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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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A340660 A000079 is the first row. For the second row, subtract A001045. For the third row, subtract A001045 from the second one, etc. The resulting array is read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 1, 0, 3, 8, 1, -1, 2, 5, 16, 1, -2, 1, 2, 11, 32, 1, -3, 0, -1, 6, 21, 64, 1, -4, -1, -4, 1, 10, 43, 128, 1, -5, -2, -7, -4, -1, 22, 85, 256, 1, -6, -3, -10, -9, -12, 1, 42, 171, 512, 1, -7, -4, -13, -14, -23, -20, -1, 86, 341, 1024
Offset: 0

Views

Author

Paul Curtz, Jan 15 2021

Keywords

Comments

Every row has the signature (1,2).
(Among consequences: a(n) read by antidiagonals is
1,
1, 2,
1, 1, 4,
1, 0, 3, 8,
1, -1, 2, 5, 16
1, -2, 1, 2, 11, 32,
1, -3, 0, -1, 6, 21, 64,
... .
The row sums and their first two difference table terms are
1, 3, 6, 12, 23, 45, 88, ... = A086445(n+1) - 1
2, 3, 6, 11, 22, 43, 86, ... = A005578(n+2)
1, 3, 5, 11, 21, 43, 85, ... = A001045(n+2).
The antidiagonal sums are
b(n) = 1, 1, 3, 2, 5, 3, 9, 4, 15, 5, 27, 6, 49, 7, ... .)

Examples

			Square array:
1,  2,  4,   8,  16,  32,  64,  128, ... = A000079(n)
1,  1,  3,   5,  11,  21,  43,   85, ... = A001045(n+1)
1,  0,  2,   2,   6,  10,  22,   42, ... = A078008(n)
1, -1,  1,  -1,   1,  -1,   1,   -1, ... = A033999(n)
1, -2,  0,  -4,  -4, -12, -20,  -44, ... = -A084247(n)
1, -3, -1,  -7,  -9, -23, -41,  -87, ... = (-1)^n*A140966(n+1)
1, -4, -2, -10, -14, -34, -62, -130, ... = -A135440(n)
1, -5, -3, -13, -19, -45, -83, -173, ... = -A155980(n+3) or -A171382(n+1)
...

Crossrefs

Cf. A000079, A001045, A033999, A078008, A084247, A135440, A140966, A155980, A171382.
Cf. A005578, A086445.

Programs

  • Maple
    A:= (n, k)-> (<<0|1>, <2|1>>^k. <<1, 2-n>>)[1$2]:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 21 2021
  • Mathematica
    A340660[m_, n_] := LinearRecurrence[{1, 2}, {1, m}, {n}]; Table[Reverse[Table[A340660[m, n + m - 2] // First, {m, 2, -n + 3, -1}]], {n, 1, 11}] // Flatten (* Robert P. P. McKone, Jan 28 2021 *)
  • PARI
    T(n, k) = 2^k - n*(2^k - (-1)^k)/3;
matrix(10,10,n,k,T(n-1,k-1)) \\ Michel Marcus, Jan 19 2021

Formula

A(n,k) = 2^k - n*round(2^k/3).

A366987 Triangle read by rows: T(n, k) = -(2^(n - k)*(-1)^n + 2^k + (-1)^k)/3.

Original entry on oeis.org

-1, 0, 0, -2, -1, -2, 2, 1, -1, -2, -6, -3, -3, -3, -6, 10, 5, 1, -1, -5, -10, -22, -11, -7, -5, -7, -11, -22, 42, 21, 9, 3, -3, -9, -21, -42, -86, -43, -23, -13, -11, -13, -23, -43, -86, 170, 85, 41, 19, 5, -5, -19, -41, -85, -170, -342, -171, -87, -45, -27, -21, -27, -45, -87, -171, -342
Offset: 0

Views

Author

Paul Curtz and Thomas Scheuerle, Oct 31 2023

Keywords

Examples

			Triangle T(n, k) starts:
   -1
    0   0
   -2  -1  -2
    2   1  -1  -2
   -6  -3  -3  -3  -6
   10   5   1  -1  -5 -10
  -22 -11  -7  -5  -7 -11 -22
   42  21   9   3  -3  -9 -21 -42
   ...
Note the symmetrical distribution of the absolute values of the terms in each row.

Links

Crossrefs

Rows sums: -A282577(n+2), if the conjectures from Colin Barker in A282577 are true.
First column: -(-1)^n * A078008(n).
Second column: (-1)^n * A001045(n).
Third column: -A140966(n).
Fourth column: (-1)^n * A155980(n+2).
Cf. A000079, A020989, A048573, A053088, A059570, A077898, A140966.

Programs

  • Maple
    T := (n, k) -> -(2^(n-k)*(-1)^n + 2^k + (-1)^k)/3:
seq(seq(T(n, k), k = 0..n), n = 0..10);  # Peter Luschny, Nov 02 2023
  • Mathematica
    A366987row[n_]:=Table[-(2^(n-k)(-1)^n+2^k+(-1)^k)/3,{k,0,n}];Array[A366987row,15,0] (* Paolo Xausa, Nov 07 2023 *)
  • PARI
    T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ Thomas Scheuerle, Nov 01 2023

Formula

T(n, 0) = -((-2)^n + 2)/3.
T(n, k+1) - T(n, k) = T(n-1, k) + (-1)^k.
T(2*n+1, n) = A001045(n).
T(2*n+1, n+1) = -A001045(n).
T(2*n, n+1) = -A048573(n-1), for n > 0.
Note that the definition of T extends to negative parameters:
T(2*n-2, n-1) = -A001045(n).
-2^n*Sum_{k=0..n} (-1)^k*T(-n, -k) = A059570(n+1).
-4^n*Sum_{k=0..2*n} T(-2*n, -k) = A020989(n).
-Sum_{k=0..n} (-1)^k*T(n, k) = A077898(n). See also A053088.
Sum_{k = 0..2*n} |T(2*n, k)| = (4^(n+1) - 1)/3.
Sum_{k = 0..2*n+1} |T(2*n+1, k)| = (1 + (-1)^n - 2^(2 + n) + 2^(1 + 2*n))/3.
G.f.: (-1 - x + x*y)/((1 - x)*(1 + 2*x)*(1 + x*y)*(1 - 2*x*y)). - Stefano Spezia, Nov 03 2023

Extensions

a(42) corrected by Paolo Xausa, Nov 07 2023

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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