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 1-3 of 3 results.

A048907 Indices of 9-gonal numbers which are also triangular.

Original entry on oeis.org

1, 10, 154, 2449, 39025, 621946, 9912106, 157971745, 2517635809, 40124201194, 639469583290, 10191389131441, 162422756519761, 2588572715184730, 41254740686435914, 657487278267789889, 10478541711598202305, 166999180107303446986, 2661508340005256949466
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Entries are == 1 (mod 3). - N. J. A. Sloane, Sep 22 2007
lim(n -> Infinity, a(n)/a(n-1)) = 8 + 3*sqrt(7). - Ant King, Nov 03 2011

Links

Crossrefs

Cf. A001080, A073352, A048908, A048909.

Programs

  • Mathematica
    LinearRecurrence[{17, -17, 1}, {1, 10, 154}, 17]; (* Ant King, Nov 03 2011 *)
  • PARI
    Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-16*x+1)) + O(x^20)) \\ Colin Barker, Jun 22 2015

Formula

G.f.: x*(1-7*x+x^2)/((1-x)*(1-16*x+x^2)).
a(n+2) = 16*a(n+1)-a(n)-5, a(n+1) = 8*a(n)-2.5+1.5*(28*a(n)^2-20*a(n)+1)^0.5. - Richard Choulet, Sep 22 2007
From Ant King, Nov 03 2011: (Start)
a(n) = 17*a(n-1) - 17*a(n-2) + a(n-3).
a(n) = ceiling(3/28*(3-sqrt(7))*(8 + 3*sqrt(7))^n).
(End)
a(n) = A097830(n-1)-7*A097830(n-2)+A097830(n-3). - R. J. Mathar, Jul 04 2024

A007752 Odd bisection of A007750.

Original entry on oeis.org

1, 24, 391, 6240, 99457, 1585080, 25261831, 402604224, 6416405761, 102259887960, 1629741801607, 25973608937760, 413948001202561, 6597194410303224, 105141162563649031, 1675661406608081280
Offset: 1

Views

Author

John C. Hallyburton, Jr. (hallyb(AT)vmsdev.enet.dec.com)

Keywords

References

  • Mentioned in a problem on p. 334 of Two-Year College Math. Jnl., Vol. 25, 1994.

Links

Crossrefs

Cf. A007750, A007751.

Programs

  • GAP
    a:=[1,24,391];; for n in [4..30] do a[n]:=17*a[n-1]-17*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Mar 04 2020
  • Magma
    I:=[1,24,391]; [n le 3 select I[n] else 17*Self(n-1) -17*Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Mar 04 2020
  • Maple
    seq( simplify( (4*ChebyshevU(n,8) - 53*ChebyshevU(n-1,8) -4)/7), n=1..20); # G. C. Greubel, Mar 04 2020
  • Mathematica
    Table[(4*ChebyshevU[n, 8] -53*ChebyshevU[n-1, 8] -4)/7, {n,20}] (* G. C. Greubel, Mar 04 2020 *)
  • PARI
    a(n)=local(w); w=8+3*quadgen(28); imag(1/w^n)+4*(real(1/w^n)-1)/7
  • PARI
    vector(30, n, (4*polchebyshev(n,2,8) -53*polchebyshev(n-1,2,8) -4)/7 ) \\ G. C. Greubel, Mar 04 2020
  • Sage
    [(4*chebyshev_U(n,8) -53*chebyshev_U(n-1,8) -4)/7 for n in (1..30)] # G. C. Greubel, Mar 04 2020

Formula

G.f.: x*(1+7*x)/((1-x)*(1-16*x+x^2)).
a(n) = 16*a(n-1) - a(n-2) + 8.
a(n) = (4*ChebyshevU(n, 8) -53*ChebyshevU(n-1, 8) -4)/7. - G. C. Greubel, Mar 04 2020
E.g.f.: (exp(8*x)*(4*cosh(3*sqrt(7)*x) - sqrt(7)*sinh(3*sqrt(7)*x)) - 4*exp(x))/7. - Stefano Spezia, Mar 14 2020
a(n) = A097830(n-1)+7*A097830(n-2). - R. J. Mathar, Jul 04 2024

Extensions

Edited by Michael Somos, Jul 27 2002

A077831 Expansion of 1/(1-3*x-2*x^2-2*x^3).

Original entry on oeis.org

1, 3, 11, 41, 151, 557, 2055, 7581, 27967, 103173, 380615, 1404125, 5179951, 19109333, 70496151, 260067021, 959412031, 3539362437, 13057045415, 48168685181, 177698871247, 655548074933, 2418379337655, 8921631905325, 32912750541151, 121418274109413
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Partial sums of S(n, x), for x=1...16, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829, A097830.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-3x-2x^2-2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,2},{1,3,11},30] (* Harvey P. Dale, Feb 28 2025 *)
  • PARI
    Vec(1/(1-3*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012
Showing 1-3 of 3 results.

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

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