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.

A241595 a(n) = -(8*a(n-4)*a(n-1) - 57*a(n-3)*a(n-2))/a(n-5) with initial values -1, 0, 1, -1, -8, 57, 455.

Original entry on oeis.org

-1, 0, 1, -1, -8, 57, 455, -22352, -47767, 69739671, -3385862936, -1747973613295, 632038062613231, 319807929289790304, -778756000716629557903, -186509371006506937278833, 7581885296067966478838810840, -17592286469464275110206466526327, -594744699237794019378328459208828297
Offset: 0

Views

Author

N. J. A. Sloane, May 18 2014

Keywords

Links

Crossrefs

See A241594 for another version.
Cf. A360381.

Programs

  • Magma
    I:=[-22352, -47767, 69739671, -3385862936, -1747973613295]; [-1, 0, 1, -1, -8, 57, 455] cat [n le 5 select I[n] else -(8*Self(n-4)*Self(n-1) - 57*Self(n-3)*Self(n-2))/Self(n-5): n in [1..20]]; // G. C. Greubel, Aug 06 2018
  • Maple
    f:=proc(n) option remember;
if n <= 2 then -1+n
elif n=3 then -1
elif n=4 then -8
elif n=5 then 57
elif n=6 then 455
else -(8*f(n-4)*f(n-1)-57*f(n-3)*f(n-2))/f(n-5); fi; end;
[seq(f(n),n=0..30)];
  • Mathematica
    nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,-(8d*g-57e*f)/c}; Transpose[ NestList[ nxt,{-1,0,1,-1,-8,57,455},20]][[1]] (* Harvey P. Dale, Jun 23 2014 *)
Join[{-1, 0, 1, -1, -8, 57, 455}, RecurrenceTable[{a[n] == -(8*a[n - 4]*a[n - 1] - 57*a[n - 3]*a[n - 2])/a[n - 5], a[7] == -22352, a[8] == -47767, a[9] == 69739671, a[10] == -3385862936, a[11] == -1747973613295}, a, {n, 7, 25}]] (* G. C. Greubel, Aug 06 2018 *)
a[ n_] := With[{m = n-1}, Module[{A = Table[1, Max[4, Abs[m]]]}, A[[2]] = -1; A[[3]] = -8; A[[4]] = 57; Do[ A[[k]] = (A[[k-1]]*A[[k-3]] + 8*A[[k-2]]*A[[k-2]])/A[[k-4]], {k, 5, Length[A]}]; If[m==0, 0, Sign[m]*A[[Abs[m]]]] ]]; (* Michael Somos, Feb 05 2023 *)
a[ n_] := With[{m = 2*n-2}, Module[{A = Table[1, Max[5, Abs[m]]]}, A[[2]] = -1; A[[5]] = -7; Do[ A[[k]] = (A[[k-1]]*A[[k-4]] + A[[k-2]]*A[[k-3]])/A[[k-5]], {k, 6, Length[A]}]; If[m==0, 0, -Sign[m]*A[[Abs[m]]]] ]]; (* Michael Somos, Nov 03 2023 *)
  • PARI
    {a(n) = n--; my(A = vector(max(4, abs(n)), k, 1)); A[2] = -1; A[3] = -8; A[4] = 57; for(k=5, #A, A[k] = (A[k-1]*A[k-3] + 8*A[k-2]*A[k-2])/A[k-4]); if(n==0, 0, sign(n)*A[abs(n)])}; /* Michael Somos, Feb 05 2023 */
  • PARI
    {a(n) = n=2*n-2; my(A = vector(max(5, abs(n)), k, 1)); A[2] = -1; A[5] = -7; for(k=6, #A, A[k] = (A[k-1]*A[k-4] + A[k-2]*A[k-3])/A[k-5]); if(n==0, 0, -sign(n)*A[abs(n)])}; /* Michael Somos, Nov 03 2023 */
  • PARI
    {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); n--; subst(elldivpol(E, n), 'x, 1) * -(-1)^n}; /* Michael Somos, Mar 01 2025 */

Formula

a(n+1) = -A360381(2*n), a(n+2)*a(n-2) = a(n+1)*a(n-1) + 8*a(n)^2 for all n in Z. - Michael Somos, Feb 05 2023
a(n) = (-1)^(Mod(n,4)!=1)*A241594(n) = -a(2-n) for all n in Z.

