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.

Previous Showing 11-14 of 14 results.

A301774 Number of odd chordless cycles in the (2n+1)-prism graph.

Original entry on oeis.org

2, 12, 30, 74, 200, 522, 1362, 3572, 9350, 24474, 64080, 167762, 439202, 1149852, 3010350, 7881194, 20633240, 54018522, 141422322, 370248452, 969323030, 2537720634, 6643838880, 17393796002, 45537549122, 119218851372, 312119004990, 817138163594, 2139295485800
Offset: 1

Views

Author

Eric W. Weisstein, Mar 26 2018

Keywords

Comments

Sequence extended to a(1) using the formula/recurrence (actual count for the 3-prism is 0, which reproduces A301775).

Links

Crossrefs

Cf. A002878, A131713, A301775.

Programs

  • Mathematica
    Table[LucasL[2 n + 1] + 2 Cos[(2 n + 1) Pi/3], {n, 20}]
LinearRecurrence[{2, 1, 2, -1}, {2, 12, 30, 74}, 20]
CoefficientList[Series[-2 (-1 - 4 x - 2 x^2 + x^3)/(1 - 2 x - x^2 - 2 x^3 + x^4), {x, 0, 20}], x]

Formula

a(n) = A002878(n) + A131713(n).
a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
G.f.: -2*x*(1+x)*(x^2-3*x-1) / ( (1+x+x^2)*(x^2-3*x+1) ).

A363433 Number of (123,231)-avoiding stabilized-interval-free permutations of size n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 5, 5, 7, 7, 10, 9, 13, 12, 16, 15, 20, 18, 24, 22, 28, 26, 33, 30, 38, 35, 43, 40, 49, 45, 55, 51, 61, 57, 68, 63, 75, 70, 82, 77, 90, 84, 98, 92, 106, 100, 115, 108, 124, 117, 133, 126, 143, 135, 153, 145, 163, 155, 174, 165, 185, 176, 196
Offset: 0

Views

Author

Juan B. Gil, Jun 30 2023

Keywords

Comments

A stabilized-interval-free (SIF) permutation on [n] = {1, 2, ..., n} is one that does not stabilize any proper subinterval of [n].

Examples

			For n from 1 to 5 the six permutations (1+1+1+1+2) are 1, 21, 312, 4312, 54132, 54213.

Links

Crossrefs

Cf. A075834, A363431, A363432.

Programs

  • Maple
    A131713 := proc(n)
    op(1+modp(n,3),[1,-2,1]) ;
end proc:
A363433 := proc(n)
    if n < 3 then
        1;
    else
        16*A131713(n) +42*n-79+6*n^2-81*(-1)^n+18*n*(-1)^n;
        %/144 ;
    end if;
end proc:
seq(A363433(n),n=0..20) ; # R. J. Mathar, Jul 17 2023
  • Mathematica
    LinearRecurrence[{0,2,1,-1,-2,0,1},{1,1,1,1,1,2,3,3,5,5},100] (* Paolo Xausa, Nov 18 2023 *)
  • PARI
    Vec((x^9 + x^8 - 3*x^6 - 2*x^5 + x^4 + 2*x^3 + x^2 - x - 1)/((x^2 + x + 1)*(x + 1)^2*(x - 1)^3) + O(x^65)) \\ Michel Marcus, Jul 01 2023

Formula

G.f.: (x^9 + x^8 - 3*x^6 - 2*x^5 + x^4 + 2*x^3 + x^2 - x - 1)/((x^2 + x + 1)*(x + 1)^2*(x - 1)^3).
E.g.f.: (144 + 36*x*(2 + x) + (3*x^2 + 15*x - 80)*cosh(x) + 8*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (3*x^2 + 33*x + 1)*sinh(x))/72. - Stefano Spezia, Jul 01 2023
144*a(n) = 16*A131713(n) +42*n -79 +6*n^2 -81*(-1)^n +18*n*(-1)^n , for n>=3. - R. J. Mathar, Jul 17 2023

A115054 G.f.: (x^3+6*x+2)^2/(x^2+x+1)^2.

