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 1-10 of 12 results. Next

A018896 a(n) = ( a(n-1)*a(n-7) + a(n-4)^2 ) / a(n-8); a(0) = ... = a(7) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 9, 18, 34, 93, 180, 348, 724, 3033, 9666, 24986, 83761, 261033, 1023728, 3923791, 26128126, 105734485, 381740209, 1895904805, 14058722881, 97964968321, 517832518189, 4364261070929, 25225712161101, 181840424632390
Offset: 0

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Author

Keywords

Comments

From Vladimir Shevelev, Apr 04 2016: (Start)
For k >= 0, an infinite sequence {a(k,n)} of Somos's sequences (n>=0) is:
a(k,0) = a(k,1)= ... = a(k,2*k+1) = 1;
and then for n >= 2*k+2,
a(k,n) = (a(k,n-1)*a(k,n-2*k-1) + a(k,n-k-1)^2)/a(k,n-2*k-2).
In particular, {a(0,n)}=A006125, {a(1,n)}=A006720, {a(2,n)}=A102276, {a(3,n)}=A018896.
One can prove that the sequence {a(k,n)} has the first 4k+2 simple differences: 2k+1 zeros, after that k+1 1's and after that k consecutive squares, beginning with 2^2.
Further we have nontrivial differences. The first of them for k=0,1,2,... are 6, 16, 33, 59, 96, 146, 211, 293, 394, 516, ... that is, {k^3/3 + 5*k^2/2 + 43*k/6 + 6}.
(End)

Crossrefs

Programs

  • Haskell
    a018896 n = a018896_list !! n
    a018896_list = replicate 8 1 ++ f 8 where
       f x = ((a018896 (x - 1) * a018896 (x - 7) + a018896 (x - 4) ^ 2)
             `div` a018896 (x - 8)) : f (x + 1)
    -- Reinhard Zumkeller, Oct 01 2012
    
  • Magma
    [n le 8 select 1 else (Self(n-1)*Self(n-7)+Self(n-4)^2 ) / Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 08 2016
  • Maple
    f:= proc(n) option remember;
      if n <= 7 then 1 else
      (procname(n-1)*procname(n-7)+procname(n-4)^2)/procname(n-8)
      fi
    end proc:
    seq(f(n),n=0..50); # Robert Israel, Apr 04 2016
  • Mathematica
    RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]==a[8]==1, a[n]==(a[n-1]a[n-7]+ a[n-4]^2)/a[n-8]},a[n],{n,50}] (* Harvey P. Dale, May 02 2011 *)
    k = 3; Set[#, 1] & /@ Map[a[k, #] &, Range[0, 2 k + 1]]; a[k_, n_] /; n >= 2 k + 2 := (a[k, n - 1] a[k, n - 2 k - 1] + a[k, n - k - 1]^2)/ a[k, n - 2 k - 2]; Table[a[k, n], {n, 0, 35}] (* Michael De Vlieger, Apr 04 2016 *)

Extensions

More terms from Harvey P. Dale, May 02 2011

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

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Crossrefs

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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
    

A271341 Somos's sequence {a(4,n)} defined in comment in A018896.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 10, 19, 35, 60, 156, 284, 499, 930, 1836, 7116, 21586, 52869, 115344, 356076, 972840, 3350009, 11844969, 37689894, 215136930, 785604755, 2444023816, 7985904285, 36968693334, 230985863335, 1429813280831, 6838592493455, 27144055289355, 191201731942399
Offset: 0

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Author

Keywords

Comments

The sequence of the first differences begins from 9 zeros, 5 1's and 4 consecutive squares, beginning with 2^2.
A generalization see in the comment in A018896.

Crossrefs

Programs

  • Magma
    [n le 10 select 1 else (Self(n-1)*Self(n-9) + Self(n-5)^2 )/Self(n-10): n in [1..40]]; // G. C. Greubel, Feb 21 2018
  • Mathematica
    k = 4; Set[#, 1] & /@ Map[a[k, #] &, Range[0, 2 k + 1]]; a[k_, n_] /;
    n >= 2 k + 2 := (a[k, n - 1] a[k, n - 2 k - 1] + a[k, n - k - 1]^2) / a[k, n - 2 k - 2]; Table[a[k, n], {n, 0, 42}] (* Michael De Vlieger, Apr 04 2016 *)
    a[n_ /; 0 <= n <= 10] = 1; a[n_]:= a[n] = (a[n-1]*a[n-9] + a[n-5]^2)/a[n -10]; Table[a[n], {n,0,40}] (* G. C. Greubel, Feb 21 2018 *)
  • PARI
    {a(n) = if(n< 10, 1, (a(n-1)*a(n-9) + a(n-5)^2)/a(n-10))};
    for(n=0,40, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018
    

