Michael Somos - cp's OEIS frontend

A385121 a(n+1) = 12*a(n) - a(n-1), a(0) = a(1) = 2, a(n) = a(1-n).

2, 2, 22, 262, 3122, 37202, 443302, 5282422, 62945762, 750066722, 8937854902, 106504192102, 1269112450322, 15122845211762, 180205030090822, 2147337515878102, 25587845160446402, 304906804409478722, 3633293807753298262, 43294618888630100422

Offset: 0

Michael Somos, Jun 18 2025

If x = 2, y = 6, z = a(n), w = a(n+1), then x^2+y^2+z^2+w^2 = x*y*z*w.

			G.f. = 2 + 2*x + 22*x^2 + 262*x^3 + 3122*x^4 + 37202*x^5 + ...

Index entries for linear recurrences with constant coefficients, signature (12,-1).

Cf. A061292, A077417.

a[ n_] := Which[n<1, a[1-n], n==1, 2, True, 12*a[n-1] - a[n-2]];

{a(n) = if(n<1, a(1-n), n==1, 2, 12*a(n-1) - a(n-2))};

G.f.: (2 - 22*x)/(1 - 12*x + x^2).
0 = 40 + a(n)^2 - 12*a(n)*a(n+1) + a(n+1)^2 for all n in Z.
a(n) = 2 * A077417(n-1).
E.g.f.: 2*exp(6*x)*(7*cosh(sqrt(35)*x) - sqrt(35)*sinh(sqrt(35)*x))/7. - Stefano Spezia, Aug 29 2025

A377310 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 and point (1, 0).

Original entry on oeis.org

0, 1, 1, 1, -3, -4, -13, 23, 87, 415, -152, -8063, -38727, -142471, 2309453, 13609844, 187790979, -1743980081, -25547499185, -575984295329, 1873521429456, 217675476797921, 5045023692031697, 65853623974941521, -5934036772012185603, -157454833217800083092

Offset: 0

Michael Somos, Oct 23 2024

sign

Bisection of A210098 (even part). The other bisection is A277279.
The elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 has LMFDB label 57.a1 (Cremona label 57a1).
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = -3.

			G.f. = x + x^2 + x^3 - 3*x^4 - 4*x^5 - 13*x^6 + 23*x^7 + 87*x^8 + 415*x^9 + ...

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
LMFDB, Elliptic Curve 57a1

Cf. A210098, A277279.

b:= proc(n) option remember; `if`(n<6, [0, 1$4, 2][n+1],
      (b(n-1)*b(n-4) -b(n-2)*b(n-3)) / b(n-5))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 05 2025

a[ n_] := a[n] = Which[ n<0, -a[-n], n<5, {0, 1, 1, 1, -3}[[n+1]], True, (a[n-1]*a[n-3] - a[n-2]^2)/a[n-4]];

{a(n) = my(v); if(n<0, -a(-n), n<5, [0, 1, 1, 1, -3][n+1], v = vector(n, i, if(i<5, a(i))); for(i=5, n, v[i] = (v[i-1]*v[i-3] - v[i-2]^2)/v[i-4]); v[n])};

{a(n) = my(E = ellinit([0, -1, 1, -2, 2]), z); z = ellpointtoz(E, [1, 0]); -(-1)^n*round(ellsigma(E, n*z)/ellsigma(E, z)^(n^2))};

{a(n) = my(E = ellinit([0, -1, 1, -2, 2])); sign(n) * subst( elldivpol( E, abs(n)), x, 1)};

a(n) = A210098(2*n).
a(n) = -a(-n) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) + a(n+2)^2 for all n in Z.
0 = a(n)*a(n+5) - a(n+1)*a(n+4) - 3*a(n+2)*a(n+3) for all n in Z.
0 = a(n+1)^2*a(n+2)^2 - a(n)^2*a(n+3)^2 - a(n)*a(n+2)^3 - a(n+1)^3*a(n+3) - 2*a(n)*a(n+1)*a(n+2)*a(n+3) for all n in Z.

Duplicate term a(15)=2309453 removed by Georg Fischer, May 05 2025

A362718 Expansion of e.g.f. cos(x)exp(x^2/2) = Sum_{n>=0} a(n)x^(2n)/(2n)!.

