A081374 -id:A081374 - cp's OEIS frontend

A083322 a(n) = 2^n - A081374(n).

1, 2, 6, 11, 22, 42, 85, 170, 342, 683, 1366, 2730, 5461, 10922, 21846, 43691, 87382, 174762, 349525, 699050, 1398102, 2796203, 5592406, 11184810, 22369621, 44739242, 89478486, 178956971, 357913942, 715827882, 1431655765, 2863311530, 5726623062

Offset: 1

David Applegate, Aug 22 2003

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

Cf. A081374.

Trisections: A082365, A007613, A132804.

I:=[1,2,6,11]; [n le 4 select I[n] else 2*Self(n-1)-Self(n-3)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 08 2016

CoefficientList[Series[(1 + 2 x^2) / ((1 - 2 x) (1 + x) (1 - x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 08 2016 *)
LinearRecurrence[{2,0,-1,2},{1,2,6,11},40] (* Harvey P. Dale, Jan 30 2024 *)

G.f.: x*(1+2*x^2) / ( (1-2*x)*(1+x)*(1-x+x^2) ). - R. J. Mathar, May 27 2011
From Paul Curtz, May 27 2011: (Start)
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4).
a(n)+a(n+3) = 3*2^(n+1) = A007283(n+1).
a(n+6)-a(n) = 21*2^(n+1) = A175805(n+1).
(End)

A082311 A Jacobsthal sequence trisection.

Original entry on oeis.org

1, 5, 43, 341, 2731, 21845, 174763, 1398101, 11184811, 89478485, 715827883, 5726623061, 45812984491, 366503875925, 2932031007403, 23456248059221, 187649984473771, 1501199875790165, 12009599006321323, 96076792050570581, 768614336404564651, 6148914691236517205

Offset: 0

Paul Barry, Apr 09 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A001045, A007613, A015565, A024494, A070366, A081374, A082365, A132805.

[2*8^n/3+(-1)^n/3 : n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

f[n_] := (2*8^n + (-1)^n)/3; Array[f, 25, 0] (* Robert G. Wilson v, Aug 13 2011 *)

x='x+O('x^30); Vec((1-2*x)/((1+x)*(1-8*x))) \\ G. C. Greubel, Sep 16 2018

a(n) = (2*8^n + (-1)^n)/3 = A001045(3*n+1).
From R. J. Mathar, Feb 23 2009: (Start)
a(n) = 7*a(n-1) + 8*a(n-2).
G.f.: (1-2*x)/((1+x)*(1-8*x)). (End)
a(n) = A024494(3*n+1). a(n) = 8*a(n-1) + 3*(-1)^n. Sum of digits = A070366. - Paul Curtz, Nov 20 2007
a(n)= A007613(n) + A132805(n) = A081374(1+3*n). - Paul Curtz, Jun 06 2011
E.g.f.: (cosh(x) + 2*cosh(8*x) - sinh(x) + 2*sinh(8*x))/3. - Stefano Spezia, Jul 15 2024

A081375 a(n) is the least number k such that A081373(k) = n.

Original entry on oeis.org

1, 2, 6, 12, 30, 42, 72, 78, 84, 90, 190, 216, 222, 228, 234, 252, 270, 540, 546, 570, 630, 738, 744, 770, 792, 858, 900, 924, 930, 990, 1050, 1638, 1710, 1890, 1980, 2100, 2310, 2418, 2442, 2508, 2562, 2574, 2604, 2700, 2772, 2790, 2850, 2970, 3150

Offset: 1

Labos Elemer, Mar 24 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A008479, A081373, A081374, A067004.

f[x_] := Count[Table[EulerPhi[j]-EulerPhi[x], {j, 1, x}], 0] t=Table[0, {50}]; Do[s=f[n]; If[s<51&&t[[s]]==0, t[[s]]=n], {n, 1, 4000}]; t

lista(len) = {my(v = vector(len), c = 0, k = 1, i); while(c < len, i = #select(x -> x <= k, invphi(eulerphi(k))); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 08 2024, using Max Alekseyev's invphi.gp

A242563 a(n) = 2a(n-1) - a(n-3) + 2a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.

