A137212 -id:A137212 - cp's OEIS frontend

A137213 First differences of A137212.

0, 1, 4, 15, 52, 173, 560, 1779, 5576, 17305, 53308, 163287, 497980, 1513541, 4587944, 13878075, 41910032, 126395953, 380795380, 1146267039, 3448170436, 10367130845, 31156000928, 93599839107, 281117798360, 844121793481

Offset: 0

Paul Curtz, Mar 06 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,-3).

Cf. A000129, A132894, A137212.

[n le 3 select (n-1)^2 else 5*Self(n-1) -5*Self(n-2) -3*Self(n-3): n in [1..31]]; // G. C. Greubel, Jan 05 2022

LinearRecurrence[{5,-5,-3},{0,1,4},30] (* Harvey P. Dale, Apr 18 2019 *)

[3^n - lucas_number1(n+1,2,-1) for n in (0..30)] # G. C. Greubel, Jan 05 2022

From R. J. Mathar, Mar 17 2008: (Start)
O.g.f.: 1/(1-3*x) - 1/(1-2*x-x^2) = x*(1-x)/( (1-3*x)*(1-2*x-x^2) ).
a(n) = 3^n - A000129(n+1). (End)

More terms from R. J. Mathar, Mar 17 2008

A193519 a(n) = (2/3)Sum_{i=1..n-1} A000129(i)3^(n-i).

Original entry on oeis.org

0, 0, 2, 10, 40, 144, 490, 1610, 5168, 16320, 50930, 157546, 484120, 1480080, 4507162, 13683050, 41439200, 125259264, 378051170, 1139641930, 3432176008, 10328516880, 31062778570, 93374780426, 280574458640, 842810055360, 2531053642322, 7599494558890, 22813774416760, 68478238362384

Offset: 0

N. J. A. Sloane, Jul 29 2011

Number of ternary words of length n on {0,1,2} containing the subwords 02 or 20. - Philippe Deléham, Apr 27 2012

			a(3) = 10 because among the 3^3 = 27 ternary words of length 3 only 10, namely 002, 020, 021, 022, 102, 120, 200, 201, 202, 220 contain the subwords 02 or 20. - _Philippe Deléham_, Apr 27 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see Eq. (19)).
Index entries for linear recurrences with constant coefficients, signature (5,-5,-3).

Cf. A000129, A002203, A137212.

[n le 3 select 2*Floor((n-1)/2) else 5*Self(n-1) -5*Self(n-2) -3*Self(n-3): n in [1..31]]; // G. C. Greubel, Jan 05 2022

Table[(2*3^n - LucasL[n+1, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jan 05 2022 *)

[(2*3^n - lucas_number2(n+1, 2, -1))/2 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 2*A137212(n).
G.f.: 2*x^2/((1-3*x)*(1-2*x-x^2)). - Philippe Deléham, Apr 27 2012
a(n) = 5*a(n-1) - 5*a(n-2) - 3*a(n-3), a(0) = a(1) = 0, a(2) = 2. - Philippe Deléham, Apr 27 2012
a(n) = (1/2)*(2*3^n - A002203(n+1)). - G. C. Greubel, Jan 05 2022

cp's OEIS Frontend

A137213 First differences of A137212.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A193519 a(n) = (2/3)Sum_{i=1..n-1} A000129(i)3^(n-i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

cp's OEIS Frontend

A137213 First differences of A137212.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A193519 a(n) = (2/3)*Sum_{i=1..n-1} A000129(i)*3^(n-i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A193519 a(n) = (2/3)Sum_{i=1..n-1} A000129(i)3^(n-i).