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.

A047538 Numbers that are congruent to {0, 1, 4, 7} mod 8.

Original entry on oeis.org

0, 1, 4, 7, 8, 9, 12, 15, 16, 17, 20, 23, 24, 25, 28, 31, 32, 33, 36, 39, 40, 41, 44, 47, 48, 49, 52, 55, 56, 57, 60, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 84, 87, 88, 89, 92, 95, 96, 97, 100, 103, 104, 105, 108, 111, 112, 113, 116, 119, 120, 121, 124
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Related to a Chebyshev transform of A046055. See A074231. - Paul Barry, Oct 27 2004
Starting (1, 4, 7, ...) = partial sums of (1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, ...). - Gary W. Adamson, Jun 19 2008
The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4 etc. - Bruno Berselli, Nov 28 2012
Nonnegative m such that floor(k*(m/4)^2) = k*floor((m/4)^2), where k can assume the values from 4 to 15. See also the second comment in A047513. - Bruno Berselli, Dec 03 2015

Links

Crossrefs

Cf. A047404, A047431, A047546, A047557, A047578, A047620, A056594.
Cf. A046055, A074231.

Programs

  • Magma
    [2*n-2-(1+(-1)^n)*(-1)^((2*n-3) div 4-(-1)^n div 4) / 2 : n in [1..80]]; // Wesley Ivan Hurt, Sep 22 2015
  • Magma
    [n: n in [0..150] | n mod 8 in {0,1,4,7}]; // Vincenzo Librandi, Sep 23 2015
  • Maple
    A047538:=n->2*n-2-sin(Pi*(n-1)/2): seq(A047538(n), n=1..80); # Wesley Ivan Hurt, Sep 22 2015
  • Mathematica
    Table[2n-2-Sin[Pi*(n-1)/2], {n, 80}] (* Wesley Ivan Hurt, Sep 22 2015 *)
Select[Range[0, 150], MemberQ[{0, 1, 4, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, Sep 23 2015 *)
LinearRecurrence[{2,-2,2,-1},{0,1,4,7},100] (* Harvey P. Dale, Aug 12 2016 *)
  • PARI
    a(n) = (-4+(-I)^n+I^n+4*n)/2 \\ Colin Barker, Oct 18 2015
  • PARI
    concat(0, Vec(x^2*(1+x)^2/((1+x^2)*(1-2*x+x^2)) + O(x^100))) \\ Colin Barker, Oct 18 2015
  • Sage
    [lucas_number1(n,0,1)+2*n-4 for n in (2..57)] # Zerinvary Lajos, Jul 06 2008

Formula

From Paul Barry, Oct 27 2004: (Start)
G.f.: x^2*(1+x)^2 / ((1+x^2)*(1-2*x+x^2)).
E.g.f.: 2*x*exp(x)-sin(x).
a(n) = 2*n-2-sin(Pi*(n-1)/2).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4. (End)
a(n) = 2*n-2-(1+(-1)^n)*(-1)^((2*n-3)/4-(-1)^n/4)/2. - Wesley Ivan Hurt, Sep 22 2015
a(n) = (-4+(-i)^n+i^n+4*n)/2, where i = sqrt(-1). - Colin Barker, Oct 18 2015
Sum_{n>=2} (-1)^n/a(n) = (6-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

Extensions

More terms from Wesley Ivan Hurt, Sep 22 2015
G.f. adapted to offset by Colin Barker, Oct 18 2015

A047620 Numbers that are congruent to {0, 1, 2, 5} mod 8.

Original entry on oeis.org

0, 1, 2, 5, 8, 9, 10, 13, 16, 17, 18, 21, 24, 25, 26, 29, 32, 33, 34, 37, 40, 41, 42, 45, 48, 49, 50, 53, 56, 57, 58, 61, 64, 65, 66, 69, 72, 73, 74, 77, 80, 81, 82, 85, 88, 89, 90, 93, 96, 97, 98, 101, 104, 105, 106, 109, 112, 113, 114, 117, 120, 121, 122
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016813, A047404, A047431, A047467, A047546, A047557, A047578, A056594.

Programs

Formula

From R. J. Mathar, Oct 08 2011: (Start)
G.f.: x^2*(1+3*x^2) / ( (x^2+1)*(x-1)^2 ).
a(n) = 2*n-3+sin(n*Pi/2). (End)
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (4n-6+I^(1-n)-I^(1+n))/2 where i=sqrt(-1).
a(2n+2) = A016813(n) n>0, a(2n-1) = A047467(n). (End)
Sum_{n>=2} (-1)^n/a(n) = Pi/16 + 5*log(2)/8. - Amiram Eldar, Dec 19 2021

Extensions

More terms from Wesley Ivan Hurt, May 22 2016

A261958 Start with a single square for n=0; for the odd n-th generation add a square at each expandable vertex of the squares of the (n-1)-th generation (this is the "vertex to vertex" version); for the even n-th generation use the "side to vertex" version; a(n) is the number of squares added in the n-th generation.

Original entry on oeis.org

1, 4, 12, 16, 24, 32, 28, 36, 32, 44, 44, 56, 56, 72, 60, 76, 64, 84, 76, 96, 88, 112, 92, 116, 96, 124, 108, 136, 120, 152, 124, 156, 128, 164, 140, 176, 152, 192, 156, 196, 160, 204, 172, 216, 184, 232, 188, 236, 192, 244, 204, 256, 216
Offset: 0

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

Comments

See a comment on V-V and V-S at A249246.
The overlap rules for the expansion are: (i) overlap within generation is allowed. (ii) overlap of different generations is prohibited.
There are a total of 16 combinations as shown in the table below:
+-------------------------------------------------------+
| Even n-th version V-V S-V V-S S-S |
+-------------------------------------------------------+
| Odd n-th version |
| V-V A008574 a(n) ... ... |
| S-V ... A008574 A008574 ... |
| V-S ... A008574 A008574 ... |
| S-S ... ... ... A008574 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.

Links

Crossrefs

Cf. A008574, A047557, A249246.

Programs

  • PARI
    {e=12; o=4; print1("1, ", o, ", ", e, ", "); for(n=3, 100, if (Mod(n,2)==0, if (Mod(n,8)==4, e=e+12); if (Mod(n,8)==6, e=e+4); if (Mod(n,8)==0, e=e+4); if (Mod(n,8)==2, e=e+12); print1(e, ", "), if (Mod(n,8)==3, o=o+12); if (Mod(n,8)==5, o=o+16); if (Mod(n,8)==7, o=o+4); if (Mod(n,8)==1, o=o+8); print1(o, ", ")))}

Formula

Conjectures from Colin Barker, Sep 10 2015: (Start)
a(n) = a(n-2)+a(n-8)-a(n-10) for n>10.
G.f.: (x^10+4*x^9+3*x^8+4*x^7+4*x^6+16*x^5+12*x^4+12*x^3+11*x^2+4*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).
(End)
Showing 1-3 of 3 results.

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

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