A006720 -id:A006720 - cp's OEIS frontend

A028935 a(n) = A006720(n)^3 (cubed terms of Somos-4 sequence).

1, 1, 1, 1, 8, 27, 343, 12167, 205379, 30959144, 3574558889, 553185473329, 578280195945297, 238670664494938073, 487424450554237378792, 2035972062206737347698803, 4801616835579099275862827431

Offset: 0

N. J. A. Sloane

nonn

If initial two 1's are omitted, denominator of y-coordinate of (2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

			5P = (1/4, -5/8).

Seiichi Manyama, Table of n, a(n) for n = 0..88
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006720, A028934, A028943, A028945, A151502.

I:=[1,1,1,1,8,27,343,12167,205379]; [n le 9 select I[n] else (129*Self(n-1)*Self(n-8) - 260*Self(n-2)*Self(n-7) - 8385*Self(n-3)*Self(n-6) + 48633*Self(n-4)*Self(n-5))/Self(n-9): n in [1..30]]; // G. C. Greubel, Feb 22 2018

b[n_ /; 0 <= n <= 4] = 1; b[n_]:= b[n] = (b[n-1]*b[n-3] + b[n-2]^2)/b[n -4]; Table[(b[n])^3, {n,0,30}] (* G. C. Greubel, Feb 21 2018 *)
a[n_ /; 0 <= n <= 3] = 1; a[4]:= 8; a[5]:= 27; a[6]:= 343; a[7]:= 12167; a[8]:= 205379; a[9]:= 30959144; a[n_]:= a[n] = (129*a[n-1]*a[n-8] - 260*a[n-2]*a[n-7] - 8385*a[n-3]*a[n-6] + 48633*a[n-4]*a[n-5])/a[n-9]; Table[a[n], {n,0,30}] (* G. C. Greubel, Feb 22 2018 *)

{b(n) = if(n< 4, 1, (b(n-1)*b(n-3) + b(n-2)^2)/b(n-4))};
for(n=0,30, print1((b(n))^3, ", ")) \\ G. C. Greubel, Feb 21 2018

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) = (129*a(n-1)*a(n-8) - 260*a(n-2)*a(n-7) - 8385*a(n-3)*a(n-6) + 48633*a(n-4)*a(n-5))/a(n-9). - G. C. Greubel, Feb 22 2018

Edited by N. J. A. Sloane, May 14 2009

A028945 a(n) = A006720(n)^2 (squared terms of Somos-4 sequence).

Original entry on oeis.org

1, 1, 1, 1, 4, 9, 49, 529, 3481, 98596, 2337841, 67387681, 6941055969, 384768368209, 61935294530404, 16063784753682169, 2846153597907293521, 2237394491744632911601, 1262082793174195430038441, 1063198259901027900600665796

Offset: 0

N. J. A. Sloane

If first two 1's are omitted, denominator of x-coordinate of (2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

Seiichi Manyama, Table of n, a(n) for n = 0..107
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006720, A028935, A028941, A028944.

I:=[1,1,1,1,4,9,49]; [n le 7 select I[n] else (- 4*Self(n-6)*Self(n-1) + 29*Self(n-5)*Self(n-2) + 116*Self(n-4)*Self(n-3) )/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

b[n_ /; 0 <= n <= 4] = 1; b[n_]:= b[n] = (b[n-1]*b[n-3] + b[n-2]^2)/b[n -4]; Table[(b[n])^2, {n,0,30}] (* G. C. Greubel, Feb 21 2018 *)

{b(n) = if(n< 4, 1, (b(n-1)*b(n-3) + b(n-2)^2)/b(n-4))};
for(n=0,30, print1((b(n))^2, ", ")) \\ G. C. Greubel, Feb 21 2018

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) = (- 4 a(n - 6) a(n - 1) + 29 a(n - 5) a(n - 2) + 116 a(n - 4) a(n - 3))/a(n-7). - Bill Gosper, May 14 2009
5P = (1/4, -5/8).
0 = a(n)*a(n+6) - 5*a(n+1)*a(n+5) + 4*a(n+2)*a(n+4) - 20*a(n+3)^2 for all n in Z. - Michael Somos, Apr 12 2020

Edited by N. J. A. Sloane, May 14 2009

A220838 Tropical version of Somos-4 sequence A006720.

