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.

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A085005 A Von Koch curve related to the Golden ratio.

Original entry on oeis.org

1, 3, 4, 4, 3, 3, 4, 4, 3, 1, 0, 0, 1, 1, 0, 0, 1, 3, 4, 4, 5, 7, 10, 12, 13, 13, 14, 16, 17, 17, 16, 16, 17, 19, 20, 20, 21, 23, 26, 28, 29, 31, 34, 38, 41, 43, 44, 46, 49, 51, 52, 52, 53, 55, 56, 56, 55, 55, 56, 58, 59, 59, 60, 62, 65, 67, 68, 68, 69, 71, 72, 72, 71, 71, 72, 72, 71
Offset: 1

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Author

Benoit Cloitre, Jun 17 2003

Keywords

Links

Crossrefs

Cf. A083035, A083036, A083037, A083038, A085002, A085003, A085004.
Cf. A094200, A094201.

Programs

  • PARI
    a(n)=(-1)*sum(i=1,n,sum(j=1,i,(-1)^floor(j*(1+sqrt(5))/2)))

Formula

a(n)=(-1)*sum(i=1, n, sum(j=1, i, (-1)^floor(j*(1+sqrt(5))/2)))
a(n) = 2*sum(k = 1, n, sum(i = 1, k, b(i)))-n*(n+1)/2, where b(k) = floor(phi*k)-2*floor(phi*k/2)

A094200 a(n) = 16*n^4 + 32*n^3 + 36*n^2 + 20*n + 3.

Original entry on oeis.org

3, 107, 699, 2547, 6803, 15003, 29067, 51299, 84387, 131403, 195803, 281427, 392499, 533627, 709803, 926403, 1189187, 1504299, 1878267, 2318003, 2830803, 3424347, 4106699, 4886307, 5772003, 6773003, 7898907, 9159699, 10565747
Offset: 0

Views

Author

Benoit Cloitre, May 25 2004

Keywords

Comments

Let x(n) = (1/2)*(-(2*n + 1) + sqrt((2*n + 1)^2 + 4)) and f(n,k) = (-1)*Sum_{i=1..k} Sum_{j=1..i} (-1)^floor(j*x(n)). Then a(n) = k is the least integer k > 0 such that f(n, k) = 0. In other words, f(n, a(n)) = 0, and if f(n,k) = 0, then a(n) <= k. [Edited by Petros Hadjicostas, Jul 12 2020]

Links

Crossrefs

Cf. A085005, A094201.

Programs

  • Mathematica
    Table[16n^4+32n^3+36n^2+20n+3,{n,0,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{3,107,699,2547,6803},30] (* Harvey P. Dale, Jul 23 2013 *)
  • PARI
    a(n) = 16*n^4+32*n^3+36*n^2+20*n+3

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n >= 5. - Harvey P. Dale, Jul 23 2013
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