A271341 -id:A271341 - cp's OEIS frontend

A102276 a(n) = (a(n-1) * a(n-5) + a(n-3)^2) / a(n-6) with a(0) = ... = a(5) = 1, a(n) = a(5-n) for all n in Z.

1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, 107, 239, 1103, 3775, 14463, 55283, 256666, 2059753, 9820288, 55075036, 503857819, 4083736906, 44590046729, 335845998321, 3581731774609, 68868876045617, 782035904796497, 11680434156713849, 194342679446776442

Offset: 0

Michael Somos, Jan 02 2005

nonn

Sequence defined by recursion derived from a genus 2 curve.
Similar to the Somos-6 and Somos-7 sequences with many bilinear identities.
If a0 := a(n), a1 := a(n+1), ..., a5 := a(n+5), a6 := a(n+6) and a6 = (a5*a1 + a3^2)/a0 for all n in Z, then c := (a0^2*a1*a4*a5^2 + a0^2*a3*a4^3 + a1^3*a2*a5^2 + a0*a2^2*a3*a4^2 + a1^2*a2*a3^2*a5 + a0*a2*a3^3*a4 + a1*a2^3*a3*a5 + a2^3*a3^3)/(a0*a1*a2*a3*a4*a5) is constant. - Michael Somos, Jun 30 2024

Seiichi Manyama, Table of n, a(n) for n = 0..211
A. J. van der Poorten, Curves of Genus 2, Continued Fractions and Somos Sequences, arXiv:math/0412372 [math.NT], 2004.
A. J. van der Poorten, Curves of Genus 2, Continued Fractions and Somos Sequences, J. Integer Seqs., 8 (2005), #05.3.4.
Index entries for two-way infinite sequences

Cf. A006720, A006722, A256858, A256916, A018896, A271341, A271835, A271831, A271837, A271838, A271839.

I:=[1, 2, 3, 4, 8, 17]; [1, 1, 1, 1, 1] cat [n le 6 select I[n] else (Self(n-1)*Self(n-5) + Self(n-3)^2)/Self(n-6): n in [1..30]]; // G. C. Greubel, Aug 03 2018

Join[{1, 1, 1, 1, 1}, RecurrenceTable[{a[n] == (a[n-1]*a[n-5] + a[n-3]^2)/a[n-6], a[6] == 1, a[7] == 2, a[8] == 3, a[9] == 4, a[10] == 8, a[11] == 17}, a, {n, 6, 60}]] (* G. C. Greubel, Aug 03 2018 *)

{a(n) = my(an); if( n<0, a(5-n), n++; an = vector(n,i,1); for(k=7, n, an[k] = (an[k-1]*an[k-5] + an[k-3]^2) / an[k-6]); an[n])};

a(n) = A256858(2*n - 5) for all n in Z. - Michael Somos, Apr 13 2015
Let b(n) = A256916(n). Then 0 = a(n) * b(n) - a(n-2) * b(n+2) + a(n-3) * b(n+3) for all n in Z. - Michael Somos, Apr 13 2015
0 = a(n) * a(n+6) - a(n+1) * a(n+5) - a(n+3) * a(n+3) for all n in Z. - Michael Somos, Apr 13 2015
0 = a(n) * a(n+9) + a(n+2) * a(n+7) - a(n+3) * a(n+6) - 9 * a(n+4) * a(n+5) for all n in Z. - Michael Somos, Apr 13 2015

A271831 Somos's sequence {a(6,n)} defined in comment in A018896: a(0)=a(1)= ... = a(13) = 1; for n>=14, a(n) = (a(n-1)*a(n-13) + a(n-7)^2)/a(n-14).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 12, 21, 37, 62, 98, 147, 358, 609, 959, 1541, 2618, 4655, 8407, 28631, 81011, 186528, 376741, 706041, 1280174, 3598503, 8411236, 24021605, 74880071, 219318499, 580374907, 1400227135, 6308924342

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 15 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..416
Eric Weisstein's World of Mathematics, Somos Sequence

Cf. A018896, A006125, A006720, A102276, A271341, A271835, A271837, A271838, A271839.

