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

1,2

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).

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

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 *)

Prog

(Magma) [5*2^n/3-2*(-1)^n/3-3: n in [1..40]]; // Vincenzo Librandi, Aug 05 2011
(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
(GAP) List([1..40], n-> (5*2^n - 2*(-1)^n - 9)/3); # G. C. Greubel, Dec 01 2019

Crossrefs

Sequence in context: A145473 A335677 A248348 * A128928 A289888 A145477

Adjacent sequences: A173075 A173076 A173077 * A173079 A173080 A173081

Keyword

nonn,easy

Author

Paul Curtz, Feb 09 2010

Status

approved