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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A109599 a(n) = A070864(n+7) - 4.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 5, 3, 5, 3, 5, 3, 5, 5, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 7, 7, 9, 7, 9, 7, 9, 7, 9, 7, 9, 7, 9, 9, 9, 11, 9, 11, 9, 11, 9, 11, 9, 11, 9, 11, 9, 11, 11, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 13, 13, 15, 13, 15, 13, 15, 13, 15, 13, 15, 13, 15
Offset: 1

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Author

Creighton Dement, Jul 29 2005

Keywords

Comments

Floretion Algebra Multiplication Program, FAMP Code: 1teszapsumrokseq[( - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*(- .5'i + .5'j - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj')], Roktype is set to: Y[15] = Y[15] + 1. Sumtype is set to: (default, .5)

Crossrefs

Cf. A070864.

Extensions

Definition corrected by Georg Fischer, Jul 15 2024

A010000 a(0) = 1, a(n) = n^2 + 2 for n > 0.

Original entry on oeis.org

1, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Least k such that A070864(k) = 2n-1. - Robert G. Wilson v and Benoit Cloitre, May 20 2002
With an offset of 3, beginning with 6 (deleting first two terms) n*(n+a(n)) + 1 is a cube = (n+1)^3: 1*(1+6) + 1 = 8, 2*(2+11) + 1 = 27, etc. - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003
For n >= 2, a(n) is the maximum element in the continued fraction for Sum_{k>=1} 1/n^(2^k) (for n=2 see A006464). - Benoit Cloitre, Jun 12 2007
Equals binomial transform of [1, 2, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008
Minimum Wiener index of 3-degenerate graphs with n+2 vertices. A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are maximal 3-degenerate graphs with diameter at most 2. - Allan Bickle, Oct 14 2022
a(n-1) is the number of unit triangles enclosed by the triangular spiral drawn on a isometric grid of which the n-th side has length n. The picture in the link shows how the spiral is constructed. - Bob Andriesse, Feb 14 2023

Examples

			G.f. = 1 + 3*x + 6*x^2 + 11*x^3 + 18*x^4 + 27*x^5 + 38*x^6 + 51*x^7 + 66*x^8 + ...

Links

Crossrefs

Cf. A070864. Apart from initial terms, same as A059100.
Cf. A206399.
Cf. A002061 (minimum Wiener index of 2-degenerate graphs).

Programs

  • Mathematica
    a[1] = a[2] = 1; a[n_] := a[n] = 2 + a[n - a[n - 1]]; b = Table[0, {100}]; Do[c = (a[n] + 1)/2; If[c < 101 && b[[c]] == 0, b[[c]] = n], {n, 1, 10^4}]; b
Join[{1}, Range[50]^2 + 2] (* Bruno Berselli, Feb 08 2012 *)
a[ n_] := n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 05 2015 *)
  • Maxima
    a(n):=if n=0 then 1 else 2*n*sum((-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1),j,0,n)/(n+1)!; /* Vladimir Kruchinin, Jun 03 2013 */
  • PARI
    {a(n) = n^2 + 2 - (n==0)}; /* Michael Somos, Aug 11 2009 */

Formula

a(n) = A000217(n-2) + A000217(n+1) for n > 0. - Jon Perry, Jul 23 2003
Euler transform of length 6 sequence [ 3, 0, 1, 0, 0, -1]. - Michael Somos, Aug 11 2009
From Michael Somos, Aug 11 2009: (Start)
G.f.: (1 + x^3) / (1 - x)^3.
a(n) = a(-n) for all n in Z. (End)
E.g.f.: (x*(x+1) + 2)*e^x - 1. - Gopinath A. R., Feb 14 2012
a(n) = (2*n/(n+1)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1), n > 0, a(0)=1. - Vladimir Kruchinin, Jun 03 2013
Sum_{n>=0} 1/a(n) = 3/4+1/4*sqrt(2)*Pi*coth(Pi*sqrt 2)= 1.8610281... - R. J. Mathar, May 07 2024
Showing 1-2 of 2 results.

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