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.

A155598 a(n) = 7^n-2^n+1.

Original entry on oeis.org

1, 6, 46, 336, 2386, 16776, 117586, 823416, 5764546, 40353096, 282474226, 1977324696, 13841283106, 96889002216, 678223056466, 4747561477176, 33232930504066, 232630513856136, 1628413597648306, 11398895184848856
Offset: 0

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Author

Mohammad K. Azarian, Jan 25 2009

Keywords

Crossrefs

Programs

Formula

G.f.: 1/(1-7*x)-1/(1-2*x)+1/(1-x). E.g.f.: e^(7*x)-e^(2*x)+e^x.
a(n) = 9*a(n-1)-14*a(n-2)+6 with a(0)=1, a(1)=6 - Vincenzo Librandi, Jul 21 2010
a(0)=1, a(1)=6, a(2)=46, a(n) = 10*a(n-1)-23*a(n-2)+14*a(n-3). - Harvey P. Dale, Feb 28 2013
a(n) = A190540(n)+1. - R. J. Mathar, Mar 10 2022

A016130 Expansion of g.f. 1/((1-2*x)*(1-7*x)).

Original entry on oeis.org

1, 9, 67, 477, 3355, 23517, 164683, 1152909, 8070619, 56494845, 395464939, 2768256621, 19377800443, 135644611293, 949512295435, 6646586100813, 46526102771227, 325682719529661, 2279779036969771, 15958453259312685, 111709172816237371, 781964209715758749, 5473749468014505547
Offset: 0

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Author

Keywords

Examples

			1/((1-2*x)*(1-7*x)) = 1 + 9*x + 67*x^2 + 477*x^3 + 3355*x^4 + 23517*x^5 + 164683*x^6 + ...
		

Crossrefs

Cf. A190540.

Programs

Formula

a(n) = (7^(n+1) - 2^(n+1))/5. - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 06 2005
a(n) = 7*a(n-1) + 2^n, a(0)=1. - Vincenzo Librandi, Jun 24 2013
From Elmo R. Oliveira, Mar 27 2025: (Start)
E.g.f.: exp(2*x)*(7*exp(5*x) - 2)/5.
a(n) = A190540(n+1)/5.
a(n) = 9*a(n-1) - 14*a(n-2). (End)

A248216 a(n) = 6^n - 2^n.

Original entry on oeis.org

0, 4, 32, 208, 1280, 7744, 46592, 279808, 1679360, 10077184, 60465152, 362795008, 2176778240, 13060685824, 78364147712, 470184951808, 2821109841920, 16926659313664, 101559956406272, 609359739486208, 3656158439014400, 21936950638280704
Offset: 0

Views

Author

Vincenzo Librandi, Oct 04 2014

Keywords

Crossrefs

Sequences of the form k^n - 2^n: A001047 (k=3), A020522 (k=4), A005057 (k=5), this sequence (k=6), A190540 (k=7), A248217 (k=8), A191465 (k=9), A060458 (k=10), A139740 (k=11).

Programs

  • Magma
    [6^n-2^n: n in [0..25]];
    
  • Mathematica
    Table[6^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[4x/((1-2x)(1-6x)), {x, 0, 30}], x]
    LinearRecurrence[{8,-12},{0,4},30] (* Harvey P. Dale, Dec 21 2019 *)
  • Sage
    [2^n*(3^n -1) for n in (0..25)] # G. C. Greubel, Feb 09 2021

Formula

G.f.: 4*x/((1-2*x)*(1-6*x)).
a(n) = 8*a(n-1) - 12*a(n-2).
a(n) = 2^n*(3^n - 1) = A000079(n) * A024023(n).
E.g.f.: exp(6*x) - exp(2*x) = 2*exp(4*x)*sinh(2*x). - G. C. Greubel, Feb 09 2021
a(n) = 4*A016129(n-1). - R. J. Mathar, Mar 10 2022
a(n) = A000400(n) - A000079(n). - Bernard Schott, Mar 27 2022
Showing 1-3 of 3 results.