Original entry on oeis.org

0, 0, 2, 3, 6, 10, 21, 42, 86, 171, 342, 682, 1365, 2730, 5462, 10923, 21846, 43690, 87381, 174762, 349526, 699051, 1398102, 2796202, 5592405, 11184810, 22369622, 44739243, 89478486, 178956970, 357913941, 715827882, 1431655766, 2863311531, 5726623062, 11453246122

Offset: 0

Paul Curtz, May 17 2014

Generally, a(n) is an autosequence if its inverse binomial transform is (-1)^n*a(n). It is of the first kind if the main diagonal is 0's and the first two upper diagonals (just above the main one) are the same. It is of the second kind if the main diagonal is equal to the first upper diagonal multiplied by 2. If the first upper diagonal is an autosequence, the sequence is a super autosequence. Example: A113405. The first upper diagonal is A001045(n). Another super autosequence: 0, 0, 0 followed by A059633(n). The first upper diagonal is A000045(n).
Difference table of a(n):
   0,  0,  2, 3, 6, 10, 21, 42, ...
   0,  2,  1, 3, 4, 11, 21, 44, ...
   2, -1,  2, 1, 7, 10, 23, 41, ...
  -3,  3, -1, 6, 3, 13, 18, 45, ... .
This is an autosequence of the second kind. The main diagonal is 2*A001045(n) = A078008(n). More precisely it is a super autosequence, companion of A113405(n).
a(n+1) mod 10 = period 12: repeat 0, 2, 3, 6, 0, 1, 2, 6, 1, 2, 2, 5.
It is shifted A081374(n+1) mod 10 =
    period 12: repeat 1, 2, 2, 5, 0, 2, 3, 6, 0, 1, 2, 6.
a(n) mod 9 = period 18:
repeat 0, 0, 2, 3, 6, 1, 3, 6, 5, 0, 0, 7, 6, 3, 8, 6, 3, 4 = c(n).
c(n) + c(n+9) = 0, 0, 9, 9, 9, 9, 9, 9, 9.

			G.f. = 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 21*x^6 + 42*x^7 + 86*x^8 + ...

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

Cf. A000032, 1/(n+1), A164555/A027642 (all autosequences of 2nd kind). A007283, A175805.

a[n_] := (m = Mod[n, 6]; 1/3*(2^n + (-1)^n + 1/120*(m-6)*(m+1)*(m^3-29*m+40))); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 19 2014, a non-recursive formula, after Mathematica's RSolve *)
LinearRecurrence[{2, 0, -1, 2}, {0, 0, 2, 3},50] (* G. C. Greubel, Feb 21 2017 *)

concat([0,0], Vec(x^2*(x-2)/((x+1)*(2*x-1)*(x^2-x+1)) + O(x^100))) \\ Colin Barker, May 18 2014

a(n+3) = 3*2^n - a(n), a(0)=a(1)=0, a(2)=2.
a(n) = 2*A113405(n+1) - A113405(n).
a(n+1) = 2*a(n) + period 6: repeat 0, 2, -1, 0, -2, 1. a(0)=0.
a(n) = 2^n - A081374(n+1).
a(n+3) = a(n+1) + A130755(n).
G.f.: x^2*(x-2) / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, May 18 2014
a(n) = A024495(n) + A131531(n).
a(n+6) = a(n) + 21*2^n, a(0)=a(1)=0, a(2)=2, a(3)=3, a(4)=6, a(5)=10.
a(n) = A001045(n) - A092220(n).
a(n+12) = a(n) + 1365*2^n. First 12 values in the Data. (A024495(n+12) = A024495(n) + 1365*2^n).
a(3n)   = A132805(n) = 3*A015565(n).
a(3n+1) = A132804(n) = 6*A015565(n).
a(3n+2) = A132397(n) = 2*A082311(n).
a(n) = 1/3*((-1)^n - 2*cos((n*Pi)/3) + 2^n). - Alexander R. Povolotsky,  Jun 02 2014

More terms from Colin Barker, May 18 2014

cp's OEIS Frontend

A083322 a(n) = 2^n - A081374(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A082311 A Jacobsthal sequence trisection.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A081375 a(n) is the least number k such that A081373(k) = n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A242563 a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A242563 a(n) = 2a(n-1) - a(n-3) + 2a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.