cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A141355 The Jacobsthal sequence, dropping each third term.

Original entry on oeis.org

1, 1, 5, 11, 43, 85, 341, 683, 2731, 5461, 21845, 43691, 174763, 349525, 1398101, 2796203, 11184811, 22369621, 89478485, 178956971, 715827883, 1431655765, 5726623061, 11453246123, 45812984491, 91625968981, 366503875925, 733007751851
Offset: 0

Views

Author

Paul Curtz, Aug 03 2008

Keywords

Comments

A001045 after removal of the subsequence A132805.

Programs

  • Python
    def A141355(n): return ((1<<(n+1<<1)-(n+1>>1)-1)|1)//3 # Chai Wah Wu, Apr 19 2025

Formula

a(2n+1)-a(2n) = 6*A015565(n).
a(4n+1)=2a(4n)-1. a(4n+2)=4a(4n+1)+1. a(4n+3)=2a(4n+2)+1. a(4n+4)=4a(4n+3)-1.
a(2n)= A082311(n). a(2n+1) = A082365(n). - R. J. Mathar, Feb 23 2009
a(n)=7*a(n-2)+8*a(n-4). G.f.: (1+x-2*x^2+4*x^3)/((1-8*x^2)*(1+x^2)). - R. J. Mathar, Feb 23 2009

Extensions

Edited and extended by R. J. Mathar, Feb 23 2009

A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

Original entry on oeis.org

0, 1, 1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Jul 19 2017

Keywords

Comments

This is the inverse binomial transform of A157241.
Successive differences of A157241 begin:
0, 1, 3, 3, -5, -21, -21, 43, 171, 171, ... = A157241
1, 2, 0, -8, -16, 0, 64, 128, 0, -512, ... = A088138
1, -2, -8, -8, 16, 64, 64, -128, -512, -512, ... = A138230
-3, -6, 0, 24, 48, 0, -192, -384, 0, 1536, ...
-3, 6, 24, 24, -48, -192, -192, 384, 1536, 1536, ...
9, 18, 0, -72, -144, 0, 576, 1152, 0, -4608, ...
9, -18, -72 -72, 144, 576, 576, -1152, -4608, -4608, ...
...
a(n) is the n-th term of the first column.
Successive differences of a(n) begin:
0, 1, 1, -3, -3, 9, 9, -27, -27, 81, ...
1, 0, -4, 0, 12, 0, -36, 0, 108, 0, ...
-1, -4, 4, 12, -12, -36, 36, 108, -108, -324, ...
-3, 8, 8, -24, -24, 72, 72, -216, -216, 648, ...
11, 0, -32, 0, 96, 0, -288, 0, 864, 0, ...
-11, -32, 32, 96, -96, -288, 288, 864, -864, -2592, ...
-21, 64, 64, -192, -192, 576, 576, -1728, -1728, 5184, ...
85, 0, -256, 0, 768, 0, -2304, 0, 6912, 0, ...
...
First column appears to be a subsequence of Jacobsthal numbers A001045 (the trisection A082311 is missing), second column is A104538, and third column is A137717.
a(n) = A128019(n-2) for n > 2. - Georg Fischer, Oct 23 2018

Links

Crossrefs

Cf. A001045, A082311, A088138, A104538, A108411, A128019, A137717, A138230, A140429, A157241.

Programs

  • Mathematica
    Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
(* or, computation from b = A157241 : *)
b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]
  • PARI
    concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ Felix Fröhlich, Oct 23 2018

Formula

a(n) = -3*a(n-2) for n > 2.
E.g.f.: (1 - cos(sqrt(3)*x) + sqrt(3)*sin(sqrt(3)*x))/3. - Stefano Spezia, Jul 15 2024

A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

Original entry on oeis.org

4, -3, 5, -1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119, 22906492249
Offset: 0

Views

Author

Paul Curtz, Mar 29 2022

Keywords

Comments

Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
4, -3, 5, -1, 9, 7, 25, ...
-7, 8, -6, 10, -2, 18, 14, 50, ...
15, -14, 16, -12, 20, -4, 36, 28, 100, ...
-29, 30, -28, 32, -24, 40, -8, 72, 56, 200, ...
59, -58, 60, -56, 64, -48, 80, -16, 144, 112, 400, ...
...
The diagonals are given by D(n,n+k) = a(k)*2^n.
D(n,1) = -(-1)^n* A340627(n).
a(n) - a(n) = 0, 0, 0, 0, 0, ... (trivially)
a(n+1) + a(n) = 1, 2, 4, 8, 16, ... = 2^n (by definition)
a(n+2) - a(n) = 1, 2, 4, 8, 16, ... = 2^n
a(n+3) + a(n) = 3, 6, 12, 24, 48, ... = 2^n*3
a(n+4) - a(n) = 5, 10, 20, 40, 80, ... = 2^n*5
a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
(...)
This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
Sums of antidiagonals are A045883(n).
Main diagonal: A192382(n).
First upper diagonal: A054881(n+1).
First subdiagonal: A003683(n+1).
Second subdiagonal: A246036(n).
Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
4, 3, 5, 1, 9, -7, 25, -39, ... = c(n)
-1, 2, -4, 8, -16, 32, -64, 128, ... = -A122803(n)
3, -6, 12, -24, 48, -96, 192, -384, ... =
-9, 18, -36, 72, -144, 288, -576, 1152, ...
27, -54, 108, -216, 432, -864, 1728, -3456, ...
...
The first subdiagonal is -A000400(n). The second is A169604(n).

Links

Crossrefs

If a(0) = k then A001045 (k=0), A078008 (k=1), A140966 (k=2), A154879 (k=3), this sequence (k=4).
Essentially the same as A115335.
Cf. A000079, A002450, A340627.
Cf. A020714, A175805.
Cf. A045883, A192382, A003683, A246036.
Cf. A054881, A000400, A169604, A122803.
Cf. A132805, A082311, A082365.
Cf. A024495, A132804, A163868.

Programs

  • Maple
    a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # Peter Luschny, Mar 29 2022
A352691 := proc(n)
    (11*(-1)^n + 2^n)/3
end proc: # R. J. Mathar, Apr 26 2022
  • Mathematica
    LinearRecurrence[{1, 2}, {4, -3}, 40] (* Amiram Eldar, Mar 29 2022 *)
  • PARI
    a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022

Formula

abs(a(n)) = A115335(n-1) for n >= 1.
a(3*n) - (-1)^n*4 = A132805(n).
a(3*n+1) + (-1)^n*4 = A082311(n).
a(3*n+2) - (-1)^n*4 = A082365(n).
From Thomas Scheuerle, Mar 29 2022: (Start)
G.f.: (-4 + 7*x)/(-1 + x + 2*x^2).
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(m + 2*n-k) = a(m)*2^n.
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(1 + n-k) = -(-1)^n*A340627(n).
a(n) = (11*(-1)^n + 2^n)/3.
a(n + 2*m) = a(n) + A002450(m)*2^n.
a(2*n) = A192382(n+1) + (-1)^n*a(n).
a(n) = ( A045883(n) - Sum_{k=0..n-1}(-1)^k*a(k) )/n, for n > 0. (End)
a(n) = A001045(n) + 4*(-1)^n.
a(n+1) = 2*a(n) -11*(-1)^n.
a(n+2) = a(n) + 2^n.
a(n+4) = a(n) + A020714(n).
a(n+6) = a(n) + A175805(n).
a(2*n) = A163868(n).
a(2*n+1) = (2^(2*n+1) - 11)/3.

Extensions

Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022
Edited by M. F. Hasler, Apr 26 2022.
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