A097073 -id:A097073 - cp's OEIS frontend

A140505 Second differences of Jacobsthal sequence A001045, pairs with even and odd indices swapped.

2, -1, 4, 0, 12, 4, 44, 20, 172, 84, 684, 340, 2732, 1364, 10924, 5460, 43692, 21844, 174764, 87380, 699052, 349524, 2796204, 1398100, 11184812, 5592404, 44739244, 22369620, 178956972, 89478484, 715827884, 357913940, 2863311532, 1431655764, 11453246124

Offset: 0

Paul Curtz, Jun 30 2008

The second differences are -1, 2, 0, 4, 4, 12, 20, 44, ... (-1)^(n+1)*A084247(n), essentially A097073, which are listed here with -1 <=> 2, 0 <=> 4 etc. swapped in pairs.

a(n) = 4*A092808(n-2), n>1.
a(n+1) - 2a(n) = (-1)^n*A140504(n).
O.g.f.: (2+x-5x^2)/[(1+x)(1-2x)(1+2x)]. - R. J. Mathar, Jul 08 2008

Edited by R. J. Mathar, Jul 08 2008

A166977 Jacobsthal-Lucas numbers A014551, except a(0) = 0.

Original entry on oeis.org

0, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 2147483647

Offset: 0

Paul Curtz, Oct 26 2009

The sequence (-1)^n*a(n) is the inverse binomial transform of A166956.

The main diagonal of the table of a(n) and its higher differences in successive rows is 0,4,8,16,32,.. , 4*A131577(n).

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

Join[{0, 1}, LinearRecurrence[{1, 2}, {5, 7}, 50]] (* or *) Table[2^n + (-1)^n, {n,1,25}] (* G. C. Greubel, May 30 2016 *)

a(n) = A014551(n), n>0.
a(n) - A001045(n) = A097073(n), n>0.
a(n) - A001045(n) = 4*A001045(n-1).
a(n) = a(n-1) + 2*a(n-2), n>2.
G.f.: x*(1 + 4*x)/((1+x) * (1-2*x)).
a(n) = (-1)^n + 2^n for n>0. - Colin Barker, Jun 06 2012
E.g.f.: exp(2*x) + exp(-x) - 2. - G. C. Greubel, May 30 2016

Edited and extended by R. J. Mathar, Mar 14 2010

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.

Original entry on oeis.org

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 *)

A322783 a(n) = 1 - n + (2^(n+2) - (-1)^n)/3.

Original entry on oeis.org

2, 3, 4, 9, 18, 39, 80, 165, 334, 675, 1356, 2721, 5450, 10911, 21832, 43677, 87366, 174747, 349508, 699033, 1398082, 2796183, 5592384, 11184789, 22369598, 44739219, 89478460, 178956945, 357913914, 715827855, 1431655736

Offset: 0

Paul Curtz, Dec 26 2018

a(n) mod 10 = period 20: repeat [2, 3, 4, 9, 8, 9, 0, 5, 4, 5, 6, 1, 0, 1, 2, 7, 6, 7, 8, 3] = disordered [0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9].

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A001045, A004442(n-2), A097073, A097074.

Vec((2 - 3*x - 3*x^2 + 6*x^3) / ((1 - x)^2*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 26 2018

a(n+1) - 2*(n) = -1, -2, 1, 0, 3, 2, 5, 4, ..., n >= 0.
a(n+1) - a(n) = A097074(n).
a(n+2) - 2*a(n+1) + a(n) = A097073(n+1).
From Colin Barker, Dec 26 2018: (Start)
G.f.: (2 - 3*x - 3*x^2 + 6*x^3) / ((1 - x)^2*(1 + x)*(1 - 2*x)).
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4) for n > 3.
(End)

cp's OEIS Frontend

A140505 Second differences of Jacobsthal sequence A001045, pairs with even and odd indices swapped.

Views

Author

Keywords

Comments

Formula

Extensions

A166977 Jacobsthal-Lucas numbers A014551, except a(0) = 0.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A322783 a(n) = 1 - n + (2^(n+2) - (-1)^n)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula