A012855 a(0) = 0, a(1) = 1, a(2) = 1; thereafter a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3).

0, 1, 1, 1, 2, 7, 28, 114, 465, 1897, 7739, 31572, 128801, 525456, 2143648, 8745217, 35676949, 145547525, 593775046, 2422362079, 9882257736, 40315615410, 164471408185, 670976837021, 2737314167775, 11167134898976

Offset

0,5

Comments

Old name was "Take every 5th term of Padovan sequence A000931".

Lim_{n -> infinity} a(n+1)/a(n) = p^5 = 4.0795956..., where p is the plastic constant (A060006). - Jianing Song, Feb 04 2019

Links

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (5, -4, 1).

Formula

a(n) = A000931(5*n-12) for n >= 3. - Alois P. Heinz, Feb 04 2019

G.f. (4x^2 - x)/(x^3 - 4x^2 + 5x - 1). For n > 2, a(n) = 1 + Sum_{k=0..n-3} A012814(k). - Ralf Stephan, Jan 15 2004

a(n) = 1 + A176476(n-3) = 1 + Sum_{k=0..n-3} A000931(5*k+2) for n >= 3. - Jianing Song, Feb 04 2019

Maple

A012855 := proc(n, A, B, C) option remember; if n = 0 then A elif n = 1 then B elif n = 2 then C else 5*procname(n-1, A, B, C)-4*procname(n-2, A, B, C)+procname(n-3, A, B, C); fi; end; [ seq(A012855(i, 0, 1, 1), i = 0..40) ]; # R. J. Mathar, Dec 30 2011

Mathematica

CoefficientList[Series[(4x^2-x)/(x^3-4x^2+5x-1), {x, 0, 40}], x] (* or *) LinearRecurrence[{5, -4, 1}, {0, 1, 1}, 40] (* Harvey P. Dale, Mar 28 2013 *)

Prog

(PARI) a(n) = my(v=vector(n+1), u=[0, 1, 1]); for(k=1, n+1, v[k]=if(k<=3, u[k], 5*v[k-1] - 4*v[k-2] + v[k-3])); v[n+1] \\ Jianing Song, Feb 04 2019

Crossrefs

Cf. A000931, A012814, A012864, A060006, A176476.

Sequence in context: A068944 A215143 A289158 * A224066 A150646 A364145

Adjacent sequences: A012852 A012853 A012854 * A012856 A012857 A012858

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by N. J. A. Sloane, Feb 06 2019 at the suggestion of Jianing Song, replacing imprecise definition with formula from Harvey P. Dale, Mar 28 2013

Status

approved