Extensions

More terms from Michael De Vlieger, Apr 04 2016

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

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Author

Keywords

Crossrefs

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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
    

Extensions

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

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Author

Keywords

Crossrefs

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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

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Author

Keywords

Crossrefs

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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

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Author

Keywords

Crossrefs

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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
    

A256858 a(n) = (-a(n-1) * a(n-6) + a(n-2) * a(n-5)) / a(n-7) with a(n) = 1 if abs(n) < 4, a(11) = 4.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, -1, 2, -3, 3, -3, 4, -2, 8, 9, 17, 29, 50, 83, 107, 56, 239, -243, 1103, -2351, 3775, -7227, 14463, -18648, 55283, 54011, 256666, 698301, 2059753, 5324929, 9820288, 15128062, 55075036, -28437275, 503857819, -1438167267, 4083736906
Offset: 0

Views

Author

Michael Somos, Apr 12 2015

Keywords

Comments

Similar to the Somos-6 and Somos-7 sequences with many bilinear identities.

Crossrefs

Programs

  • Magma
    I:=[-2,8,9,17,29,50,83]; [1, 1, 1, 1, 0, 1, -1, 2, -3, 3, -3, 4] cat [n le 7 select I[n] else (-Self(n-1)*Self(n-6) + Self(n-2)*Self(n-5))/Self(n-7): n in [1..30]]; // G. C. Greubel, Aug 03 2018
  • Mathematica
    Join[{1, 1, 1, 1, 0, 1, -1, 2, -3, 3, -3, 4}, RecurrenceTable[{a[n] == (-a[n - 1]*a[n - 6] + a[n - 2]*a[n - 5])/a[n - 7], a[12] == -2, a[13] == 8, a[14] == 9, a[15] == 17, a[16] == 29, a[17] == 50, a[18] == 83}, a, {n, 12, 60}]] (* G. C. Greubel, Aug 03 2018 *)
    a[n_] := Which[n<0, a[-n], n<12, {1, 1, 1, 1, 0, 1, -1, 2, -3, 3, -3, 4}[[1+n]], True, a[n] = (-a[n-1]*a[n-6] + a[n-2]*a[n-5])/a[n-7]]; (* Michael Somos, Dec 16 2023 *)
  • PARI
    {a(n) = my(an); n = abs(n)+1; an = concat([ 1, 1, 1, 1, 0, 1, -1, 2, -3, 3, -3, 4], vector(max(0, n-12), k)); for(k=13, n, an[k] = (-an[k-1] * an[k-6] + an[k-2] * an[k-5]) / an[k-7]); an[n]};
    
  • PARI
    {a(n) = my(an); n = abs(n)+1; an = vector(n, k, 1); if( n>=5, an[5] = 0); if( n>=7, an[7] = -1); if( n>=8, an[8] = 2); for(k=9, n, an[k] = if( k==12, 4, (-an[k-1] * an[k-6] + an[k-2] * an[k-5]) / an[k-7])); an[n]};
    

Formula

a(2*n - 5) = A102276(n) for all n in Z.
a(2*n) = A256916(n) for all n in Z.
a(n) = a(-n) for all n in Z.
0 = a(n) * a(n+7) + a(n+1) * a(n+6) - a(n+2) * a(n+5) for all n in Z.
0 = a(n) * a(n+8) - a(n+2) * a(n+6) - a(n+4)^2 + (2 - mod(n,2)) * a(n+3) * a(n+5) for all n in Z.
0 = a(n) * a(n+11) + a(n+1) * a(n+10) + a(n+5) * a(n+6) for all n in Z. - Michael Somos, Apr 14 2015

A256916 a(n) = (a(n-1) * a(n-5) + a(n-3)^2) / a(n-6) with a(0) = a(1) = 1, a(2) = 0, a(3) = -1, a(4) = -3, a(8) = 29.

Original entry on oeis.org

1, 1, 0, -1, -3, -3, -2, 9, 29, 83, 56, -243, -2351, -7227, -18648, 54011, 698301, 5324929, 15128062, -28437275, -1438167267, -14356619593, -108319050672, 80689859625, 13472837856577, 268773209122329, 2678522836045616, 7565687047045511, -672545703786704803
Offset: 0

Views

Author

Michael Somos, Apr 13 2015

Keywords

Comments

Similar to the Somos-6 and Somos-7 sequences with many bilinear identities.

