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

Author

Paul Curtz, Mar 29 2022

Keywords

• sign
• easy

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

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

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.