Original entry on oeis.org

-1, 0, 0, 0, 1, 1, 2, 3, 3, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 22, 25, 27, 30, 33, 35, 39, 42, 45, 49, 52, 56, 60, 63, 68, 72, 76, 81, 85, 90, 95, 99, 105, 110, 115, 121, 126, 132, 138, 143, 150, 156, 162, 169, 175, 182, 189, 195, 203, 210, 217, 225, 232

Offset: 1

N. J. A. Sloane, Dec 23 2012

Given the generalized Somos-4 sequence with variables s(1), s(2), s(3), s(4), u, v and recursion s(n) = (u*s(n-1)*s(n-3) + v*s(n-2)^2)/s(n-4), then s(n) is a Laurent polynomial with denominator b(n) := s(1)^a(n)*s(2)^a(n-1)*s(3)^a(n-2)*s(4)^a(n-3) for all n in Z. Moreover, s(n)*b(n) is an irreducible polynomial for all n in Z. - Michael Somos, Sep 16 2023

			G.f. = -x + x^5 + x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 5*x^10 + 6*x^11 + 7*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, arXiv:1207.6072 [nlin.SI], 2012, See Example 3.6.
A. P. Fordy, Periodic Cluster Mutations and Related Integrable Maps, arXiv preprint arXiv:1403.8061 [math-ph], 2014.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. A006720, A236294.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*( x^6-x^5+x^4-x^2+2*x-1)/((1-x)^2*(1-x^8)))); // G. C. Greubel, Aug 10 2018

A118825x := proc(n)
    coeftayl((1-2*x+x^2)/(x^4+1),x=0,n) ;
end proc:
A056594 := proc(n)
    coeftayl(1/(x^2+1),x=0,n) ;
end proc:
A220838 := proc(n)
    -9/32-1/8*n+1/16*n^2+1/32*(-1)^n ;
    %+A118825x(n)/4 - A056594(n+3)/8  ;
end proc:
seq(A220838(n),n=0..80) ; # R. J. Mathar, Jan 30 2013

LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 1, -2, 1}, {-1, 0, 0, 0, 1, 1, 2, 3, 3, 5}, 62] (* Jean-François Alcover, Nov 26 2017 *)
a[ n_] := With[{m = n-1}, Floor[m^2/16] - Boole[Mod[m, 8] == 0]]; (* Michael Somos, Sep 16 2023 *)

{a(n) = if( n<1, n = 2-n); polcoeff( x * (x^6 - x^5 + x^4 - x^2 + 2*x - 1) / ( (1 - x)^2 * (1 - x^8) ) + x * O(x^n), n)} /* Michael Somos, Dec 27 2012 */

{a(n) = n--; n^2\16 - !(n%8)}; /* Michael Somos, Sep 16 2023 */

From Michael Somos, Dec 27 2012: (Start)
G.f.: x * (x^6 - x^5 + x^4 - x^2 + 2*x - 1) / ( (1 - x)^2 * (1 - x^8) ).
a(2-n) = a(n). (End)
Second difference has period 8. - Michael Somos, Dec 27 2012
a(n) = A236294(n-5) = max( a(n-1) + a(n-3), 2*a(n-2) ) - a(n-4) for all n in Z. - Michael Somos, Sep 16 2023

A129739 Primes in Somos-4 sequence (A006720).

Original entry on oeis.org

2, 3, 7, 23, 59, 8209, 620297, 1687054711, 25907979805412230144914099508240296236020415269340706571266102156690578761249, 167688864076998154482920561111926793545475633249050257599724515210137245508480818512193851652306467577687209241137

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

Cf. A006720, A129740, A129741.

Cf. A192241, primes in Dana Scott's sequence (A048736).

a(9)-a(10) from Robert G. Wilson v, Jul 04 2007

A129741 List of primitive prime divisors of the Somos-4 sequence (A006720) in their order of occurrence.

Original entry on oeis.org

2, 3, 7, 23, 59, 157, 11, 139, 8209, 9257, 620297, 983, 4003, 1847, 9803, 1687054711, 1433, 33008447, 83, 101, 113, 51563, 61, 823, 5381, 20117, 6329, 262650531833, 197, 10259, 519606349, 2621, 11887, 136667817691, 13933, 42591667, 564188663, 211, 8802371

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

Read A006720 term-by-term, factorize each term, write down any primes not seen before.

