cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A176476 Partial sums of A012814.

Original entry on oeis.org

0, 1, 6, 27, 113, 464, 1896, 7738, 31571, 128800, 525455, 2143647, 8745216, 35676948, 145547524, 593775045, 2422362078, 9882257735, 40315615409, 164471408184, 670976837020, 2737314167774, 11167134898975, 45557394660800, 185855747875875, 758216295635151
Offset: 0

Views

Author

Carmine Suriano, Apr 18 2010

Keywords

Comments

Old name was "a(n) is the minimum integer that can be expressed as the sum of n Padovan numbers (see A000931)".
Lim_{n -> infinity} a(n+1)/a(n) = p^5 = 4.0795956..., where p is the plastic constant (A060006).

Examples

			a(5) = A000931(2) + A000931(7) + A000931(12) + A000931(17) + A000931(22) + A000931(27) = 0 + 1 + 5 + 21 + 86 + 351 = 464.

Links

Crossrefs

Cf. A000931, A012814, A012855, A060006.

Programs

  • Mathematica
    LinearRecurrence[{6,-9,5,-1},{0,1,6,27},30] (* Harvey P. Dale, Feb 08 2025 *)
  • PARI
    a(n) = my(v=vector(n+1), u=[0,1,6,27]); for(k=1, n+1, v[k]=if(k<=4, u[k], 5*v[k-1] - 4*v[k-2] + v[k-3] + 1)); v[n+1] \\ Jianing Song, Feb 04 2019

Formula

a(n) = A012855(n+3) - 1. a(n) = 6*a(n-1) - 9*a(n-2) + 5*a(n-3) - a(n-4). - R. J. Mathar, Oct 18 2010
G.f.: x/(1 - 6*x + 9*x^2 - 5*x^3 + x^4). - Colin Barker, Feb 03 2012
From Jianing Song, Feb 04 2019: (Start)
a(n+3) = 5*a(n+2) - 4*a(n+1) + a(n) + 1.
a(n) = Sum_{k=0..n} A012814(k) = Sum_{k=0..n} A000931(5*k+2). (End)

Extensions

New name, more terms and a(0) = 0 prepended by Jianing Song, Feb 04 2019

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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

Crossrefs

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

Programs

  • 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 *)
  • 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

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

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

A012866 a(n+3) = 5*a(n+2)-4*a(n+1)+a(n).

Original entry on oeis.org

0, 1, 2, 6, 23, 93, 379, 1546, 6307, 25730, 104968, 428227, 1746993, 7127025, 29075380, 118615793, 483904470, 1974134558, 8053670703, 32855719753, 134038050511, 546821044246, 2230808738939, 9100797568222
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A012855.

Formula

G.f. x*(-1+3*x) / ( -1+5*x-4*x^2+x^3 ). - R. J. Mathar, Dec 22 2011
a(n) = A012814(n)-3*A012814(n-1). - R. J. Mathar, Sep 20 2012

A012880 a(n+3) = 5*a(n+2)-4*a(n+1)+a(n).

Original entry on oeis.org

0, 1, 3, 11, 44, 179, 730, 2978, 12149, 49563, 202197, 824882, 3365185, 13728594, 56007112, 228486369, 932131991, 3802721591, 15513566360, 63289077427, 258193843286, 1053326473082, 4297146069693, 17530618299423
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A012855.

Formula

G.f. x*(-1+2*x) / ( -1+5*x-4*x^2+x^3 ). a(n) = A012814(n)-2*A012814(n-1). - R. J. Mathar, Sep 20 2012

A012886 a(n+3) = 5*a(n+2)-4*a(n+1)+a(n).

Original entry on oeis.org

1, 2, 3, 8, 30, 121, 493, 2011, 8204, 33469, 136540, 557028, 2272449, 9270673, 37820597, 154292742, 629451995, 2567909604, 10476032782, 42737977489, 174353665921, 711292452431, 2901785575960, 11838111735997
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A012855.

Programs

  • Mathematica
    LinearRecurrence[{5,-4,1},{1,2,3},30] (* Harvey P. Dale, Sep 16 2023 *)

Formula

G.f. ( -1+3*x+3*x^2 ) / ( -1+5*x-4*x^2+x^3 ). a(n) = -3*A012814(n)+A012814(n+1) -3*A012814(n-1). - R. J. Mathar, Sep 20 2012
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.