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.

A047393 Numbers that are congruent to {0, 1} mod 8.

Original entry on oeis.org

0, 1, 8, 9, 16, 17, 24, 25, 32, 33, 40, 41, 48, 49, 56, 57, 64, 65, 72, 73, 80, 81, 88, 89, 96, 97, 104, 105, 112, 113, 120, 121, 128, 129, 136, 137, 144, 145, 152, 153, 160, 161, 168, 169, 176, 177, 184, 185, 192, 193, 200, 201, 208, 209, 216, 217, 224, 225, 232
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers k such that floor(k/2) = 4*floor(k/8). - Bruno Berselli, Oct 05 2017

Links

Crossrefs

Union of A008590 and A017077.

Programs

  • Maple
    a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+8 od: seq(a[n], n=0..58); # Zerinvary Lajos, Mar 16 2008
  • Mathematica
    Riffle[#, # + 1] & [Range[0, 400, 8]] (* Paolo Xausa, Mar 11 2025 *)
  • PARI
    forstep(n=0,200,[1,7],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011
  • PARI
    a(n) = 4*n - 11/2 - 3*(-1)^n/2; \\ David Lovler, Jul 25 2022
  • Python
    def A047393(n): return n-1<<2 if n&1 else (n<<2)-7 # Chai Wah Wu, Mar 11 2025

Formula

a(n) = 8*n - a(n-1) - 15 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 11/2 - 3*(-1)^n/2.
G.f.: x^2*(1+7*x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*A146541(k). - Philippe Deléham, Oct 17 2011
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/2 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021
E.g.f.: ((8*x - 3)*exp(x) + 3*exp(-x))/2 = 4*x*exp(x) - 3*sinh(x). - David Lovler, Aug 02 2022

Extensions

More terms from James Sellers, Jun 20 2000

A036543 a(n) = T(3,n), array T given by A048471.

Original entry on oeis.org

1, 9, 33, 105, 321, 969, 2913, 8745, 26241, 78729, 236193, 708585, 2125761, 6377289, 19131873, 57395625, 172186881, 516560649, 1549681953, 4649045865, 13947137601, 41841412809, 125524238433, 376572715305, 1129718145921
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+2) for n=1, 2, 3, ...
Cf. A146541 (inv. bin. transf.)

Programs

  • Magma
    [4*3^n-3: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011
  • Mathematica
    4*3^Range[0,25]-3 (* or *) LinearRecurrence[{4,-3},{1,9},25] (* Harvey P. Dale, Aug 16 2011 *)
  • PARI
    vector(30, n, n--; 4*3^n-3) \\ G. C. Greubel, Nov 23 2018
  • Sage
    [4*3^n-3 for n in range(30)] # G. C. Greubel, Nov 23 2018

Formula

Binomial transform of A084242. Second binomial transform of periodic sequence A010688. - Paul Barry, May 23 2003
From Paul Barry, May 23 2003: (Start)
a(n) = 4*3^n - 3;
G.f.: (1+5*x)/((1-x)*(1-3*x));
E.g.f.: 4*exp(3*x) - 3*exp(x). (End)
a(n) = 4*a(n-1) - 3*a(n-2); a(0)=1, a(1)=9. - Harvey P. Dale, Aug 16 2011
a(n) = 3*a(n-1) + 6. - Vincenzo Librandi, Nov 11 2011
a(n) = A171498(n) - 2. - Philippe Deléham, Apr 13 2013

A258935 Independence number of Keller graphs.

Original entry on oeis.org

4, 5, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592
Offset: 1

Views

Author

Stan Wagon, Nov 06 2015

Keywords

Examples

			For G(2), a maximum independent set is {03,10,12,13,23}.

References

  • W. Jarnicki, W. Myrvold, P. Saltzman, S. Wagon, Properties, proved and conjectured, of Keller, queen, and Mycielski graphs, Ars Mathematica Contemporanea 13:2 (2017) 427-460.

Links

Crossrefs

Cf. A006946, A135831, A135907.
Essentially the same as A143858, A240951, A198633, A171497, A151821, A146541 and A077552.

Programs

Formula

a(n) = 2^n except a(1) = 4 and a(2) = 5.
G.f.: x*(x*(3+2*x)-4)/(2*x-1), e.g.f.: exp(2*x)+x^2/2+2*x-1. - Benedict W. J. Irwin, Jul 15 2016
Showing 1-3 of 3 results.

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

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