A028294 -id:A028294 - cp's OEIS frontend

A075690 a(n) = (n-1)*(n-2)^4 - A028294(n), for n > 4, with a(1) = a(2) = 0, a(3) = 2, and a(4) = 48.

0, 0, 2, 48, 304, 999, 2393, 4791, 8542, 14039, 21719, 32063, 45596, 62887, 84549, 111239, 143658, 182551, 228707, 282959, 346184, 419303, 503281, 599127, 707894, 830679, 968623, 1122911, 1294772, 1485479, 1696349, 1928743, 2184066, 2463767

Offset: 1

Jon Perry, Oct 12 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Ed Pegg Jr., Pancakes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A028294.

[0,0,2,48] cat [(11*n^4+19*n^3-632*n^2+2012*n-1686)/6: n in [4..50]]; // G. C. Greubel, Jan 03 2024

LinearRecurrence[{5,-10,10,-5,1}, {0,0,2,48,304,999,2393,4791,8542}, 50] (* G. C. Greubel, Jan 03 2024 *)

[0,0,2,48] + [(11*n^4+19*n^3-632*n^2+2012*n-1686)/6 for n in range(4,51)] # G. C. Greubel, Jan 03 2024

From G. C. Greubel, Jan 03 2024: (Start)
a(n) = (n-1)*(n-2)^4 - A028294(n) + 46*[n=1] - 23*[n=2] - 9*[n=3] + [n=4].
a(n) = (11*n^4 + 19*n^3 - 632*n^2 + 2012*n - 1686)/6 + 46*[n=1] - 23*[n=2] - 9*[n=3] + [n=4].
G.f.: x^3*(2 + 38*x + 84*x^2 - 61*x^3 - 32*x^4 + 14*x^5 - x^6)/(1-x)^5.
E.g.f.: (1/6)*(-1686 + 1410*x - 498*x^2 + 85*x^3 + 11*x^4)*exp(x) + 281 + 46*x - 23*x^2/2 - 9*x^3/3! + x^4/4!. (End)

More terms from David Wasserman, Jan 22 2005

Name clarified by G. C. Greubel, Jan 03 2024

A006368 The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 9, 5, 12, 7, 15, 8, 18, 10, 21, 11, 24, 13, 27, 14, 30, 16, 33, 17, 36, 19, 39, 20, 42, 22, 45, 23, 48, 25, 51, 26, 54, 28, 57, 29, 60, 31, 63, 32, 66, 34, 69, 35, 72, 37, 75, 38, 78, 40, 81, 41, 84, 43, 87, 44, 90, 46, 93, 47, 96, 49, 99, 50, 102, 52, 105, 53

Offset: 0

N. J. A. Sloane and J. H. Conway

A permutation of the nonnegative integers.
There is a famous open question concerning the closed trajectories under this map - see A217218, A028393, A028394, and Conway (2013).
This is lodumo_3 of A131743. - Philippe Deléham, Oct 24 2011
Multiples of 3 interspersed with numbers other than multiples of 3. - Harvey P. Dale, Dec 16 2011
For n>0: a(2n+1) is the smallest number missing from {a(0),...,a(2n-1)} and a(2n) = a(2n-1) + a(2n+1). - Bob Selcoe, May 24 2017
From Wolfdieter Lang, Sep 21 2021: (Start)
The permutation P of positive natural numbers with P(n) = a(n-1) + 1, for n >= 1, is the inverse of the permutation given in A265667, and it maps the index n of A178414 to the index of A047529: A178414(n) = A047529(P(n)).
Thus each number {1, 3, 7} (mod 8) appears in the first column A178414 of the array A178415 just once. For the formulas see below. (End)
Starting at n = 1, the sequence equals the smallest unused positive number such that a(n)-a(n-1) does not appear as a term in the current sequence. - Scott R. Shannon, Dec 20 2023

			9 is odd so a(9) = round(3*9/4) = round(7-1/4) = 7.