Original entry on oeis.org

1, 0, -2, -16, -132, -1216, -12440, -138048, -1601264, -18108928, -161934624, 404007680, 92590134208, 4221314202624, 159324751301248, 5730872535686144, 205239818509082880, 7450322829180649472, 276342876017093172736, 10509280308463090102272

Offset: 0

Michael Somos, Apr 30 2023

sign

Cf. A001464.

a[ n_] := If[ n<0, 0, (2*n)! * SeriesCoefficient[ Cos[x] * Exp[x^2/2], {x, 0, 2*n}]];

{a(n) = my(A); if( n<0, 0, A = x*O(x^(2*n)); (2*n)! * polcoef( cos(x + A)*exp(x^2/2 + A), 2*n))};

def egfExpand(f, step, size) -> list[int]:
    x = LazyPowerSeriesRing(QQ, "x").gen()
    return [f(x)[step*n] * factorial(step*n) for n in range(size+1)]
def egf(x): return cos(x)*exp(x^2/2)
print(egfExpand(egf, 2, 19))  # Peter Luschny, May 02 2023

a(n) = (-1)^n * A001464(2*n).

0 = a(n)*(360*a(n+2) -600*a(n+3) +230*a(n+4) -28*a(n+5) +a(n+6)) +a(n+1)*(216*a(n+2) -296*a(n+3) +84*a(n+4) -6*a(n+5)) +a(n+2)*(66*a(n+2) -56*a(n+3) +15*a(n+4)) -10*a(n+3)^2 for all n >= 0.

A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (mx + 2)a(m+1) - a(m) for all m in Z.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 10, 16, 6, 1, 20, 70, 76, 24, 1, 35, 224, 496, 428, 120, 1, 56, 588, 2260, 3808, 2808, 720, 1, 84, 1344, 8140, 23008, 32152, 21096, 5040, 1, 120, 2772, 24772, 107328, 245560, 298688, 178848, 40320

Offset: 0

Michael Somos, Mar 15 2023

This sequence is essentially A204024 except for extra row, alternating signs and reversed rows.

The sequence of polynomials a(m) satisfies a(m)*a(m-2) = a(m-1) * (a(m-1) + x*a(m-2) + a(m-3)) - a(m-2)^2 for all m > 3.

			a(3) = 1, a(4) = 1 + x, a(5) = 1 + 4*x + 2*x^2.
Triangular array T(n, k) starts:
n\k | 0   1   2   3   4   5
--- + - --- --- --- --- ---
 0  | 1
 1  | 1   1
 2  | 1   4   2
 3  | 1  10  16   6
 4  | 1  20  70  76  24
 5  | 1  35 224 496 428 120

Cf. A000292, A040977, A058797, A093986, A204024.

T[ n_, k_] := If[ n<0, 0, Module[{a = Table[1, n+3], x}, Do[ a[[m]] = a[[m-1]] *(a[[m-1]] + x*a[[m-2]] + a[[m-3]])/a[[m-2]] - a[[m-2]] //Factor//Expand, {m, 4, n+3}]; Coefficient[ a[[n+3]], x, k]]];

{T(n, k) = if( n<0, 0, my(a = vector(n+3, i, 1)); for(m = 4, n+3, a[m] = a[m-1]*(a[m-1] + 'x*a[m-2] + a[m-3])/a[m-2] - a[m-2]); polcoeff( a[n+3], k))};

If x=1, then a(n) = A058797(n+2) = Sum_{k=0..n} T(n, k).
If x=2, then a(n) = A093986(n+2).
T(n, n) = n!, T(n, 0) = 1, T(n, 1) = A000292(n). T(n, 2) = 2*A040977(n-2).

A360381 Generalized Somos-5 sequence a(n) = (a(n-1)a(n-4) + a(n-2)a(n-3))/a(n-5) = -a(-n), a(1) = 1, a(2) = -1, a(3) = a(4) = 1, a(5) = -7.