A241594 a(n) = -(8*a(n-4)*a(n-1)+57*a(n-3)*a(n-2))/a(n-5) with initial values 1, 0, -1, 1, 8, 57, -455.

Original entry on oeis.org

1, 0, -1, 1, 8, 57, -455, 22352, 47767, 69739671, 3385862936, 1747973613295, -632038062613231, 319807929289790304, 778756000716629557903, 186509371006506937278833, -7581885296067966478838810840, -17592286469464275110206466526327, 594744699237794019378328459208828297
Offset: 0

Views

Author

N. J. A. Sloane, May 18 2014

Keywords

Links

Crossrefs

See A241595 for another version.
Cf. A360381.

Programs

  • Magma
    I:=[22352, 47767, 69739671, 3385862936, 1747973613295]; [1, 0, -1, 1, 8, 57, -455] cat [n le 5 select I[n] else -(8*Self(n-4)*Self(n-1) + 57*Self(n-3)*Self(n-2))/Self(n-5): n in [1..20]]; // G. C. Greubel, Aug 08 2018
  • Maple
    f:=proc(n) option remember;
if n <= 2 then 1-n
elif n=3 then 1
elif n=4 then 8
elif n=5 then 57
elif n=6 then -455
else -(8*f(n-4)*f(n-1)+57*f(n-3)*f(n-2))/f(n-5); fi; end;
[seq(f(n),n=0..30)];
  • Mathematica
    a[n_] := a[n] = If[n <= 6, {1, 0, -1, 1, 8, 57, -455}[[n + 1]], -(8*a[n - 4]*a[n - 1] + 57*a[n - 3]*a[n - 2])/a[n - 5]];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 02 2017 *)
nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,-(8b e+57c d)/a}; Join[{1,0},NestList[nxt,{-1,1,8,57,-455},20] [[;;,1]]] (* Harvey P. Dale, Aug 14 2023 *)
  • PARI
    {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); n--; subst(elldivpol(E, n), 'x, 1) * (-1)^(n%4!=2)}; /* Michael Somos, Mar 01 2025 */
  • PARI
    {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); n--; subst(elldivpol(E, 2*n), 'x, 2) * (-1)^(n + (n%4==2))/6^(n^2 + 1)}; /* Michael Somos, Mar 02 2025 */

Formula

-a(n+1) = A360381(2*n)*(-1)^(n%4!=0). a(n+2)*a(n-2) = -(-1)^n*a(n+1)*a(n-1) + 8*a(n)^2 for all n in Z. - Michael Somos, Mar 01 2025

A360537 Areas of primitive Heron triangles with two rational medians from the infinite family based on Somos-5 sequences.

Original entry on oeis.org

420, 55440, 23931600, 142334216640, 2137147184560080, 4323341954766548553840, 18705358317240372854759881380, 1333577710124626249998068999458413600, 248363720675646323338068819310182950300884320, 4199805494977793853528867974891927438920668319491840
Offset: 1

Views

Author

Andrey Zabolotskiy, Feb 10 2023

Keywords

Links

Crossrefs

Cf. A006721, A360381.
This is a subsequence of A223941.

Programs

  • Mathematica
    t[1|3|4] = 1; t[2] = -1; t[5] = -7;
s[-2|-1|0|1|2] = 1;
Do[f[n_] := f[n] = (f[n-1] f[n-4] + f[n-2] f[n-3]) / f[n-5], {f, {t, s}}];
a[n_] := Abs@Product[f[n] f[n+1] f[n+2]^2 f[n+3] f[n+4], {f, {s, t}}];
Table[a[n], {n, 10}]

Formula

a(n) = |S(n)*S(n+1)*S(n+2)^2*S(n+3)*S(n+4)*T(n)*T(n+1)*T(n+2)^2*T(n+3)*T(n+4)|, where S(n) = A006721(n+2) and T(n) = A360381(n) [Hone, Eq. (1.21)].
Showing 1-3 of 3 results.

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

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