J. H. Conway, Unpredictable iterations, in Proc. Number Theory Conf., Boulder, CO, 1972, pp. 49-52.
R. K. Guy, Unsolved Problems in Number Theory, E17.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Zumkeller, Table of n, a(n) for n = 0..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7..
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198. [Introduces the name "amusical permutation".]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
S. Schreiber & N. J. A. Sloane, Correspondence, 1980
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for two-way infinite sequences
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Inverse mapping to A006369.
Cf. A047529, A178414, A178415, A265667.
Cf. A028393, A028294, A028397, A180853, A180864, A182205, A217218.
Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

a006368 n | u' == 0   = 3 * u
          | otherwise = 3 * v + (v' + 1) `div` 2
          where (u,u') = divMod n 2; (v,v') = divMod n 4
-- Reinhard Zumkeller, Apr 18 2012

[n mod 2 eq 1 select Round(3*n/4) else 3*n/2: n in [0..80]]; // G. C. Greubel, Jan 03 2024

f:=n-> if n mod 2 = 0 then 3*n/2 elif n mod 4 = 1 then (3*n+1)/4 else (3*n-1)/4; fi; # N. J. A. Sloane, Jan 21 2011
A006368:=(1+3*z+z**2+3*z**3+z**4)/(1+z**2)/(z-1)**2/(1+z)**2; # [Conjectured (correctly, except for the offset) by Simon Plouffe in his 1992 dissertation.]

Table[If[EvenQ[n],(3n)/2,Floor[(3n+2)/4]],{n,0,80}] (* or *) LinearRecurrence[ {0,1,0,1,0,-1},{0,1,3,2,6,4},80] (* Harvey P. Dale, Dec 16 2011 *)

PARI
```
a(n)=(3*n+n%2)\(2+n%2*2)
```
PARI
```
a(n)=if(n%2,round(3*n/4),3*n/2)
```

def a(n): return 0 if n == 0 else 3*n//2 if n%2 == 0 else (3*n+1)//4
print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 12 2021

If n even, then a(n) = 3*n/2, otherwise, a(n) = round(3*n/4).
G.f.: x*(1+3*x+x^2+3*x^3+x^4)/((1-x^2)*(1-x^4)). - Michael Somos, Jul 23 2002
a(n) = -a(-n).
From Reinhard Zumkeller, Nov 20 2009: (Start)
a(n) = A006369(n) - A168223(n).
A168221(n) = a(a(n)).
A168222(a(n)) = A006369(n). (End)
a(n) = a(n-2) + a(n-4) - a(n-6); a(0)=0, a(1)=1, a(2)=3, a(3)=2, a(4)=6, a(5)=4. - Harvey P. Dale, Dec 16 2011
From Wolfdieter Lang, Sep 21 2021: (Start)
Formulas for the permutation P(n) = a(n-1) + 1 mentioned above:
P(n) = n + floor(n/2) if n is odd, and n - floor(n/4) if n is even.
P(n) = (3*n-1)/2 if n is odd; P(n) = (3*n+2)/4 if n == 2 (mod 4); and P(n) = 3*n/4 if n == 0 (mod 4). (End)

Edited by Michael Somos, Jul 23 2002

I replaced the definition with the original definition of Conway and Guy. - N. J. A. Sloane, Oct 03 2012

A003777 a(n) = n^3 + n^2 - 1.

Original entry on oeis.org

1, 11, 35, 79, 149, 251, 391, 575, 809, 1099, 1451, 1871, 2365, 2939, 3599, 4351, 5201, 6155, 7219, 8399, 9701, 11131, 12695, 14399, 16249, 18251, 20411, 22735, 25229, 27899, 30751, 33791, 37025, 40459, 44099, 47951, 52021, 56315, 60839, 65599, 70601, 75851

Offset: 1

N. J. A. Sloane, Jun 14 1998

This sequence in related to A095794 by a(n) = n*A095794(n) - Sum_{i=1..n-1} A095794(i), for n > 1. - Bruno Berselli, Dec 28 2010
a(n) is the area of a triangle with vertices at points (n-1,(n-1)^2), (n,n^2), and ((n+1)^2,n+1). - J. M. Bergot, Jun 03 2014
Old name was: "Number of stacks of n pikelets, distance 3 flips from a well-ordered stack".

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A001107, A002411, A028294, A028295, A095794.

