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 1-4 of 4 results.

A006769 Elliptic divisibility sequence associated with elliptic curve "37a1": y^2 + y = x^3 - x and multiples of the point (0,0).

Original entry on oeis.org

0, 1, 1, -1, 1, 2, -1, -3, -5, 7, -4, -23, 29, 59, 129, -314, -65, 1529, -3689, -8209, -16264, 83313, 113689, -620297, 2382785, 7869898, 7001471, -126742987, -398035821, 1687054711, -7911171596, -47301104551, 43244638645
Offset: 0

Views

Author

Michael Somos, Jul 16 1999

Keywords

Comments

This sequence has a recursion same as the Somos-4 sequence recursion.
a(n+1) is the Hankel transform of A178072. - Paul Barry, May 19 2010
The recurrence formulas in [Kimberling, p. 16] are missing square and cube exponents. - Michael Somos, Jul 07 2014
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = 1.
From Helmut Ruhland, Nov 28 2023: (Start)
This sequence and its two subsequences with even/odd indices satisfy the Somos-4 recursion.
The even subsequence is A051138, here called r[ ]. The odd subsequence is the classical Somos-4 A006720, here called s[ ].
These two subsequences interleaved as follows, recover the original sequence which is now: r[0], s[2], r[1], -s[3], r[2], s[4], r[3], -s[5], ..., all Somos-4 s[ ] with odd index with a minus sign. (End)

References

  • G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 11 and 164.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006720, A028941, A050512, A051138, A178072, A278314.

Programs

  • Haskell
    a006769 n = a050512_list !! n
a006769_list = 0 : 1 : 1 : (-1) : 1 : zipWith div (zipWith (+) (zipWith (*)
   (drop 4 a006769_list) (drop 2 a006769_list))
     (map (^ 2) (drop 3 a006769_list))) (tail a006769_list)
-- Reinhard Zumkeller, Nov 02 2011
  • Mathematica
    a[n_] := If[n < 0, -a[-n], If[n == 0, 0, ClearAll[an]; an[] = 1; an[3] = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]]]; Table[a[n], {n, 0, 32}] (* _Jean-François Alcover, Dec 14 2011, after first Pari program *)
Join[{0},RecurrenceTable[{a[1]==a[2]==1,a[3]==-1,a[4]==1,a[n]==(a[n-1] a[n-3]+ a[n-2]^2)/a[n-4]},a,{n,40}]] (* Harvey P. Dale, May 04 2018 *)
a[ n_] := Which[n<0, -a[-n], n<5, {0, 1, 1, -1, 1}[[1+n]], True, (a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]]; (* Michael Somos, Aug 20 2024 *)
  • PARI
    {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(3, n), i, 1); an[3] = -1; for( k=5, n, an[k] = (an[k-1] * an[k-3] + an[k-2]^2) / an[k-4]); an[n]))};
  • PARI
    {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = Vec((-1 - 2*x + sqrt(1 + 4*x - 4*x^3 + O(x^n))) / (2 * x^2)); matdet( matrix((n-1)\2, (n-1)\2, i, j, if(i + j - 1 - n%2<0, 0, an[i + j -n%2])))))};
  • PARI
    {a(n) = my(E, z); E = ellinit([0, 0, -1, -1, 0]); z = ellpointtoz(E, [0, 0]); round( ellsigma(E, n*z) / ellsigma(E, z)^(n^2))}; /* Michael Somos, Oct 22 2004 */
  • PARI
    {a(n) = sign(n) * subst( elldivpol( ellinit([0, 0, -1, -1, 0]), abs(n)), x, 0)}; /* Michael Somos, Dec 16 2014 */

Formula

a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4) for all n != 4.
a(n) = (-a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5) for all n != 5.
a(-n) = -a(n) for all n.
a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3, a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2 for all n.
A006720(n) = (-1)^n * a(2*n - 3), A028941(n) = a(n)^2 for all n.
a(2*n) = A051138(n). - Michael Somos, Feb 10 2015
a(2*n+1) = a(n-1)*a(n)^2*a(n+3) - a(n-2)*a(n+1)^2*a(n+2) for all n. - Michael Somos, Aug 20 2024

A025250 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 1, 1, 0.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 3, 3, 6, 11, 15, 31, 50, 85, 161, 267, 490, 883, 1548, 2863, 5127, 9307, 17116, 31021, 57123, 104963, 192699, 356643, 658034, 1218517, 2262079, 4196895, 7812028, 14549655, 27126118, 50671255, 94697293, 177220411, 332015747
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Number of lattice paths from (0,0) to (n-3,0) that stay weakly in the first quadrant and such that each step is either U=(1,1),D=(2,-1), or H=(2,0). E.g. a(10)=6 because we have HHUD, HUDH, HUHD, UDHH, UHDH and UHHD. - Emeric Deutsch, Dec 23 2003
Hankel transform of a(n+2) is Somos-4 variant A050512. - Paul Barry, Jul 05 2009