Original entry on oeis.org

0, 1, -1, 1, 1, -7, 8, -1, -57, 391, -455, -2729, 22352, -175111, 47767, 8888873, -69739671, 565353361, 3385862936, -195345149609, 1747973613295, -4686154246801, -632038062613231, 34045765616463119, -319807929289790304, -11453004955077020783

Offset: 0

Michael Somos, Feb 04 2023

This has the same recurrence as Somos-5 (A006721) with different initial values.
The elliptic curve y^2 + xy = x^3 + x^2 - 2x (LMFDB label 102.a1) has infinite order point P = (2, 2). The x and y coordinates of n*P have denominators a(n)^2 and |a(n)^3| respectively.
If b(2*n) = 6^(1/4)*a(2*n), b(2*n+1) = a(2*n+1), then b(n) is a generalized Somos-4 sequence with b(n+2)*b(n-2) = 6^(1/2)*b(n+1)*b(n-1) - b(n)*b(n) for all n in Z.
This is the sequence T_n in the Hone 2022 paper.

			5*P = (50/49, 20/343) and a(5) = -7, 6*P = (121/64, -1881/512) and a(6) = 8.

Andrew N. W. Hone, Heron triangles with two rational medians and Somos-5 sequences, European Journal of Mathematics, 8 (2022), 1424-1486; arXiv:2107.03197 [math.NT], 2021-2022.
Andrew N. W. Hone, Heron triangles and the hunt for unicorns, arXiv:2401.05581 [math.NT], 2024.
LMFDB, Elliptic Curve 102.a1 (Cremona label 102a1)

Cf. A006721, A241595.

a[0] = 0; a[1] = a[3] = a[4] = 1; a[2] = -1; a[5] = -7;
a[n_?Negative] := -a[-n];
a[n_] := a[n] = (a[n-1] a[n-4] + a[n-2] a[n-3]) / a[n-5]; (* Andrey Zabolotskiy, Feb 05 2023 *)
a[ n_] := Module[{A = Table[1, Max[5, Abs[n]]]}, A[[2]] = -1; A[[5]] = -7; Do[ A[[k]] = (A[[k-1]]*A[[k-4]] + A[[k-2]]*A[[k-3]])/A[[k-5]], {k, 6, Length[A]}]; If[n==0, 0, Sign[n]*A[[Abs[n]]] ]];

{a(n) = my(A = vector(max(5, abs(n)), k, 1)); A[2] = -1; A[5] = -7; for(k=6, #A, A[k] = (A[k-1]*A[k-4] + A[k-2]*A[k-3])/A[k-5]); if(n==0, 0, sign(n)*A[abs(n)])};

{a(n) = my(E = ellinit([1, 1, 0, -2, 0])); subst(elldivpol(E, n), 'x, 2) *(-1)^(n-1) / 6^((n-1)%2 + n^2\4)}; /* Michael Somos, Mar 01 2025 */

a(2*n) = -A241595(n+1), a(n) = -a(-n) for all n in Z.
From Michael Somos, Aug 19 2025: (Start)
Let S(n) = A006721(n+2) as in Hone. We have for all n in Z:
S(2*n) = S(n-1)*S(n)*a(n-1)*a(n+2) - S(n-2)*S(n+1)*a(n)*a(n+1).
S(2*n+1) = S(n)*S(n+1)*a(n-1)*a(n+2) - S(n-1)*S(n+2)*a(n)*a(n+1).
a(2*n) = a(n)*(a(n-2)*a(n+1)^2 - a(n+2)*a(n-1)^2).
a(2*n+1) = a(n-1)*a(n)^2*a(n+3) - a(n+2)*a(n+1)^2*a(n-2).
S(n-3)*S(n) = S(n-2)*S(n-1) - a(n-2)*a(n-1).
a(n-3)*a(n) = S(n-2)*S(n-1) + a(n-2)*a(n-1).
(End)

A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)a(n-4) + 42a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 1, 3, 13, 113, 1525, 57123, 2165017, 262621633, 42422452969, 14070212996451, 7658246457672229, 10650393355715621873, 15512114571284835412957, 75606222210863532170808003, 452005526897888844293504165425

Offset: 0

Michael Somos, Jan 29 2023

nonn

The elliptic curve y^2 = x^3 - 2*x (LMFDB label 256.b1) has infinite order point P = (2, 2) and 2-torsion point T = (0, 0). The x and y coordinates of n*P + T have denominators a(n)^2 and a(n)^3 respectively.

