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.

Previous Showing 11-15 of 15 results.

A187362 Pell trisection: Pell(3*n+2), n >= 0.

Original entry on oeis.org

2, 29, 408, 5741, 80782, 1136689, 15994428, 225058681, 3166815962, 44560482149, 627013566048, 8822750406821, 124145519261542, 1746860020068409, 24580185800219268, 345869461223138161, 4866752642924153522, 68480406462161287469, 963592443113182178088, 13558774610046711780701
Offset: 0

Views

Author

Wolfdieter Lang, Mar 09 2011

Keywords

Comments

For the general trisection of a sequence see a Wolfdieter Lang comment under A187357.

Links

Crossrefs

Cf. A142588 (Pell(3n)), A187361 (Pell(3n+1)).

Programs

  • Mathematica
    Table[Fibonacci[3n + 2, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
  • PARI
    Vec((2+x)/(1-14*x-x^2) + O(x^20)) \\ Colin Barker, Jan 25 2016

Formula

a(n) = Pell(3*n+2), n >= 0, with Pell(n):=A000129(n).
O.g.f.: (2+x)/(1-14*x-x^2).
a(n) = 14*a(n-1) + a(n-2), a(-1)=1, a(0)=2.
a(n) = (((7-5*sqrt(2))^n*(-3+2*sqrt(2)) + (3+2*sqrt(2))*(7+5*sqrt(2))^n)) / (2*sqrt(2)). - Colin Barker, Jan 25 2016

A187442 A trisection of A001405 (central binomial coefficients): binomial(3*n,floor(3*n/2)), n>=0.

Original entry on oeis.org

1, 3, 20, 126, 924, 6435, 48620, 352716, 2704156, 20058300, 155117520, 1166803110, 9075135300, 68923264410, 538257874440, 4116715363800, 32247603683100, 247959266474052, 1946939425648112, 15033633249770520, 118264581564861424
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For the trisection of sequences see a comment and a reference under A187357.

Crossrefs

A187443 binomial(3*n+1,floor((3*n+1)/2)),
A187444 binomial(3*n+2,floor((3*n+2)/2))/2.

Programs

  • Mathematica
    Table[Binomial[3n,Floor[(3n)/2]],{n,0,20}] (* Harvey P. Dale, Dec 23 2012 *)

Formula

a(n) = binomial(3*n,floor(3*n/2)), n>=0.
O.g.f.: G0(x^2) + 3*x*G2(x^2) with G0(x) and G2(x) the o.g.f.s of A187363 and A187365, respectively.

A187443 A trisection of A001405 (central binomial coefficients): binomial(3n+1,floor((3n+1)/2)), n>=0.

Original entry on oeis.org

1, 6, 35, 252, 1716, 12870, 92378, 705432, 5200300, 40116600, 300540195, 2333606220, 17672631900, 137846528820, 1052049481860, 8233430727600, 63205303218876, 495918532948104, 3824345300380220, 30067266499541040, 232714176627630544, 1832624140942590534, 14226520737620288370, 112186277816662845432, 873065282167813104916
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For trisection of sequences see a comment and a reference under A187357.

Crossrefs

Cf. A187442: binomial(3n,floor(3n/2)), A187444: binomial(3n+2,floor((3n+2)/2))/2.
Cf. A187357, A187364, A187365.
Cf. A001405.

Programs

  • Mathematica
    Table[Binomial[3n+1,Floor[(3n+1)/2]],{n,0,30}] (* Harvey P. Dale, Jan 13 2021 *)

Formula

a(n) = binomial(3*n+1,floor((3*n+1)/2)), n>=0.
O.g.f.: 3!*x*G2(x^2) + G1(x^2), with G2(x) and G1(x) the o.g.f.s of A187365 and A187364, respectively.

Extensions

Corrected and extended by Harvey P. Dale, Jan 13 2021

A187444 A trisection of A001405 (central binomial coefficients): binomial(3n+2,floor((3n+2)/2))/2, n>=0.

Original entry on oeis.org

1, 5, 35, 231, 1716, 12155, 92378, 676039, 5200300, 38779380, 300540195, 2268783825, 17672631900, 134564468610, 1052049481860, 8061900920775, 63205303218876, 486734856412028, 3824345300380220, 29566145391215356, 232714176627630544, 1804857108504066435
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For the trisection of sequences see a comment and a reference under A187357.

Crossrefs

Cf. A187442: binomial(3*n,floor(3*n/2)), A187443: binomial(3*n+1,floor((3*n+1)/2)).

Programs

  • PARI
    vector(30, n, n--; binomial(3*n+2,(3*n+2)\2)/2) \\ Michel Marcus, Jun 11 2015

Formula

a(n) = binomial(3*n+2,floor((3*n+2)/2))/2, n>=0.
O.g.f.: G1(x^2) + x*G2(x^2), with G1(x) and G2(x) the o.g.f.s of A187364 and A187366, respectively.

A187366 One half of a trisection of A001700: binomial(6n+5,3(n+1))/2, n>=0.

Original entry on oeis.org

5, 231, 12155, 676039, 38779380, 2268783825, 134564468610, 8061900920775, 486734856412028, 29566145391215356, 1804857108504066435, 110628135069209194801, 6804253717299758003900, 419727621552972772561830, 25956855321888352842417780
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For trisection of a sequence see a comment and a reference under A187357.

Crossrefs

Cf. A187364 binomial(2(3n)+1,3n+1),
A187365 binomial(2(3n+1)+1,(3n+1)+1)/3.

Formula

a(n)= binomial(2*(3*n+2)+1,(3*n+2)+1)/2 = binomial(6*n+5,3*(n+1))/2 , n>=0.
O.g.f.: (cb(x^(1/3)) - 3 + sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3)) + 1 + 2*x^(1/3)))/(12*x),
with cb(x):=1/sqrt(1-4*x) (o.g.f. of A000984) and P(x):=P(-1/2,4*x)=1/sqrt(1+4*x+16*x^2) (o.g.f. of A116091, with P(x,z)the o.g.f. of the Legendre polynomials).
Previous Showing 11-15 of 15 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.