Original entry on oeis.org

4, 16, -8, -36, 72, -36, -63, 126, -63, -90, 180, -90, -117, 234, -117, -144, 288, -144, -171, 342, -171, -198, 396, -198, -225, 450, -225, -252, 504, -252, -279, 558, -279, -306, 612, -306, -333, 666, -333, -360, 720, -360, -387, 774, -387, -414, 828, -414, -441, 882, -441, -468, 936, -468, -495, 990, -495
Offset: 0

Views

Author

Roger L. Bagula, Feb 28 2006

Keywords

Comments

q=3 coefficient expansion of hierarchical lattice renormalization polynomial.
Auto-convolution of the sequence 2,4,-6,3,3,-6,3,3,.. (period length 3). [From R. J. Mathar, Mar 09 2009]

References

  • Peitgen and Richter, The Beauty of Fractals, Springer-Verlag, New York, 1986, page 146

Links

Programs

  • Maple
    G:=(x^3+6*x+2)^2/(x^2+x+1)^2: Gser:=series(G,x=0,55): seq(coeff(Gser,x,n),n=0..50);
  • Mathematica
    q=3 b = 9*Flatten[{{4/9}, Abs[Table[Coefficient[ Series[((x^3 + 3*(q - 1)*x + (q - 1)*(q - 2))/(3*x^2 + 3*( q - 2)*x + q^2 - 3*q + 3))^2, {x, 0, 30}], x^n], {n, 1, 30}]]}]

Formula

a(n) = 18*A131713(n)-27*(-1)^n*A099254(n), n>2. [From R. J. Mathar, Mar 09 2009]

Extensions

Edited by N. J. A. Sloane, Apr 16 2006

A165189 Partial sums of partial sums of (A001840 interleaved with zeros).

Original entry on oeis.org

1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132, 171, 210, 264, 318, 390, 462, 556, 650, 770, 890, 1040, 1190, 1375, 1560, 1785, 2010, 2280, 2550, 2871, 3192, 3570, 3948, 4389, 4830, 5341, 5852, 6440, 7028, 7700, 8372, 9136, 9900, 10764, 11628, 12600, 13572
Offset: 1

Views

Author

Alford Arnold, Sep 16 2009

Keywords

Comments

Also convolution of period six sequence 1,0,0,0,0,0,1,... (A079979) with sequence 1,2,5,8,14,20,30,40,... (A006918 without initial zero).

Examples

			A001840 interleaved with zeros is
1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 9, 0, 12, 0, 15, 0, ...
Partial sums thereof are
1, 1, 3, 3, 6, 6, 11, 11, 18, 18, 27, 27, 39, 39, 54, 54, ...
This equals A014125 interleaved with itself.
Partial sums thereof are
1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132, 171, 210, 264, 318, ...

Links

Crossrefs

Cf. A001840 (expansion of x/((1-x)^3*(1+x+x^2))), A001840 (expansion of x/((1-x)^2*(1-x^3))), A079979, A006918, A014125.

Programs

  • Mathematica
    Drop[Accumulate[Accumulate[Riffle[LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},30],0]]],2] (* or *) LinearRecurrence[{2,1,-4,1,2,0,-2,-1,4,-1,-2,1},{1,2,5,8,14,20,31,42,60,78,105,132},50] (* Harvey P. Dale, Jun 08 2018 *)
  • PARI
    /* first computes u = A001840 interleaved with zeros, then v = partial sums, then w = second partial sums */ {m=50; u=vector(m, n, polcoeff(x/((1-x^2)^3*(1+x^2+x^4))+x*O(x^(n)),n)); v=vector(m); a=u[1]; v[1]=a; for(n=2, m, a+=u[n]; v[n]=a); w=vector(m-1); a=v[1]; w[1]=a; for(n=2, m-1, a+=v[n]; w[n]=a); w} \\ Klaus Brockhaus, Sep 21 2009

Formula

G.f.: x/((1-x)^5*(1+x)^3*(1-x+x^2)*(1+x+x^2)).
54*a(n) = 631/64 +405/16*n +3/32*n^4 +15/8*n^3 +381/32*n^2 -(-1)^n*( 9/32*n^2 +45/16*n +375/64) -A131713(n) -3*A057079(n). - R. J. Mathar, Jun 16 2018

Extensions

Edited and corrected by R. J. Mathar, Klaus Brockhaus and N. J. A. Sloane, Sep 21 2009 - Sep 25 2009
Previous Showing 11-14 of 14 results.

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

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