			2*P + T = (-8/9, -28/27). 3*P + T  = (-1/169, 239/2197).

LMFDB, Elliptic Curve 256.b1 (Cremona label 256b1)

Cf. A166929.

a[ m_] := With[{n = Abs[m]}, If[ n<3, {1, 1, 3}[[n+1]], (-a[n-1]*a[n-4] + 42*a[n-2]*a[n-3])/a[n-5]]];

{a(n) = my(E = ellinit([-2, 0])); sqrtint(denominator(elladd(E, [0, 0], ellmul(E, [2, 2], n))[1]))};

{a(n) = my(A); n = abs(n); A = vector(max(4, n+1), k, 1); A[3] = 3; A[4] = 13; for(k = 4, n, A[k+1] = (if(k%2, 4, 8)*A[k]*A[k-2] + A[k-1]^2)/A[k-3]); A[n+1]};

a(2*n-1) = A166929(n) for all n in Z.

A357537 a(n) = 2*A080635(n) if n > 0. a(0) = 1.

Original entry on oeis.org

1, 2, 2, 6, 18, 78, 378, 2214, 14562, 108702, 897642, 8171766, 81066258, 871695918, 10091490138, 125189658054, 1656458307522, 23288226400062, 346663764078282, 5447099463010326, 90094171024954098, 1564653992673809358, 28467075416816935098, 541467979789775621094

Offset: 0

Michael Somos, Oct 02 2022

nonn

			G.f. = 1 + 2*x + 2*x^2 + 6*x^3 + 18*x^4 + 78*x^5 + 378*x^6 + ...
E.g.f. = 1 + 2*x + x^2 + x^3 + 3/4*x^4 + 13/20*x^5 + 21/40*x^6 + ...

Cf. A080635.

a[ n_] := If[ n < 0, 0, n! Simplify@SeriesCoefficient[ Sqrt[3] Tan[ Pi/6 + x Sqrt[3]/2], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, Nest[Expand[(1 + x + x^2) D[#, x]]&, 1 + 2 x, n] /. x->0];

{a(n) = my(A); if(n<0, 0, A = 1 + 2*x; for( k=1, n, A = A' * (1 + x + x^2)); polcoeff(A, 0))};

E.g.f.: sqrt(3) tan(Pi/6 + x sqrt(3)/2).
E.g.f. A(x) satisfies 2*A' = 3 + A^2, A'' = A*A'.
Let f(x) = 1 + x + x^2. Then a(n+1) = (f(x)*d/dx)^n f'(x) evaluated at x = 0.

A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - xy - x^2y*f(x, y+1)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 2, 6, 6, 1, 0, 5, 16, 20, 10, 1, 0, 15, 51, 71, 50, 15, 1, 0, 52, 186, 281, 231, 105, 21, 1, 0, 203, 759, 1223, 1114, 616, 196, 28, 1, 0, 877, 3409, 5795, 5701, 3564, 1428, 336, 36, 1, 0, 4140, 16655, 29634, 31011, 21187, 9780

Offset: 1

Michael Somos, Sep 27 2022

Row sums are A000110.

			Triangle starts:
  1,
  0,   1,
  0,   1,    1,
  0,   1,    3,    1,
  0,   2,    6,    6,    1,
  0,   5,   16,   20,   10,    1,
  0,  15,   51,   71,   50,   15,    1,
  0,  52,  186,  281,  231,  105,   21,  1,
  0, 203,  759, 1223, 1114,  616,  196,  28,  1,
  0, 877, 3409, 5795, 5701, 3564, 1428, 336, 36, 1,
  ...

Cf. A000110, A049347, A074664.

T[ n_, k_] := If[n < 0, 0, Coefficient[SeriesCoefficient[ Nest[ 1/(1 - x*y - x^2*y*(#/.y -> y+1))&, 1 + O[x], Ceiling[n/2]], {x, 0, n}], y, k]];

{T(n, k) = if(n < 0, 0, f = 1 + O(x); forstep(i=1, n, 2, f = 1/(1 - x*y - x^2*y*subst(f, y, y+1))); polcoef(polcoef(f, n), k))};

f(x, -1) = 1/(1 + x + x^2).

x + x^2*f(x, 2) = 1 - 1/f(x, 1) is g.f. for A074664.