[n le 14 select 1 else (Self(n-1)*Self(n-13) + Self(n-7)^2)/Self(n-14): n in [1..50]]; // G. C. Greubel, Feb 21 2018

a[n_ /; 0 <= n <= 14] = 1; a[n_]:= a[n] = (a[n-1]*a[n-13] + a[n-7]^2)/a[n -14]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 21 2018 *)

{a(n) = if(n<= 14, 1, (a(n-1)*a(n-13) + a(n-7)^2)/a(n-14))};
for(n=1,50, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

A271835 Somos's sequence {a(5,n)} defined in comment in A018896: a(0)=a(1)= ... = a(11) = 1; for n>=12, a(n) = (a(n-1)*a(n-11) + a(n-6)^2)/a(n-12).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 11, 20, 36, 61, 97, 243, 425, 700, 1199, 2183, 4115, 14902, 43515, 102827, 214168, 418685, 1223440, 3053628, 9484929, 31351174, 95335734, 260010845, 1305343146, 4437434637, 12553187856, 35704506092

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 15 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..368
Eric Weisstein's World of Mathematics, Somos Sequence

Cf. A018896, A006125, A006720, A102276, A271341, A271831, A271837, A271838, A271839.

[n le 12 select 1 else (Self(n-1)*Self(n-11) + Self(n-6)^2 )/Self(n-12): n in [1..50]]; // G. C. Greubel, Feb 21 2018

a[k_,n_]:=a[k,n]= If[n>2k+1,(a[k,(n-1)]*a[k,(n-2k-1)]+(a[k,(n-k-1)])^2 )/a[k,(n-2k-2)],1]; Map[a[5,#]&,Range[0,43]] (* Peter J. C. Moses, Apr 15 2016 *)
RecurrenceTable[{Table[a[i]==1,{i,0,11}],a[n]==(a[n-1]a[n-11]+a[n-6]^2)/ a[n-12]},a,{n,50}](* Harvey P. Dale, Sep 24 2021 *)

{a(n) = if(n< 12, 1, (a(n-1)*a(n-11) + a(n-6)^2)/a(n-12))};
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

More terms from Alois P. Heinz, Apr 15 2016

A271837 Somos's sequence {a(7,n)} defined in comment in A018896: a(0)=a(1)= ... = a(15) = 1; for n>=16, a(n) = (a(n-1)*a(n-15)+ a(n-8)^2)/a(n-16).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 22, 38, 63, 99, 148, 212, 505, 842, 1284, 1966, 3153, 5312, 9200, 15968, 51401, 141522, 319386, 631223, 1149722, 2003800, 3442200, 9402302, 20908517, 55671036, 164685883, 466783858

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 15 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..463
Eric Weisstein's World of Mathematics, Somos Sequence

Cf. A018896, A006125, A006720, A102276, A271341, A271831, A271835, A271838, A271839.

[n le 16 select 1 else (Self(n-1)*Self(n-15) + Self(n-8)^2 )/Self(n-16): n in [1..50]]; // G. C. Greubel, Feb 21 2018

a[k_,n_]:=a[k,n] = If[n>2*k+1,(a[k,(n-1)]*a[k,(n-2*k-1)]+(a[k,(n-k-1)])^2 )/a[k,(n-2*k-2)],1]; Map[a[7,#]&,Range[0,50]] (* Peter J. C. Moses, Apr 15 2016 *)

{a(n) = if(n< 16, 1, (a(n-1)*a(n-15) + a(n-8)^2)/a(n-16))};
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

A271838 Somos's sequence {a(8,n)} defined in comment in A018896: a(0)=a(1)= ... = a(17) = 1; for n>=18, a(n) = (a(n-1)*a(n-17)+ a(n-9)^2)/a(n-18).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 23, 39, 64, 100, 149, 213, 294, 688, 1130, 1683, 2484, 3800, 6100, 10143, 17082, 28584, 87352, 234714, 521145, 1013424, 1809100, 3067659, 5075784, 8375940, 22379904, 47848348

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 15 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..510
Eric Weisstein's World of Mathematics, Somos Sequence

Cf. A018896, A006125, A006720, A102276, A271341, A271835, A271831, A271837, A271839.

