A006720 -id:A006720 - cp's OEIS frontend

A028941 Denominator of X-coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

1, 1, 1, 1, 4, 1, 9, 25, 49, 16, 529, 841, 3481, 16641, 98596, 4225, 2337841, 13608721, 67387681, 264517696, 6941055969, 12925188721, 384768368209, 5677664356225, 61935294530404, 49020596163841, 16063784753682169

Offset: 1

N. J. A. Sloane

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives denominators of the x_n. - N. J. A. Sloane, Jan 27 2022

Squares of terms in A006769 (or A006720).

G. Everest, A. J. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences: Examples and Applications, AMS Monographs, 2003
A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Seiichi Manyama, Table of n, a(n) for n = 1..212
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028940, A028942, A028943.

nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,((c*2e)-f*b+2d^2)/a}; NestList[nxt,{1,1,1,1,4,1},30][[;;,1]] (* Harvey P. Dale, Dec 26 2024 *)

PARI
```
- see A028940.
```

This sequence satisfies the quadratic recurrence relation a(n)a(n-6)=-a(n-1)a(n-5)+2a(n-2)a(n-4)+2a(n-3)^2 which is a generalized Somos-6 relation. - Graham Everest (g.everest(AT)uea.ac.uk), Dec 16 2002

P=(0, 0), 2P=(1, 0), if kP=(a, b) then (k+1)P=(a'=(b^2-a^3)/a^2, b'=-1-b*a'/a).

A056010 Number of words of length n in a simple grammar.

Original entry on oeis.org

1, 1, 3, 8, 23, 68, 207, 644, 2040, 6558, 21343, 70186, 232864, 778550, 2620459, 8872074, 30195288, 103246502, 354508628, 1221846856, 4225644866, 14659644348, 51002664023, 177909901566, 622093882290, 2180123564130, 7656055966092

Offset: 0

Michael Somos, Aug 01 2000

nonn

The grammar defines a language L consisting of words from the alphabet S = {e, w, n, s}. If each letter in S is regarded as an integer lattice step, e = (1,0), w = (-1,0), n = (0,1), s = (0,-1), then each word is a path in the two-dimensional integer lattice starting from (0,0), never going below the x-axis and ending on the x-axis. Thus, this is a variant of Motzkin paths with two kinds of level steps. The algebraic definition is L = 1 + Le + Lw + LnLs - w where each word is regarded as a noncommutative monomial with variables in S. Replacing each letter in S by x and L by the g.f. A(x) leads to x + A(x) = (1 + x*A(x))^2. If we let y = x + x*x*A, then y^2 - y = x^3 - x which is an elliptic curve. - Michael Somos, Mar 28 2020

The Hankel number wall for the sequence L(0), L(1), ... has a zigzag diagonal sequence b(0) = 1, b(1) = 1. b(2) = e, b(3) = ew+ns, b(4) = na(ee-ew-ns), ... which is a generalized Somos-5 sequence with b(i)*b(i+5) = -n*s*b(i+1)*b(i+4) + e*n*s*b(i+2)*b(i+3). Define sequence c(0) = 0, c(1) = 1, c(i) = b(i-2) for i>1, and c(i) = -(-n*s)^(-i)*c(-i) if i<0. Then c(i)*c(i+5) = -n*s*c(i+1)*c(i+4) + e*n*s*c(i+2)*c(i+3) for all i in Z. If e=w=n=s=1, then c(i) = A006769(i) * (-1)^[mod(i,4)=3]. - Michael Somos, Oct 14 2024

			L(0) = 1, L(1) = e, L(2) = ee + ew + ns, L(3) = eee + ewe + nse + eew + eww + nsw + nes + ens.
G.f. = 1 + x + 3*x^2 + 8*x^3 + 23*x^4 + 68*x^5 + 207*x^6 + 644*x^7 + ...

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
Michael Somos, Number Walls in Combinatorics.

Cf. A001006, A006720, A006769, A025262.

