A173078 a(n) = (5*2^n - 2*(-1)^n - 9)/3.

1, 3, 11, 23, 51, 103, 211, 423, 851, 1703, 3411, 6823, 13651, 27303, 54611, 109223, 218451, 436903, 873811, 1747623, 3495251, 6990503, 13981011, 27962023, 55924051, 111848103, 223696211, 447392423, 894784851, 1789569703, 3579139411

Offset: 1

Author

Paul Curtz, Feb 09 2010

Keywords

• nonn
• easy

Comments

The sequence and higher-order differences in subsequent rows are

1, 3, 11, 23, 51, 103, 211, 423, 851, 1703, 3411, 6823, 13651

2, 8, 12, 28, 52, 108, 212, 428, 852, 1708, 3412, 6828, 13652

6, 4, 16, 24, 56, 104, 216, 424, 856, 1704, 3416, 6824, 13656

-2, 12, 8, 32, 48, 112, 208, 432, 848, 1712, 3408, 6832, 13648

14, -4, 24, 16, 64, 96, 224, 416, 864, 1696, 3424, 6816, 13664

-18, 28, -8, 48, 32, 128, 192, 448, 832, 1728, 3392, 6848, 1363

46, -36, 56, -16, 96, 64, 256, 384, 896, 1664, 3456, 6784, 1369

The main diagonal 1,8,16,... is essentially A000079.

A subdiagonal is 2, 4, 8, 16, ... A155559.

Other diagonals are 3, 12, 24, 48, ... = 3*A151821, 6, 12, 24, ... = A082505 and -2, -4, -8, -16, ..., a negated variant of A171449.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Programs

GAP

List([1..40], n-> (5*2^n - 2*(-1)^n - 9)/3); # G. C. Greubel, Dec 01 2019

Magma

[5*2^n/3-2*(-1)^n/3-3: n in [1..40]]; // Vincenzo Librandi, Aug 05 2011

Maple

seq( (5*2^n -2*(-1)^n -9)/3, n=1..40); # G. C. Greubel, Dec 01 2019

Mathematica

LinearRecurrence[{2,1,-2},{1,3,11},40] (* Harvey P. Dale, Oct 01 2018 *)

PARI

vector(40, n, (5*2^n - 2*(-1)^n - 9)/3) \\ G. C. Greubel, Dec 01 2019

Sage

[(5*2^n - 2*(-1)^n - 9)/3 for n in (1..40)] # G. C. Greubel, Dec 01 2019

Formula

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).

a(n+1) - 2*a(n) = A010686(n-1).

a(n) = A084214(n+1) - 3.

G.f.: x*(1 + x + 4*x^2) / ( (1-x)*(1-2*x)*(1+x) ).

a(2n+3) - a(2n+1) = 10*A000302(n).

E.g.f.: (-2*exp(-x) + 6 - 9*exp(x) + 5*exp(2*x))/3. - G. C. Greubel, Dec 01 2019