[n le 18 select 1 else (Self(n-1)*Self(n-17) + Self(n-9)^2 )/Self(n-18): n in [1..40]]; // G. C. Greubel, Feb 21 2018

a[k_,n_]:=a[k,n] = If[n>2*k+1,(a[k,(n-1)]*a[k,(n-2*k-1)]+(a[k,(n-k-1)])^2 )/a[k,(n-2*k-2)],1]; Map[a[8,#]&,Range[0,50]] (* Peter J. C. Moses, Apr 15 2016 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]==a[8]==a[9]== a[10]==a[11]==a[12]==a[13]==a[14]==a[15]==a[16]==a[17]==1,a[n]==(a[n-1]a[n-17]+ a[n-9]^2)/a[n-18]},a,{n,60}] (* Harvey P. Dale, Jun 30 2023 *)

{a(n) = if(n< 18, 1, (a(n-1)*a(n-17) + a(n-9)^2)/a(n-18))};
for(n=0,40, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

A271839 Somos's sequence {a(9,n)} defined in comment in A018896: a(0)= a(1) = ... = a(19) = 1; for n >= 20, a(n) = (a(n-1)*a(n-19) + a(n-10)^2)/a(n-20).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 24, 40, 65, 101, 150, 214, 295, 395, 911, 1479, 2164, 3105, 4571, 7033, 11252, 18383, 30095, 48707, 141866, 372815, 816479, 1567804, 2757573, 4585139, 7385515

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 15 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..555
Eric Weisstein's World of Mathematics, Somos Sequence

Cf. A018896, A006125, A006720, A102276, A271341, A271835, A271831, A271837, A271838.

[n le 20 select 1 else (Self(n-1)*Self(n-19) + Self(n-10)^2 )/Self(n-20): n in [1..50]]; // G. C. Greubel, Feb 21 2018

a[k_,n_]:=a[k,n]=If[n>2k+1,(a[k,(n-1)]*a[k,(n-2k-1)]+(a[k,(n-k-1)])^2)/a[k,(n-2k-2)],1];
Map[a[9,#]&,Range[0,70]] (* Peter J. C. Moses, Apr 15 2016 *)

{a(n) = if(n< 20, 1, (a(n-1)*a(n-19) + a(n-10)^2)/a(n-20))};
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

cp's OEIS Frontend

A102276 a(n) = (a(n-1) * a(n-5) + a(n-3)^2) / a(n-6) with a(0) = ... = a(5) = 1, a(n) = a(5-n) for all n in Z.

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A271831 Somos's sequence {a(6,n)} defined in comment in A018896: a(0)=a(1)= ... = a(13) = 1; for n>=14, a(n) = (a(n-1)*a(n-13) + a(n-7)^2)/a(n-14).

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A271835 Somos's sequence {a(5,n)} defined in comment in A018896: a(0)=a(1)= ... = a(11) = 1; for n>=12, a(n) = (a(n-1)*a(n-11) + a(n-6)^2)/a(n-12).

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A271837 Somos's sequence {a(7,n)} defined in comment in A018896: a(0)=a(1)= ... = a(15) = 1; for n>=16, a(n) = (a(n-1)*a(n-15)+ a(n-8)^2)/a(n-16).

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A271838 Somos's sequence {a(8,n)} defined in comment in A018896: a(0)=a(1)= ... = a(17) = 1; for n>=18, a(n) = (a(n-1)*a(n-17)+ a(n-9)^2)/a(n-18).

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Magma

Mathematica

PARI

A271839 Somos's sequence {a(9,n)} defined in comment in A018896: a(0)= a(1) = ... = a(19) = 1; for n >= 20, a(n) = (a(n-1)*a(n-19) + a(n-10)^2)/a(n-20).

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Links

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Programs

Magma

Mathematica

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