T. D. Noe, Table of n, a(n) for n = 1..121 (from the first 49 terms)
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

Cf. A006720, A129739, A129740.

Cf. A227199 (primes in this sequence).

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = (a[n - 1] a[n - 3] + a[n - 2]^2)/a[n - 4]; t = Array[a, 30]; t2 = {}; ps = {}; Do[f = Transpose[FactorInteger[t[[n]]]][[1]]; c = Complement[f, ps]; t2 = Join[t2, c]; ps = Union[ps, c], {n, 4, 30}]; t2 (* T. D. Noe, Nov 19 2013 *)
DeleteDuplicates[DeleteCases[Flatten[FactorInteger[#][[;;,1]]&/@RecurrenceTable[{a[0]==a[1]== a[2]==a[3]==1,a[n]==(a[n-1]a[n-3]+a[n-2]^2)/a[n-4]},a,{n,30}]],1]] (* Harvey P. Dale, May 25 2024 *)

Order of some of the terms corrected by T. D. Noe, Nov 19 2013

A254316 Hankel transform of a(n) is A006720(n+1). Hankel transform of a(n+1) is A006720(n+3).

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 299, 1172, 4677, 18947, 77746, 322545, 1350906, 5704822, 24265651, 103872254, 447146683, 1934538301, 8407277728, 36685185300, 160663301053, 705974374128, 3111584887543, 13752592535137, 60939737103636, 270672216346769, 1204862348053296

Offset: 0

Michael Somos, Jan 28 2015

nonn

			G.f. = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 78*x^5 + 299*x^6 + 1172*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A006720.

CoefficientList[Series[(1-2*x+x^2-Sqrt[(1-4*x+x^2)^2-4*x^3])/(2*x*(1 - x)), {x, 0, 60}], x] (* G. C. Greubel, Aug 04 2018 *)

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

Given g.f. A(x), 0 = (x^2-x)*A(x)^2 + (x^2-2*x+1)*A(x) + (2*x-1).
G.f.: (1 - 2*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 )) / (2*x*(1 - x)).
Conjecture: +(n+1)*a(n) +(-8*n+3)*a(n-1) +(18*n-29)*a(n-2) +(-12*n+31)*a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Jun 07 2016

A129740 Indices of primes in Somos-4 sequence (A006720).

Original entry on oeis.org

4, 5, 6, 7, 8, 11, 13, 16, 43, 52, 206, 647

Offset: 1

N. J. A. Sloane, May 13 2007

Sequence is probably finite - see A006720.

G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

Cf. A006720, A129739, A129741.

More terms (based on the entry A006720) from Stefan Steinerberger, Jun 06 2007

A151502 a(n) = A006720(n)^4 (fourth powers of Somos-4 sequence).

Original entry on oeis.org

1, 1, 1, 1, 16, 81, 2401, 279841, 12117361, 9721171216, 5465500541281, 4541099550557761, 48178257964790528961, 148046697174216601867681, 3835980708567891638880403216, 258045180612631702971803868544561, 8100590302880631846481071607248577441

Offset: 0

N. J. A. Sloane, May 14 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..76

Cf. A006720(n)^k; A006720 (k=1), A028945 (k=2), A028935 (k=3), this sequence (k=4).

I:=[1,1,1,1]; [n le 4 select I[n] else ((Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4))^4: n in [1..15]]; // G. C. Greubel, Sep 25 2018

b[n_ /; 0 <= n <= 4] = 1; b[n_]:= b[n] = (b[n-1]*b[n-3] + b[n-2]^2)/b[n -4]; Table[(b[n])^4, {n, 0, 20}] (* G. C. Greubel, Sep 25 2018 *)

{b(n) = if(n<4, 1, (b(n-1)*b(n-3) + b(n-2)^2)/b(n-4))};
for(n=0, 20, print1((b(n))^4, ", ")) \\ G. C. Greubel, Sep 25 2018

b=vector(20); b[1]=b[2]=b[3]=1;b[4]=2; for(n=5, #b, b[n]=(b[n-1]*b[n-3]+b[n-2]^2)/b[n-4]); concat(1, vector(20, n, b[n]^4)) \\ Altug Alkan, Sep 25 2018

a(n) = A028945(n)^2 = A006720(n)^4. - Seiichi Manyama, Nov 20 2016

A227199 Primes that divide some term of A006720.

