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 1-4 of 4 results.

A181532 a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2; a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 10, 18, 31, 55, 96, 169, 296, 520, 912, 1601, 2809, 4930, 8651, 15182, 26642, 46754, 82047, 143983, 252672, 443409, 778128, 1365520, 2396320, 4205249, 7379697, 12950466, 22726483, 39882198, 69988378, 122821042, 215535903, 378239143, 663763424
Offset: 0

Views

Author

Gary W. Adamson, Oct 28 2010

Keywords

Comments

Essentially the same as A060945: a(0)=0 and a(n)=A060945(n-1) for n>=1.
lim(n->infinity) a(n+1)/a(n) = A109134 = 1.754877666..., the square of the absolute value of one of the complex-valued roots of the characteristic polynomial. [R. J. Mathar, Nov 01 2010]
The Ze4 sums, see A180662 for the definition of these sums, of the ‘Races with Ties’ triangle A035317 lead to this sequence. [Johannes W. Meijer, Jul 20 2011]

Examples

			a(7) = 18 = a(6) + a(5) + a(3) = 10 + 6 + 2.
a(7) = 18 = (1 0, 2, 0, 2, 0, 3) dot (10, 6, 3, 2, 1, 1, 1) = (10 + 3 + 2 + 3).

Links

Crossrefs

All of A060945, A077930, A181532 are variations of the same sequence. - N. J. A. Sloane, Mar 04 2012

Programs

  • Mathematica
    LinearRecurrence[{1,1,0,1},{0,1,1,2},40] (* Harvey P. Dale, Jun 20 2015 *)

Formula

a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2; a(n) = a(n-1) + a(n-2) + a(n-4).
G.f.: x/(1-x-x^2-x^4). [Franklin T. Adams-Watters, Feb 25 2011]
a(n) = |A077930(n)| = ( |A056016(n+2)|-(-1)^n)/5. [R. J. Mathar, Oct 29 2010]
a(n) = A060945(n-1), n>1. [R. J. Mathar, Nov 03 2010]

Extensions

Values from a(9) on changed by R. J. Mathar, Oct 29 2010
Edited and a(0) added by Franklin T. Adams-Watters, Feb 25 2011

A108140 a(n) = 4*a(n-1) -3*a(n-2) -2*a(n-3) +a(n-4), n>8.

Original entry on oeis.org

1, 1, 1, 1, 0, 4, 17, 55, 161, 449, 1220, 3266, 8667, 22879, 60203, 158107, 414728, 1087064, 2848061, 7459703, 19535229, 51152749, 133933964, 350666854, 918095255, 2403665279, 6292975607, 16475382935, 43133369616, 112925043724
Offset: 0

Views

Author

Roger L. Bagula, Jun 05 2005

Keywords

Links

Crossrefs

Cf. A000045, A056015, A056016.

Programs

  • Mathematica
    F[1] = 1; F[2] = 1; F[3] = 1; F[4] = 1; F[n__] := F[n] = 4*F[n - 1] - 3*F[n - 2] - 2*F[n - 3] + F[n - 4] a = Table[Abs[F[n]], {n, 1, 50}]

Formula

a(n) = -A000045(n+2)/2 + A001906(n-1)/2, n>3. [Sep 28 2009]
G.f.: (-1+2*x^7-2*x^6-8*x^5-2*x^3+3*x)/((x^2+x-1)*(x^2-3*x+1)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009]

Extensions

Definition replaced by recurrence by the Associate Editors of the OEIS, Sep 28 2009

A108142 a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 1; a[5] = 1; a[6] = 1; for n >= 7, a[n] = 6*a[n - 1] - 5*a[n - 2] - 4*a[n - 3] - 3*a[ n - 4] + 2*a[n - 5] + a[n - 6]; then take absolute values.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 27, 151, 759, 3679, 17599, 83767, 397943, 1889059, 8964891, 42539855, 201849743, 957752095, 4544385823, 21562354767, 102309686479, 485441784803, 2303337053819, 10928934112423, 51855892302151
Offset: 1

Views

Author

Roger L. Bagula, Jun 05 2005

Keywords

Comments

The 2nd countdown sequence.

References

  • Roger Bagula, Factoring Double Fibonacci Sequences, 2000

Links

Crossrefs

Cf. A056015, A056016.

Programs

  • Mathematica
    F[1] = 1; F[2] = 1; F[3] = 1; F[4] = 1; F[5] = 1; F[6] = 1; F[n__] := F[n] = 6*F[n - 1] - 5*F[n - 2] - 4*F[n - 3] - 3*F[ n - 4] + 2*F[n - 5] + F[n - 6] a = Table[Abs[F[n]], {n, 1, 50}]
LinearRecurrence[{6,-5,-4,-3,2,1},{1,1,1,1,1,1,3,27,151,759,3679,17599},30] (* Harvey P. Dale, Apr 25 2018 *)

Extensions

Edited by N. J. A. Sloane, Jun 08 2007

A108143 a(n)= 5*a(n-1) -a(n-2) -a(n-3).

Original entry on oeis.org

1, 1, 1, 3, 13, 61, 289, 1371, 6505, 30865, 146449, 694875, 3297061, 15643981, 74227969, 352198803, 1671122065, 7929183553, 37622596897, 178512678867, 847011613885, 4018922793661, 19069089675553, 90479513970219
Offset: 0

Views

Author

Roger L. Bagula, Jun 05 2005

Keywords

References

  • Roger Bagula, Factoring Double Fibonacci Sequences, 2000

Links

Crossrefs

Cf. A056015, A056016.

Programs

  • Mathematica
    F[1] = 1; F[2] = 1; F[3] = 1; F[n__] := F[n] = 5*F[n - 1] - F[n - 2] - F[n - 3] a = Table[Abs[F[n]], {n, 1, 50}]
LinearRecurrence[{5,-1,-1},{1,1,1},30] (* Harvey P. Dale, Jan 21 2023 *)

Formula

G.f.: (1-4*x-3*x^2)/(1-5*x+x^2+x^3) [Sep 28 2009]

Extensions

Definition replaced by recurrence by the Associate Editors of the OEIS, Sep 28 2009
Showing 1-4 of 4 results.

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