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.

A054113 T(2*n,n), array T as in A054110.

Original entry on oeis.org

1, 5, 17, 57, 197, 701, 2549, 9413, 35153, 132393, 501905, 1912769, 7321081, 28122281, 108355481, 418590521, 1620751301, 6287963741, 24438234341, 95128761941, 370821819581, 1447337568461, 5655535495901, 22122396951101, 86617604317301, 339438817192805, 1331275883089013
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A054110, A079309.

Programs

  • Mathematica
    CoefficientList[Series[( -1+4x+2*Sqrt[1-4x])/((1-x)(1-4x)),{x,0,26}],x] (* Stefano Spezia, May 11 2024 *)

Formula

G.f.: (-1+4*x+2*sqrt(1-4*x))/((1-x)*(1-4*x)). - Ralf Stephan, Apr 03 2004
a(n) = 4*s(n) + 1, with s(n) = A079309(n) = Sum_{k=1..n} C(2k+1, k+1). - Ralf Stephan, Apr 03 2004

A054111 Row sums of array T as in A054110.

Original entry on oeis.org

1, 2, 7, 18, 39, 82, 167, 338, 679, 1362, 2727, 5458, 10919, 21842, 43687, 87378, 174759, 349522, 699047, 1398098, 2796199, 5592402, 11184807, 22369618, 44739239, 89478482, 178956967, 357913938, 715827879, 1431655762
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Formula

G.f.: [1+2x^2+4x^3]/[(1-x^2)(1-2x)]. - Ralf Stephan, Apr 03 2004
a(n+1) = 2^(n+4)/3 -7/2 +(-1)^n/6, n>0. - R. J. Mathar, May 20 2013

A054114 T(2n+1,n), array T as in A054110.

Original entry on oeis.org

1, 8, 28, 98, 350, 1274, 4706, 17576, 66196, 250952, 956384, 3660540, 14061140, 54177740, 209295260, 810375650, 3143981870, 12219117170, 47564380970, 185410909790, 723668784230, 2827767747950, 11061198475550, 43308802158650, 169719408596402, 665637941544506
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Apart from the initial term, identical to A066796(n+1).

Crossrefs

Cf. A054110, A066796.

Programs

  • Mathematica
    CoefficientList[Series[(-1+3*x+5*x^2-4*x^3+Sqrt[1-4*x])/(x*(1-x)(1-4*x)), {x,0,30}],x] (* Georg Fischer, Apr 09 2020 *)

Formula

G.f.: (-1+3*x+5*x^2-4*x^3+sqrt(1-4*x))/(x*(1-x)*(1-4*x)). - Ralf Stephan, Apr 03 2004; corrected by Georg Fischer, Apr 09 2020
For n>0, a(n) = sum(k=1, n+1, C(2k, k)). - Ralf Stephan, Apr 03 2004

Extensions

Corrected by Franklin T. Adams-Watters, Oct 25 2006
a(23)-a(25) corrected by Georg Fischer, Apr 09 2020

A054112 T(n,n-3), array T as in A054110.

Original entry on oeis.org

1, 10, 28, 57, 98, 152, 222, 310, 418, 548, 702, 882, 1090, 1328, 1598, 1902, 2242, 2620, 3038, 3498, 4002, 4552, 5150, 5798, 6498, 7252, 8062, 8930, 9858, 10848, 11902, 13022, 14210, 15468, 16798, 18202, 19682, 21240
Offset: 3

Views

Author

Clark Kimberling

Keywords

Formula

For n>6, a(n) = (n^3 - 7n)/3. - Ralf Stephan, Apr 03 2004
Showing 1-4 of 4 results.

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

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