Links

Programs

  • GAP
    List([0..45], n-> Sum([0..n], k-> Binomial(k+1,n-2*k-1)*Binomial(n-k-2,k)/(k+1) )); # G. C. Greubel, Feb 23 2019
  • Haskell
    a025250 n = a025250_list !! (n-1)
a025250_list = 0 : 1 : 1 : f 1 [1,1,0] where
   f k xs = x' : f (k+1) (x':xs) where
     x' = sum $ zipWith (*) a025250_list $ take k xs
-- Reinhard Zumkeller, Nov 03 2011
  • Magma
    m:=45; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( (1 +x^2 -Sqrt(1-2*x^2-4*x^3+x^4))/2 )); // G. C. Greubel, Feb 23 2019
  • Mathematica
    Rest[CoefficientList[Series[(1+x^2-Sqrt[1-2*x^2-4*x^3+x^4])/2, {x,0,40}],x]]  (* Harvey P. Dale, Apr 05 2011 *)
Rest@CoefficientList[Series[x^2+ContinuedFractionK[-x^3,x^2-1,{k, 0, 40}],{x,0,40}], x] (* Benedict W. J. Irwin, Oct 13 2016 *)
  • Maxima
    a(n):=sum((binomial(k+1,n-2*k-1)*binomial(n-k-2,k))/(k+1),k,0,n); /* Vladimir Kruchinin, Nov 22 2014 */
  • PARI
    a(n)=polcoeff((x^2-sqrt(1-2*x^2-4*x^3+x^4+x*O(x^n)))/2,n)
  • Sage
    a=((1+x^2 -sqrt(1-2*x^2-4*x^3+x^4))/2).series(x, 45).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 23 2019

Formula

G.f.: (1 +x^2 -sqrt(1-2*x^2-4*x^3+x^4))/2. - Michael Somos, Jun 08 2000
G.f.: x^2+x^3*(1/(1-x^2))c(x^3/(1-x^2)^2) where c(x) is the g.f. of A000108. - Paul Barry, May 20 2009
a(n+2) = Sum_{k=0..n} binomial((n+k)/2, 2*k)*(1+(-1)^(n-k))*A000108(k)/2. - Paul Barry, Jul 06 2009
a(n) = Sum_{k=0..n} binomial(k+1,n-2*k-1)*binomial(n-k-2,k)/(k+1). - Vladimir Kruchinin, Nov 22 2014
G.f.: K_{k>=0} (-x^3)/(x^2-1), where K is the Gauss notation for a continued fraction. - Benedict W. J. Irwin, Oct 11 2016
a(n) ~ sqrt(1 - r^2 - r^3) * (2*r + 4*r^2 - r^3)^n / (2*sqrt(Pi)*n^(3/2)), where r = 0.51361982956383128341133963576515885989214886200017191578885... is the root of the equation 1 - 2*r^2 - 4*r^3 + r^4 = 0. - Vaclav Kotesovec, Jul 03 2021
Shifts left 3 places under the INVERT transform. - J. Conrad, Mar 08 2023

A160569 a(n)=9*(a(n-1)a(n-3)-a(n-2)^2)/a(n-4), a(1)=a(2)=a(3)=1, a(4)=-9.

Original entry on oeis.org

1, 1, 1, -9, -90, -1539, 51759, 7026831, 1349328699, -119669884380, -462804380329131, -818173230956710191, -775566981773559330471, 23349606875054800260574779, 383206716517298992863985977570
Offset: 0

Views

Author

Paul Barry, May 19 2009

Keywords

Comments

a(n+1) is the Hankel transform of A160568.

Links

Crossrefs

Cf. A050512, A160566.

Programs

  • Mathematica
    nxt[{a_,b_,c_,d_}]:={b,c,d,(9(d*b-c^2))/a}; NestList[nxt,{1,1,1,-9},20][[All,1]] (* Harvey P. Dale, Feb 21 2022 *)

A160567 a(n)=4*(a(n-1)a(n-3)-a(n-2)^2)/a(n-4), a(1)=a(2)=a(3)=1, a(4)=-4.

Original entry on oeis.org

1, 1, 1, -4, -20, -144, 704, 34816, 1101824, 12124160, -4499439616, -586464362496, -99309307559936, 33947352788107264, 26466630265219317760, 25787111566764821970944, -7045529183328701486465024
Offset: 0

Views

Author

Paul Barry, May 19 2009

Keywords

Comments

a(n+1) is the Hankel transform of A160565.

Crossrefs

Cf. A050512.

Programs

  • Maple
    A160567 := proc(n)
    option remember;
    if n < 5 then
        op(n,[1,1,1,-4]);
    else
        4*(procname(n-1)*procname(n-3)-procname(n-2)^2)/procname(n-4) ;
    end if;
end proc: # R. J. Mathar, Feb 25 2015
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.