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

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

0,1

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

Table of n, a(n) for n=0..36.

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

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.

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 *)

Prog

(PARI) a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022

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.

Sequence in context: A246665 A274260 A011397 * A339579 A081665 A131911

Adjacent sequences: A352689 A352690 A352691 * A352693 A352694 A352695

Keyword

sign,easy

Author

Paul Curtz, Mar 29 2022

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.

Status

approved