Crossrefs

Programs

  • Magma
    I:=[83, 56, -243, -2351, -7227, -18648]; [1,1,0,-1,-3,-3,-2,9,29] 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
  • Mathematica
    Join[{1,1,0,-1,-3,-3,-2,9,29}, RecurrenceTable[{a[n] == (a[n-1]*a[n-5] + a[n-3]^2)/a[n-6], a[9] == 83, a[10] == 56, a[11] == -243, a[12] == -2351, a[13] == -7227, a[14] == -18648}, a, {n, 9, 60}]] (* G. C. Greubel, Aug 03 2018 *)
  • PARI
    {a(n) = my(an); n = abs(n)+1; an = concat([ 1, 1, 0, -1, -3, -3, -2, 9, 29], vector(max(0, n-9), k)); for(k=10, n, an[k] = (an[k-1] * an[k-5] + an[k-3]^2) / an[k-6]); an[n]};
    

Formula

a(n) = a(-n) for all n in Z.
a(n) = A256858(2*n) for all n in Z.
Let b(n) = A102276(n). Then 0 = a(n) * b(n) - a(n+2) * b(n-2) + a(n+3) * b(n-3) for all n in Z.
0 = a(n) * a(n+6) - a(n+1) * a(n+5) - a(n+3) * a(n+3) for all n in Z.
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.

A276529 a(n) = (a(n-1) * a(n-5) + 1) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 20, 27, 34, 41, 89, 137, 185, 233, 281, 610, 939, 1268, 1597, 1926, 4181, 6436, 8691, 10946, 13201, 28657, 44113, 59569, 75025, 90481, 196418, 302355, 408292, 514229, 620166, 1346269, 2072372, 2798475, 3524578, 4250681, 9227465
Offset: 0

Views

Author

Seiichi Manyama, Nov 16 2016

Keywords

Comments

Thanks to the linear recurrence signature, we see that this is actually five separate linear recurrence sequences, each with signature (7,-1), interwoven together. - Greg Dresden, Oct 16 2021

Crossrefs

5th-sections: A049685, A033891, A033889.

Programs

  • Mathematica
    LinearRecurrence[{0,0,0,0,7,0,0,0,0,-1}, {1,1,1,1,1,1,2,3,4,5}, 50] (* G. C. Greubel, Nov 18 2016 *)
    RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==1,a[n]==(a[n-1]a[n-5]+ 1)/a[n-6]},a,{n,50}] (* Harvey P. Dale, Oct 08 2020 *)
    Flatten[Table[{LucasL[4 n - 2]/3, Fibonacci[4 n - 1], LucasL[4 n + 2]/3 - Fibonacci[4 n], LucasL[4 n - 2]/3 + Fibonacci[4 n], Fibonacci[4 n + 1]}, {n, 0, 10}]] (* Greg Dresden, Oct 16 2021 *)
  • PARI
    Vec((1 +x +x^2 +x^3 +x^4 -6*x^5 -5*x^6 -4*x^7 -3*x^8 -2*x^9)/(1 -7*x^5 +x^10) + O(x^50)) \\ Colin Barker, Nov 16 2016
  • Ruby
    def A(k, m, n)
      a = Array.new(2 * k, 1)
      ary = [1]
      while ary.size < n + 1
        i = a[-1] * a[1] + a[k] ** m
        break if i % a[0] > 0
        a = *a[1..-1], i / a[0]
        ary << a[0]
      end
      ary
    end
    def A276529(n)
      A(3, 0, n)
    end
    

Formula

a(n) + a(n+10) = 7*a(n+5).
a(5-n) = a(n).
G.f.: (1 +x +x^2 +x^3 +x^4 -6*x^5 -5*x^6 -4*x^7 -3*x^8 -2*x^9) / (1 -7*x^5 +x^10). - Colin Barker, Nov 16 2016
From Greg Dresden, Oct 16 2021: (Start)
a(5*n) = L(4*n-2)/3 = A049685(n-1),
a(5*n+1) = F(4*n-1) = A033891(n-1),
a(5*n+2) = L(4*n+2)/3 - F(4*n),
a(5*n+3) = L(4*n-2)/3 + F(4*n),
a(5*n+4) = F(4*n+1) = A033889(n). (End)
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