CoefficientList[Series[(1 - 2 x - Sqrt[1 - 4 x + 4 x^3])/(2 x^2), {x, 0, 26}], x] (* Michael De Vlieger, Oct 30 2019 *)
a[ n_] := SeriesCoefficient[ (2 - 2*x)/(1 - 2*x + (1 - 4*x + 4*x^3)^(1/2)), {x, 0, n}]; (* Michael Somos, Oct 27 2024 *)
a[ n_] := If[ n<0, 0, SeriesCoefficient[Nest[(1 + x*#)^2 - x&, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, Oct 27 2024 *)

{a(n) = if( n<0, 0, polcoef( (1 - 2*x - sqrt( 1 - 4*x + 4*x^3 + x^3 * O(x^n)) ) / (2*x^2), n))};

{a(n) = if( n<0, 0, polcoef( (2 - 2*x)/(1 - 2*x + (1 - 4*x + 4*x^3 + x*O(x^n))^(1/2)), n))}; /* Michael Somos, Oct 27 2024 */

L = 1 + Le + Lw + LnLs - w.
a(n) = 2*a(n-1) + a(0)*a(n-2) + ... + a(n-2)*a(0) for n>1.
The Somos-4 sequence A006720(n+2) is the Hankel transform of a(n-1). See A001906 for definition of Hankel transform.
Let s(n)= A006769(n). Then 0 = f( -s(n-1) * s(n+1) / s(n)^2, -s(n) * s(n+2) / s(n+1)^2 ) where f(u, v) = u + v - (1 + u*v)^2.
G.f. A(x) satisfies 0 = f(x, A(x)) where f(u, v) = u + v - (1 + u*v)^2.
G.f.: (1 - 2*x - sqrt( 1 - 4*x + 4*x^3) ) / (2*x^2).
From Paul Barry, Mar 04 2010: (Start)
G.f.: ((1-x)/(1-2x))c(x^2(1-x)/(1-2x)^2), c(x) the g.f. of A000108;
a(n) = Sum_{k=0..floor(n/2)} (A000108(k) * Sum_{i=0..k+1} C(k+1,i)*C(n-i,n-2k-i)*(-1)^i*2^(n-2k-i)). (End)
a(n) = A025262(n+2) if n >= 0.
0 = a(n)*(+16*a(n+1) - 64*a(n+3) + 22*a(n+4)) + a(n+1)*(+32*a(n+2) - 14*a(n+3)) + a(n+2)*(+16*a(n+3) - 10*a(n+4)) + a(n+3)*(+2*a(n+3) + a(n+4)) if n>=0. - Michael Somos, Jan 18 2015

A157003 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

Original entry on oeis.org

1, 1, 2, 4, 10, 27, 78, 234, 722, 2274, 7280, 23617, 77466, 256481, 856016, 2876940, 9728090, 33072228, 112974592, 387580856, 1334821448, 4613225722, 15994465796, 55615889745, 193904367362, 677709772035, 2374027931492, 8333765738127

Offset: 0

Paul Barry, Feb 20 2009

Image of the Catalan numbers A000108 by the Riordan array (1, x*(1-x^2)). Hankel transform is A006720(n+2).
Partial sums of A157002.
Empirical: number of Dyck n-paths that avoid any one of {UDUDD, UUDDD, UUDUD, UUUDD}. e.g. of the 5 Dyck 3-paths UUDUDD contains UDUDD so a(3)=4. Also, number of Dyck n-paths that avoid DUD that ends at height of form 3*k+1, or that avoid UDU that ends at height of form 3*k-1. e.g. of the 5 Dyck 3-paths UUDUDD contains DUD ending at height 1 so a(3)=4. - David Scambler, Mar 24 2011
Apparently: number of Dyck n-paths with no descent length equal to twice the preceding ascent length. - David Scambler, May 11 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
P. Gawrychowski and P. K. Nicholson, Encodings of Range Maximum-Sum Segment Queries and Applications, arXiv:1410.2847 [cs.DS], 2014-2015.

