 25 2016
(Magma) [Modexp(2, n, 7): n in [0..100]]; // Vincenzo Librandi, Mar 25 2016
(GAP) List([0..83], n->PowerMod(2, n, 7)); # Muniru A Asiru, Jan 31 2019
CROSSREFS
Cf. A145642. - Artur Jasinski, Oct 15 2008
Cf. A178233 (decimal expansion of (11+sqrt(229))/18). - Klaus Brockhaus, May 23 2010
Appears in A179132. - Johannes W. Meijer, Jul 01 2010
KEYWORD
nonn,easy
AUTHOR
Jon Perry, Jan 14 2003
STATUS
approved
A144437 Period 3: repeat [3, 3, 1]. +10
15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Table of n, a(n) for n=1..86.
Index entries for linear recurrences with constant coefficients, signature (0,0,1).
FORMULA
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)
MAPLE
seq(op([3, 3, 1]), n=1..50); # Wesley Ivan Hurt, Jul 02 2016
MATHEMATICA
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 *)
PROG
(PARI) a(n)=if(n%3, 3, 1) \\ Charles R Greathouse IV, Sep 28 2015
(Magma) &cat [[3, 3, 1]^^30]; // Wesley Ivan Hurt, Jul 02 2016
CROSSREFS
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), ...
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Oct 05 2008
EXTENSIONS
Edited by R. J. Mathar, May 21 2009
STATUS
approved
A176979 Decimal expansion of (15+sqrt(365))/10. +10
2
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Continued fraction expansion of (15+sqrt(365))/10 is A052901.
LINKS
Table of n, a(n) for n=1..105.
EXAMPLE
(15+sqrt(365))/10 = 3.41049731745428001791...
CROSSREFS
Cf. A176980 (decimal expansion of sqrt(365)), A052901 (repeat 3, 2, 2).
KEYWORD
cons,nonn
AUTHOR
Klaus Brockhaus, Apr 30 2010
STATUS
approved
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. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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).)
LINKS
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).
FORMULA
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)
MATHEMATICA
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 *)
PROG
(PARI) 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
(PARI) vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ Altug Alkan, Nov 24 2015
(Magma) 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
CROSSREFS
Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Nov 22 2015
STATUS
approved
A267068 a(n) = (n+1) / A189733(n). +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
A189733(n) is the denominator of an autosequence of the first kind (the main diagonal is A000004).
LINKS
Table of n, a(n) for n=0..69.
FORMULA
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).
MATHEMATICA
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 *)
CROSSREFS
Cf. A000004, A047238, A047245, A052901, A130196, A189733.
KEYWORD
nonn
AUTHOR
Paul Curtz, Jan 10 2016
STATUS
approved
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Last modified August 29 21:13 EDT 2024. Contains 375518 sequences. (Running on oeis4.)