Original entry on oeis.org

2, 3, 7, 11, 23, 41, 47, 53, 59, 61, 71, 73, 83, 97, 101, 113, 127, 139, 149, 157, 173, 179, 181, 191, 197, 199, 211, 223, 229, 239, 257, 263, 271, 277, 281, 307, 331, 337, 347, 359, 373, 379, 383, 389, 397, 409, 419, 433, 439, 443, 463, 467, 479, 499, 509

Offset: 1

Jeremy Rouse, Sep 18 2013

The density of primes in this sequence is 11/21.

If p divides A006720(n) for some n, then the smallest such n is <= (1/2)*((sqrt(p)+1)^2+3).

			11 is in the sequence because 11 divides A006720(10) = 1529.

Jeremy Rouse, Table of n, a(n) for n = 1..10000
R. Jones and J. Rouse, Galois Theory of Iterated Endomorphisms, Proceedings of the London Mathematical Society, 100, no. 3 (2010), 763-794.

[ n : n in [2..500] | IsPrime(n) and (n ne 37) and (Order(EllipticCurve([GF(n)!0,0,1,-1,0])![0,0,1]) mod 2 eq 1) ];

A141604 Triangle, read by rows, T(n,k) = round(A006720(n)/(A006720(n-k)*A006720(k))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 4, 7, 4, 2, 1, 1, 3, 8, 12, 12, 8, 3, 1, 1, 3, 8, 20, 15, 20, 8, 3, 1, 1, 5, 14, 45, 52, 52, 45, 14, 5, 1, 1, 5, 26, 66, 109, 170, 109, 66, 26, 5, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 21 2008

The round-function in the definition is round-to-nearest, not Mathematica's round-to-even. - R. J. Mathar, Jul 12 2012

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   1,   1,   1;
  1,   2,   2,   2,   1;
  1,   2,   3,   3,   2,   1;
  1,   2,   4,   7,   4,   2,   1;
  1,   3,   8,  12,  12,   8,   3,   1;
  1,   3,   8,  20,  15,  20,   8,   3,   1;
  1,   5,  14,  45,  52,  52,  45,  14,   5,   1;
  1,   5,  26,  66, 109, 170, 109,  66,  26,   5,   1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A006720.

A006720:= [n le 4 select 1 else (Self(n-1)*Self(n-3)+Self(n-2)^2)/Self(n-4): n in [1..30]];
A141604:= func< n,k | Round(A006720[n+1]/(A006720[k+1]*A006720[n-k+1])) >;
[A141604(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 21 2024

A141604 := proc(n,m)
        round(A006720(n)/A006720(n-m)/A006720(m)) ;
end proc:
seq(seq(A141604(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 12 2012

f[n_]:= f[n]= If[n<4, 1, (f[n-1]*f[n-3] +f[n-2]^2)/f[n-4]]; (* A006720 *)
A141604[n_, k_]:= Round[f[n]/(f[k]*f[n-k])];
Table[A141604[n,k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Sep 21 2024 *)

def f(n): # f = A006720
    if n<4: return 1
    else: return (f(n-1)*f(n-3) +f(n-2)^2)/f(n-4)
def A141604(n,k): return round(f(n)/(f(k)*f(n-k)))
flatten([[A141604(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 21 2024

cp's OEIS Frontend

A028935 a(n) = A006720(n)^3 (cubed terms of Somos-4 sequence).

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A028945 a(n) = A006720(n)^2 (squared terms of Somos-4 sequence).

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A220838 Tropical version of Somos-4 sequence A006720.

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A129739 Primes in Somos-4 sequence (A006720).

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A129741 List of primitive prime divisors of the Somos-4 sequence (A006720) in their order of occurrence.

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A254316 Hankel transform of a(n) is A006720(n+1). Hankel transform of a(n+1) is A006720(n+3).

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A129740 Indices of primes in Somos-4 sequence (A006720).

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A151502 a(n) = A006720(n)^4 (fourth powers of Somos-4 sequence).