A062092 -id:A062092 - cp's OEIS frontend

A321373 Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.

2, 2, -1, 2, 0, 3, 2, 1, 4, 1, 2, 2, 5, 4, 7, 2, 3, 6, 7, 12, 9, 2, 4, 7, 10, 17, 20, 23, 2, 5, 8, 13, 22, 31, 44, 41, 2, 6, 9, 16, 27, 42, 65, 84, 87, 2, 7, 10, 19, 32, 53, 86, 127, 172, 169, 2, 8, 11, 22, 37, 64, 107, 170, 257, 340, 343

Offset: 0

Paul Curtz, Nov 08 2018

Array:
2, -1,  3,  1,  7,  9,  23,  41,  87, ... = (-1)^n*A140966(n)
2,  0,  4,  4, 12, 20,  44,  84, 172, ... = abs(A084247(n+1))
2,  1,  5,  7, 17, 31,  65, 127, 257, ... = A014551(n)
2,  2,  6, 10, 22, 42,  86, 170, 342, ... = A078008(n+2) = A014113(n+1)
2,  3,  7, 13, 27, 53, 107, 213, 427, ... = A048573(n)
2,  4,  8, 16, 32, 64, 128, 256, 512, ... = A000079(n+1)
2,  5,  9, 19, 37, 75, 149, 299, 597, ... = A062092(n)
2,  6, 10, 22, 42, 86, 170, 342, 682, ... = A078008(n+3) = A014113(n+2).
T(n+1,k) = (-1)^k*A140966(k) + (n+1)*A001045(k).
Every row T(n+1,k) has the signature (1,2).
T(0,k) = 2, -2, 2, -2, ... = (-1)^n*2.
T(n+1,k) - T(0,k) = (n+1)*A001045(n).
5*A001045(n) is not in the OEIS.

			Triangle a(n):
  2;
  2, -1;
  2,  0,  3;
  2,  1,  4,  1;
  2,  2,  5,  4,  7;
  2,  3,  6,  7, 12,  9;
  2,  4,  7, 10, 17, 20, 23;
  etc.
Row sums: 2, 1, 5, 8, 20, 39, 83, 166, 338, 677, 1361, 2724, ... = b(n+2).
With b(0) = 2 and b(1) = 0, b(n) = b(n-1) + 2*b(n-2)  + n - 4, n > 1.
b(n) = A001045(n) - A097065(n-1).
b(n) = b(n-2) + A000225(n-2).

Cf. A000079, A001045, A014113, A014551, A048573, A062092, A078008, A084247, A092297, A097073, A140360, A140966.

Cf. A000225, A097065.

T[_, 0] = 2;
T[0, k_] := (2^k + 5(-1)^k)/3;
T[n_ /; n>0, k_ /; k>0] := T[n, k] = T[n-1, k] + (2^k + (-1)^(k+1))/3;
T[, ] = 0;
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

A328881 a(n+3) = 2^n - a(n), a(0)=a(2)=1, a(1)=0 for n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 8, 14, 29, 56, 114, 227, 456, 910, 1821, 3640, 7282, 14563, 29128, 58254, 116509, 233016, 466034, 932067, 1864136, 3728270, 7456541, 14913080, 29826162, 59652323, 119304648, 238609294, 477218589, 954437176, 1908874354, 3817748707

Offset: 0

Paul Curtz, Oct 29 2019

The array of a(n) and its repeated differences:
    1,   0,   1,   0,   2,   3,   8,  14, ...
   -1,   1,  -1,   2,   1,   5,   6,  15, ...
    2,  -2,   3,  -1,   4,   1,   9,  12, ...
   -4,   5,  -4,   5,  -3,   8,   3,  19, ...
    9,  -9,   9,  -8,  11,  -5,  16,   5, ...
  -18,  18, -17,  19, -16,  21, -11,  32, ...
   36, -35,  36, -35,  37, -32,  43, -21, ...
  -71,  71, -71,  72, -69,  75, -64,  85, ...
  ...
The recurrence is the same for every row.
From Jean-François Alcover, Nov 28 2019: (Start)
It appears that, when odd, a(n) is never a multiple of 5.
Main and 3rd upper diagonals of the difference array are A001045 (Jacobsthal numbers); first upper diagonal is negated A001045; second upper diagonal is A000079 (powers of 2); 4th upper diagonal is A062092.
(End)

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

Cf. A024495, A062092, A131714.

Cf. A015565, A092220, A113405.

a[0] = a[2] = 1; a[1] = 0; a[n_] := a[n] = 2^(n - 3) - a[n - 3]; Array[a, 36, 0] (* Amiram Eldar, Nov 06 2019 *)

Vec((1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Oct 29 2019

a(n+1) - 2*a(n) = period 6: repeat [-2, 1, -2, 2, -1, 2].
a(n+12) - a(n) = 455*2^n.
From Colin Barker, Oct 29 2019: (Start)
G.f.: (1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)).
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.
(End)
a(n+2) - a(n) = A024495(n).
a(n+6) - a(n) = 7*2^n.
a(n+9) + a(n) = 57*2^n.
a(n) = A113405(n) + A092220(n+5).
9*a(n) = 2^n + 5*(-1)^n + 3*A010892(n). - R. J. Mathar, Nov 28 2019

A360033 Table T(n,k), n >= 1 and k >= 0, read by antidiagonals, related to Jacobsthal numbers A001045.

Original entry on oeis.org

1, 2, 1, 3, 3, 3, 4, 5, 7, 5, 5, 7, 11, 13, 11, 6, 9, 15, 21, 27, 21, 7, 11, 19, 29, 43, 53, 43, 8, 13, 23, 37, 59, 85, 107, 85, 9, 15, 27, 45, 75, 117, 171, 213, 171, 10, 17, 31, 53, 91, 149, 235, 341, 427, 341, 11, 19, 35, 61, 107, 181, 299, 469

Offset: 1

Philippe Deléham, Jan 22 2023

			The array T(n,k), for n <= 1 and k >= 0, begins:
n = 1: 1,  1,  3,  5,  11,  21,  43, ... -> A001045(k+1)
n = 2: 2,  3,  7, 13,  27,  53, 107, ... -> A048573(k)
n = 3: 3,  5, 11, 21,  43,  85, 171, ... -> A001045(k+3)
n = 4: 4,  7, 15, 29,  59, 117, 235, ... -> ?
n = 5: 5,  9, 19, 37,  75, 149, 299, ... -> A062092(k+1)
n = 6: 6, 11, 23, 45,  91, 181, 363, ... -> ?
n = 7: 7, 13, 27, 53, 107, 213, 427, ... -> A048573(k+2)

Cf. A001045, A048573, A062092.

Columns: A000027, A005408, A004767, A004770, A106839 for k = 0, 1, 2, 3, 4.

T(n,k) = T(1,k) + (n-1)*2^k.
T(n,k) = 2*T(n, k-1) + (-1)^k.
T(n,k) = T(n-1,k) + 2^k.
T(n,k) = 2^k * n - A001045(k).
T(n,k) = T(n,k-1) +2*T(n,k-2).

cp's OEIS Frontend

A321373 Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.

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A328881 a(n+3) = 2^n - a(n), a(0)=a(2)=1, a(1)=0 for n >= 0.

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A360033 Table T(n,k), n >= 1 and k >= 0, read by antidiagonals, related to Jacobsthal numbers A001045.

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