A352625 A (25,-29) Somos-4 sequence.

Original entry on oeis.org

1, 2, 7, 59, 1529, 83313, 7869898, 1687054711, 1123424582771, 1662315215971057, 4257998884448335457, 23385756731869683322514, 397068399296019032727466599, 15886280085653574502219650145963, 1107464108502549897934954766675333353, 157131202095317153373302215985417166354641

Offset: 0

Michael Somos, Mar 24 2022

nonn

Hankel transform of A188314 with first term omitted.

			G.f.: 1 + 2*x + 7*x^2 + 59*x^3 + 1529*x^4 + 83313*x^5 + ...
a(2) = 7 = 2*16 - 5*5 = det([2, 5; 5, 16]).

Cf. A006720, A188313, A188314.

b[ n_] := If[OddQ[n], a[-(n-1)/2], a[n/2-1]]; a[ n_] := If[-3<=n<=1, {23, 3, 1, 1, 2}[[n+4]], 2*b[1-n]^3*b[2-n] + b[-n]^2*(b[2-n]*b[3-n] - b[1-n]*b[4-n])];

a(n) = (25*a(n-1)*a(n-3) - 29*a(n-2)^2)/a(n-4) for all n in Z.
a(n) = (29*a(n-1)*a(n-4) - 13*a(n-2)*a(n-3))/a(n-5) for all n in Z.
a(n) = b(1-2*n) = b(2*n+2) = A188313(-1-n) for all n in Z where b(n) = A006720(n).

A338218 Number of terms in polynomial sequence s(n) = xyz(s(n-1)s(n-3) + s(n-2)^2)/s(n-4), with s(1) = x, s(2) = s(3) = 1, s(4) = y.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 12, 23, 43, 80, 140, 233, 387, 612, 930, 1411, 2067, 2936, 4170, 5768, 7796, 10537, 13960, 18163, 23639, 30285, 38249, 48322, 60285, 74340, 91706, 111967, 135403

Offset: 1

Michael Somos, Jan 29 2021

The Somos-4 polynomial sequence is s(n).

			a(6) = 3 because s(6) = x*y^3*z^2 + x*y^3*z + x*y^2*z^2 has 3 terms.

Michael Somos, Somos polynomials

Cf. A333260.

a[ n_] := If[1 <= n <= 4, 1, RecurrenceTable[{s[m]*s[m - 4] == x*y*z*(s[m - 1]*s[m - 3] + s[m - 2]^2), s[1] == x, s[2] == 1, s[3] == 1, s[4] == y}, s, {m, Max[n, 5 - n]}] // Last // Factor // Expand // Length];

a(n) = a(5-n) for all n in Z.

a(31)-a(34) from Jinyuan Wang, Feb 14 2021

cp's OEIS Frontend

User: Michael Somos

A385121 a(n+1) = 12*a(n) - a(n-1), a(0) = a(1) = 2, a(n) = a(1-n).

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A377310 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 and point (1, 0).

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A362718 Expansion of e.g.f. cos(x)*exp(x^2/2) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.

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A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.

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A360381 Generalized Somos-5 sequence a(n) = (a(n-1)*a(n-4) + a(n-2)*a(n-3))/a(n-5) = -a(-n), a(1) = 1, a(2) = -1, a(3) = a(4) = 1, a(5) = -7.

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A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)*a(n-4) + 42*a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.

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A357537 a(n) = 2*A080635(n) if n > 0. a(0) = 1.

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A362718 Expansion of e.g.f. cos(x)exp(x^2/2) = Sum_{n>=0} a(n)x^(2n)/(2n)!.

A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (mx + 2)a(m+1) - a(m) for all m in Z.

A360381 Generalized Somos-5 sequence a(n) = (a(n-1)a(n-4) + a(n-2)a(n-3))/a(n-5) = -a(-n), a(1) = 1, a(2) = -1, a(3) = a(4) = 1, a(5) = -7.

A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)a(n-4) + 42a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.

A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - xy - x^2y*f(x, y+1)).

A338218 Number of terms in polynomial sequence s(n) = xyz(s(n-1)s(n-3) + s(n-2)^2)/s(n-4), with s(1) = x, s(2) = s(3) = 1, s(4) = y.