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 21-23 of 23 results.

A174192 Expansion of (1-x+2x^2)/ ((1-x)*(1-2x-x^2)).

Original entry on oeis.org

1, 2, 7, 18, 45, 110, 267, 646, 1561, 3770, 9103, 21978, 53061, 128102, 309267, 746638, 1802545, 4351730, 10506007, 25363746, 61233501, 147830750, 356895003, 861620758, 2080136521, 5021893802, 12123924127
Offset: 0

Views

Author

Clark Kimberling, Mar 11 2010

Keywords

Examples

			a(2) = 2*a(1)+a(0)+2 = 2*2+1+2 = 7
a(3) = 2*a(2)+a(1)+2 = 2*7+2+2 = 18.

Links

Crossrefs

Cf. A174191, A048655 (first differences)

Programs

  • Mathematica
    CoefficientList[Series[(1-x+2x^2)/((1-x)*(1-2x-x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-1},{1,2,7},30] (* Harvey P. Dale, Jul 18 2019 *)

Formula

a(n)=2*a(n-1)+a(n-2)+2, with a(0)=1, a(1)=2.

Extensions

Terms corrected by R. J. Mathar, Oct 26 2011

A048755 Partial sums of A048693.

Original entry on oeis.org

1, 7, 20, 52, 129, 315, 764, 1848, 4465, 10783, 26036, 62860, 151761, 366387, 884540, 2135472, 5155489, 12446455, 30048404, 72543268, 175134945, 422813163, 1020761276, 2464335720, 5949432721, 14363201167, 34675835060, 83714871292, 202105577649, 487926026595
Offset: 0

Views

Author

Barry E. Williams

Keywords

Links

Crossrefs

Cf. A048693, A048654, A048655.

Programs

  • Mathematica
    Accumulate[LinearRecurrence[{2,1},{1,6},30]] (* or *) LinearRecurrence[ {3,-1,-1},{1,7,20},40] (* Harvey P. Dale, Mar 29 2013 *)

Formula

a(n)=2*a(n-1)+a(n-2)+5; a(0)=1, a(1)=6.
a(n)=[ {(6+(7/2)*sqrt(2))(1+sqrt(2))^n - (6-(7/2)*sqrt(2))(1-sqrt(2))^n}/ 2*sqrt(2) ]-5/2.
G.f. ( 1+4*x ) / ( (x-1)*(x^2+2*x-1) ). - R. J. Mathar, Nov 08 2012
a(0)=1, a(1)=7, a(2)=20, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, Mar 29 2013

Extensions

More terms from Harvey P. Dale, Mar 29 2013

A048770 Partial sums of A048694.

Original entry on oeis.org

1, 8, 23, 60, 149, 364, 883, 2136, 5161, 12464, 30095, 72660, 175421, 423508, 1022443, 2468400, 5959249, 14386904, 34733063, 83853036, 202439141, 488731324, 1179901795, 2848534920, 6876971641, 16602478208, 40081928063
Offset: 0

Views

Author

Barry E. Williams

Keywords

Links

Crossrefs

Cf. A001333, A000129, A048654, A048655, A048694.

Programs

  • Mathematica
    Accumulate[LinearRecurrence[{2,1},{1,7},40]] (* Harvey P. Dale, Jul 22 2011 *)
LinearRecurrence[{3, -1, -1},{1, 8, 23},27] (* Ray Chandler, Aug 03 2015 *)

Formula

a(n) = ((7+4*sqrt(2))*(1+sqrt(2))^n-(7-4*sqrt(2))*(1-sqrt(2))^n)/(2*sqrt(2))-3.
a(n) = 2*a(n-1)+a(n-2)+6 with n>1, a(0)=1, a(1)=8.
a(n) = 3*a(n-1)-a(n-2)-a(n-3). G.f.: (1+5*x)/((1-x)*(1-2*x-x^2)). - Colin Barker, Jun 23 2012
a(n) = 3*A000129(n)+4*A000129(n+1)-3. - R. J. Mathar, Sep 27 2012

Extensions

More terms from James Sellers, Jun 20 2000
Previous Showing 21-23 of 23 results.

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

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