A006723 -id:A006723 - cp's OEIS frontend

A058058 Generalized Somos-7 sequence: a(n)a(n+7) = 3a(n+1)a(n+6) - 4a(n+2)* a(n+5) + 4a(n+3)a(n+4).

1, 1, 1, 1, 1, 1, 1, 3, 9, 19, 33, 131, 681, 3921, 23801, 132881, 598121, 7466321, 141401273, 1865484899, 19358862929, 314151573363, 7831607063961, 237725833277411, 8937694547422641, 293153245305595201, 7098035759907924561, 310194702846756799041, 35075042744420641281841

Offset: 1

Robert G. Wilson v, Nov 20 2000

nonn

N. Elkies, posting to the NMBRTHRY(AT)LISTSERV.NODAK.EDU newsgroup, Nov. 2000.

G. C. Greubel, Table of n, a(n) for n = 1..170

Cf. A006720, A006721, A006722 and A006723.

I:=[1,1,1,1,1,1,1]; [n le 7 select I[n] else (3*Self(n-1)*Self(n-6) - 4*Self(n-2)*Self(n-5) + 4*Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

a[1] =a[2] =a[3] =a[4] =a[5] =a[6] =a[7] =1; a[n_]:= a[n] = (3*a[n-1]*a[n-6] - 4*a[n-2]*a[n-5] + 4*a[n-3]*a[n-4])/a[n-7]; Table[ a[n], {n, 1, 35}]

{a(n) = if (n <=7, 1, (3*a(n-1)*a(n-6) - 4*a(n-2)*a(n-5) + 4*a(n-3)*a(n-4))/a(n-7))};
for(n=1, 30, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

A064268 a(n) = (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7). a(1) = ... = a(7) = 1. Somos-7 variation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 13, 43, 113, 521, 1241, 3681, 23657, 177721, 679505, 4674203, 27273277, 275517767, 3496390229, 37043734803, 226196947873, 4391322667601, 81041508965617, 1433151398896001, 25397505914206225, 472652420405241521, 9156799134584424289, 499597377081528480243

Offset: 1

Michael Somos, Sep 24 2001

In general, suppose a(n)*a(n-7) = c1*a(n-1)*a(n-6) + c2*a(n-3)*a(n-4) for all n and constants c1,c2. Define u(n) = a(n)*a(n+5)/(a(n+2)*a(n+3)) which satisfies the generalized Lyness recursion u(n) = (c1*u(n-1) + c2)/u(n-2) for all n. For this sequence c1=1, c2=2, u(n) is (1, 1, 3, 5, 7/3, 13/15, 43/35, ...) and satisfies u(n) = (u(n-1) + 2)/u(n-2). See A076839 for Lyness references. - Michael Somos, Sep 26 2022

Harry J. Smith, Table of n, a(n) for n = 1..100
Index entries for two-way infinite sequences

Cf. A006723, A076839.

I:=[1,1,1,1,1,1,1]; [n le 7 select I[n] else (Self(n-1)*Self(n-6) + 2*Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]==1,a[n] == (a[n-1]a[n-6]+2a[n-3]a[n-4])/a[n-7]},a,{n,30}] (* Harvey P. Dale, Nov 26 2015 *)

{a(n) = if( n<1, a(8-n), if( n<8, 1, (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7)))};

{ a7=a6=a5=a4=a3=a2=a1=a=1; for (n=1, 100, if (n>7, a=(a1*a6 + 2*a3*a4)/a7; a7=a6; a6=a5; a5=a4; a4=a3; a3=a2; a2=a1; a1=a); write("b064268.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 10 2009

a(8-n) = a(n).

A275695 a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1; for n>6, a(n) = ( a(n-1)+a(n-3)+a(n-5) )*( a(n-2)+a(n-4)+a(n-6) ) / a(n-7).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 33, 385, 13825, 5474305, 75873853441, 415386585427968001, 3501887406773528570406162401, 44079910680970588907541344275243042224979209, 400942556117903539711475671972145122347091674105174721165559627509313

Offset: 0

Bruno Langlois, Aug 21 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..21

Cf. A006723, A276130.

I:=[1,1,1,1,1,1,1]; [n le 7 select I[n] else (Self(n-1) + Self(n-3) + Self(n-5))*(Self(n-2) + Self(n-4) + Self(n-6))/Self(n-7): n in [1..17]]; // G. C. Greubel, Feb 21 2018

RecurrenceTable[{a[n] == (a[n - 1] + a[n - 3] + a[n - 5]) (a[n - 2] + a[n - 4] + a[n - 6])/a[n - 7], a[0] == a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == 1}, a, {n, 0, 16}] (* Michael De Vlieger, Aug 25 2016 *)
nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,((g+e+c)(f+d+b))/a}; NestList[ nxt,{1,1,1,1,1,1,1},20][[All,1]] (* Harvey P. Dale, May 04 2019 *)

a(n) = if (n <=6, 1, (a(n-1)+a(n-3)+a(n-5))*(a(n-2)+a(n-4)+a(n-6))/a(n-7)); \\ Michel Marcus, Aug 25 2016

def A(m, n)
  a = Array.new(2 * m + 1, 1)
  ary = [1]
  while ary.size < n + 1
    i = (1..m).inject(0){|s, i| s + a[2 * i - 1]} * (1..m).inject(0){|s, i| s + a[2 * i]}
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A275695(n)
  A(3, n)
end # Seiichi Manyama, Aug 27 2016

a(n) = (8-4*(-1)^n)*a(n-1)*a(n-3)*a(n-5) - a(n-2) - a(n-4) - a(n-6).

cp's OEIS Frontend

A058058 Generalized Somos-7 sequence: a(n)a(n+7) = 3a(n+1)a(n+6) - 4a(n+2)* a(n+5) + 4a(n+3)a(n+4).

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A064268 a(n) = (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7). a(1) = ... = a(7) = 1. Somos-7 variation.

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A275695 a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1; for n>6, a(n) = ( a(n-1)+a(n-3)+a(n-5) )*( a(n-2)+a(n-4)+a(n-6) ) / a(n-7).

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cp's OEIS Frontend

A058058 Generalized Somos-7 sequence: a(n)*a(n+7) = 3*a(n+1)*a(n+6) - 4*a(n+2)* a(n+5) + 4*a(n+3)*a(n+4).

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A064268 a(n) = (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7). a(1) = ... = a(7) = 1. Somos-7 variation.

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Magma

Mathematica

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PARI

Formula

A275695 a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1; for n>6, a(n) = ( a(n-1)+a(n-3)+a(n-5) )*( a(n-2)+a(n-4)+a(n-6) ) / a(n-7).

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Magma

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Formula

A058058 Generalized Somos-7 sequence: a(n)a(n+7) = 3a(n+1)a(n+6) - 4a(n+2)* a(n+5) + 4a(n+3)a(n+4).