Cf. A000108.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x*(1-x^2)))/(2*x*(1-x^2)) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[(1-Sqrt[1-4*x*(1-x^2)])/(2*x*(1-x^2)),{x,0,20}],x] (* Vaclav Kotesovec, Jan 27 2015 *)

my(x='x+O('x^30)); Vec((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1-x^2))) \\ G. C. Greubel, Feb 26 2019

((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1-x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

G.f.: c(x*(1-x^2)) where c(x) is the g.f. of A000108;
a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2))*A000108(k)/2.
Conjecture: (n+1)*a(n) +(n+2)*a(n-1) +(-21*n+29)*a(n-2) +(3*n-16)*a(n-3) +40*(n-3)*a(n-4) +2*(-2*n+7)*a(n-5) +10*(-2*n+9)*a(n-6)=0. - R. J. Mathar, Nov 19 2014
Recurrence: (n+1)*a(n) = 2*(2*n-1)*a(n-1) + (n+1)*a(n-2) - 8*(n-2)*a(n-3) + 2*(2*n-7)*a(n-5). - Vaclav Kotesovec, Feb 01 2015
a(n) ~ sqrt(3-8*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*sin(arccos(-3^(3/2)/8)/3 - Pi/6)/sqrt(3). - Vaclav Kotesovec, Jun 05 2022

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

Vladimir Shevelev and Peter J. C. Moses, Apr 04 2016

nonn

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.

Seiichi Manyama, Table of n, a(n) for n = 0..318

Cf. A018896, A006720, A102276, A271835, A271831, A271837, A271838, A271839.

[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

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 *)

{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

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

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

A157005 A Somos-4 variant.

Original entry on oeis.org

1, 2, 8, 24, 112, 736, 3776, 40192, 391424, 4203008, 85312512, 1270368256, 32235102208, 1038278549504, 27640704385024, 1549962593927168, 73624753456480256, 4273828146025070592, 435765959975516766208

Offset: 0

Paul Barry, Feb 20 2009

Hankel transform of A157004.

G. C. Greubel, Table of n, a(n) for n = 0..145

Cf. A157101, A162546, A162547.

a:=[1,2,8,24];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019

I:=[1,2,8,24]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019

RecurrenceTable[{a[0]==1,a[1]==2,a[2]==8,a[3]==24,a[n]==(a[n-1] a[n-3]+a[n-2]^2)/a[n-4]},a,{n,20}]  (* Harvey P. Dale, Apr 30 2011 *)

m=20; v=concat([1,2,8,24], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019

def a(n):
    if (n==0): return 1
    elif (n==1): return 2
    elif (n==2): return 8
    elif (n==3): return 24
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), with a(0)=1, a(1)=2, a(2)=8, a(3)=24.

a(n) = 2^n*A006720(n+2).

A028937 Denominator of x-coordinate of (2n)*P where P = (0,0) is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 1, 1, 25, 16, 841, 16641, 4225, 13608721, 264517696, 12925188721, 5677664356225, 49020596163841, 158432514799144041, 62586636021357187216, 1870098771536627436025, 41998153797159031581158401, 15402543997324146892198790401

Offset: 1

N. J. A. Sloane

			a(4) = 25 where 8P = (21/25, -69/125).

Seiichi Manyama, Table of n, a(n) for n = 1..106
LMFDB, Elliptic Curve 37.a1 (Cremona label 37a1)
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006720, A051138, A028936 (numerator), A028938, A028939, A028941.

a(n)=denominator(ellmul(E,[0,0],2*n)[1]) \\ Charles R Greathouse IV, Mar 23 2022

P=(0, 0), 2P=(1, 0); if kP=(a, b) then (k+1)P = (a' = (b^2 - a^3)/a^2, b' = -1 - b*a'/a).
a(n) = A028941(2n). - Seiichi Manyama, Nov 19 2016
a(n) = a(-n) = b(n)*b(n+3) - b(n+1)*b(n+2) for all n in Z where b(n) = A006720(n). - Michael Somos, Mar 23 2022

cp's OEIS Frontend

A028941 Denominator of X-coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

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A056010 Number of words of length n in a simple grammar.

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A157003 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

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

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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