[(n^3+n^2-1): n in [1..50]]; // Vincenzo Librandi, Apr 06 2011

A003777:=n->n^3+n^2-1; seq(A003777(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014

Table[n^3+n^2-1,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
CoefficientList[Series[(1 + 7 x - 3 x^2 + x^3)/(1-x)^4, {x,0,50}], x] (* Vincenzo Librandi, Jun 20 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,11,35,79},50] (* Harvey P. Dale, Jul 20 2024 *)

a(n)=n^3+n^2-1 \\ Charles R Greathouse IV, Oct 07 2015

[n^3+n^2-1 for n in range(1,51)] # G. C. Greubel, Jan 03 2024

G.f.: x*(1+7*x-3*x^2+x^3)/(1-x)^4. Also, a(n) = 2*A002411(n) - 1 = A000578(n-1) + A001107(n). - Bruno Berselli, Dec 28 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. - Wesley Ivan Hurt, Oct 08 2017
E.g.f.: 1 + (-1 + 2*x + 4*x^2 + x^3)*exp(x). - G. C. Greubel, Jan 03 2024

More terms from Wesley Ivan Hurt, Jun 04 2014

Entry revised by N. J. A. Sloane, Jun 15 2014

A028295 a(n) = n^6 - (883/60)n^5 + (157/3)n^4 + (2155/12)n^3 - (4570/3)n^2 + (42767/15)*n - 967.

Original entry on oeis.org

133, 1903, 10561, 38015, 106461, 252737, 533397, 1030505, 1858149, 3169675, 5165641, 8102491, 12301949, 18161133, 26163389, 36889845, 51031685, 69403143, 92955217, 122790103, 160176349, 206564729, 263604837, 333162401, 417337317, 518482403, 639222873

Offset: 6

R. K. Guy

Old name was: "Number of stacks of n pikelets, distance 6 flips from a well-ordered stack".

G. C. Greubel, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A003777, A003878, A028294.

[(60*n^6 -883*n^5 +3140*n^4 +10775*n^3 -91400*n^2 +171068*n -58020)/60: n in [6..46]]; // G. C. Greubel, Jan 03 2024

(* Codes from Robert G. Wilson v, Jul 29 2018: Start *)
a[n_]:= n^6 - (883/60)*n^5 + (157/3)*n^4 + (2155/12)*n^3 - (4570/3)*n^2 + (42767/15)*n - 967; Table[a[n], {n,6,36}]
CoefficientList[ Series[x^6 (3x^6 -2x^5 -187x^4 +604x^3 -33x^2 -972x - 133)/(x-1)^7, {x,0,36}], x]
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {133,1903,10561,38015,106461, 252737,533397}, 36]
(* End *)

[(60*n^6 -883*n^5 +3140*n^4 +10775*n^3 -91400*n^2 +171068*n -58020)/60 for n in range(6,47)] # G. C. Greubel, Jan 03 2024

G.f.: x^6*(133 + 972*x + 33*x^2 - 604*x^3 + 187*x^4 + 2*x^5 - 3*x^6) / (1-x)^7. - R. J. Mathar, Jun 21 2011

E.g.f.: (1/5!)*(116040 - 69480*x - 30540*x^2 - 2340*x^3 + 95*x^4 + 3*x^5 - (116040 - 185520*x + 96960*x^2 - 25880*x^3 + 3580*x^4 - 34*x^5 - 120*x^6)*exp(x)). - G. C. Greubel, Jan 03 2024

Entry revised by N. J. A. Sloane, Jun 15 2014

a(17)-a(32) from Robert G. Wilson v, Jul 29 2018

cp's OEIS Frontend

A075690 a(n) = (n-1)*(n-2)^4 - A028294(n), for n > 4, with a(1) = a(2) = 0, a(3) = 2, and a(4) = 48.

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A006368 The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

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A003777 a(n) = n^3 + n^2 - 1.

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A028295 a(n) = n^6 - (883/60)*n^5 + (157/3)*n^4 + (2155/12)*n^3 - (4570/3)*n^2 + (42767/15)*n - 967.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A028295 a(n) = n^6 - (883/60)n^5 + (157/3)n^4 + (2155/12)n^3 - (4570/3)n^2 + (42767/15)*n - 967.