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 1-5 of 5 results.

A106273 Discriminant of the polynomial x^n - x^(n-1) - ... - x - 1.

Original entry on oeis.org

1, 5, -44, -563, 9584, 205937, -5390272, -167398247, 6042477824, 249317139869, -11597205023744, -601139006326619, 34383289858207744, 2151954708695291177, -146323302326154543104, -10742330662077208945103, 846940331265064719417344, 71373256668946058057974997
Offset: 1

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Author

T. D. Noe, May 02 2005

Keywords

Comments

This polynomial is the characteristic polynomial of the Fibonacci and Lucas n-step sequences. These discriminants are prime for n=2, 4, 6, 26, 158 (A106274). It appears that the term a(2n+1) always has a factor of 2^(2n). With that factor removed, the discriminants are prime for odd n=3, 5, 7, 21, 99, 405. See A106275 for the combined list.
a(n) is the determinant of an r X r Hankel matrix whose entries are w(i+j) where w(n) = x1^n + x2^n + ... + xr^n where x1,x2,...xr are the roots of the titular characteristic polynomial. E.g., A000032 for n=2, A001644 for n=3, A073817 for n=4, A074048 for n=5, A074584 for n=6, A104621 for n=7, ... - Kai Wang, Jan 17 2021
Luca proves that a(n) is a term of the corresponding k-nacci sequence only for n=2 and 3. - Michel Marcus, Apr 12 2025

Crossrefs

Cf. A086797 (discriminant of the polynomial x^n-x-1), A000045, A000073, A000078, A001591, A001592 (Fibonacci n-step sequences), A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755 (Lucas n-step sequences), A086937, A106276, A106277, A106278 (number of distinct zeros of these polynomials for n=2, 3, 4, 5).

Programs

  • Mathematica
    Discriminant[p_?PolynomialQ, x_] := With[{n=Exponent[p, x]}, Cancel[((-1)^(n(n-1)/2) Resultant[p, D[p, x], x])/Coefficient[p, x, n]^(2n-1)]]; Table[Discriminant[x^n-Sum[x^i, {i, 0, n-1}], x], {n, 20}]
  • PARI
    {a(n)=(-1)^(n*(n+1)/2)*((n+1)^(n+1)-2*(2*n)^n)/(n-1)^2}  \\ Max Alekseyev, May 05 2005
    
  • PARI
    a(n)=poldisc('x^n-sum(k=0,n-1,'x^k)); \\ Joerg Arndt, May 04 2013

Formula

a(n) = (-1)^(n*(n+1)/2) * ((n+1)^(n+1)-2*(2*n)^n)/(n-1)^2. - Max Alekseyev, May 05 2005

A106282 Primes p such that the polynomial x^3-x^2-x-1 mod p has no zeros; i.e., the polynomial is irreducible over the integers mod p.

Original entry on oeis.org

3, 5, 23, 31, 37, 59, 67, 71, 89, 97, 113, 137, 157, 179, 181, 191, 223, 229, 251, 313, 317, 331, 353, 367, 379, 383, 389, 433, 443, 449, 463, 467, 487, 509, 521, 577, 619, 631, 641, 643, 647, 653, 661, 691, 709, 719, 727, 751, 797, 823, 829, 839, 859, 881
Offset: 1

Views

Author

T. D. Noe, May 02 2005

Keywords

Comments

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step sequences, A000073 and A001644.
Primes of the form 3x^2+2xy+4y^2 with x and y in Z. - T. D. Noe, May 08 2005

Crossrefs

Primes in A028952.
Cf. A106276 (number of distinct zeros of x^3-x^2-x-1 mod prime(n)), A106294, A106302 (period of Lucas and Fibonacci 3-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

  • Mathematica
    t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 200}];Prime[Flatten[Position[t, 0]]]
  • PARI
    forprime(p=2,1000,if(#polrootsmod(x^3-x^2-x-1,p)==0,print1(p,", ")));
    /* Joerg Arndt, Jul 19 2012 */

A106279 Primes p such that the polynomial x^3-x^2-x-1 mod p has 3 distinct zeros.

Original entry on oeis.org

47, 53, 103, 163, 199, 257, 269, 311, 397, 401, 419, 421, 499, 587, 599, 617, 683, 757, 773, 863, 883, 907, 911, 929, 991, 1021, 1087, 1109, 1123, 1181, 1237, 1291, 1307, 1367, 1433, 1439, 1543, 1567, 1571, 1609, 1621, 1697, 1699, 1753, 1873, 1907, 2003
Offset: 1

Views

Author

T. D. Noe, May 02 2005

Keywords

Comments

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step sequences, A000073 and A001644. The periods of the sequences A000073(k) mod p and A001644(k) mod p have length less than p. For a given p, let the zeros be a, b and c. Then A001644(k) mod p = (a^k+b^k+c^k) mod p. This sequence is the same as A033209 except for the initial term.

Crossrefs

Cf. A106276 (number of distinct zeros of x^3-x^2-x-1 mod prime(n)), A106294, A106302 (periods of the Fibonacci and Lucas 3-step sequences mod prime(n)).

Programs

  • Mathematica
    t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 500}];Prime[Flatten[Position[t, 3]]]

A030200 Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q.

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, 0, 0, 2, 1, 0, 2, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 2, 0, 0, 0, 1, -1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, -1, 0, -2, 0, 0, -1, 0, 0, 0, 1, -1, -2, 0, 2, 1, 0, 1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 0, 0, 0, 2, 1, 0, 0, 0, 0
Offset: 0

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Author

Keywords

Comments

Number 52 of the 74 eta-quotients listed in Table I of Martin (1996).
In [Klein and Fricke 1892] on page 586 equation (3) first line left side has A_0 and the right side the power series r^{1/2} (1 - r - r^2 + r^5 + r^7 + ...) which is the g.f. of this sequence. A_0 and the other A_1, A_3, A_9, A_5, A_4 (in a permuted order) correspond to the nonzero 11-sections of the g.f. of this sequence. - Michael Somos, Nov 12 2014

Examples

			G.f. = 1 - x - x^2 + x^5 + x^7 - x^11 + x^13 - x^15 - x^16 - x^18 + 2*x^23 + ...
G.f. = q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2*q^47 +...
		

References

  • F. Klein and R. Fricke, Vorlesungen ueber die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1892, Vol. 2, see p. 586.
  • H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 203. MR1471703 (98g:14032)

Crossrefs

Programs

  • Magma
    Basis( CuspForms( Gamma1(44), 1), 162) [1]; /* Michael Somos, Nov 13 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11], {x, 0, n}]; (* Michael Somos, Nov 12 2014 *)
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; qfrep( [1, 0; 0, 11], n)[n] - qfrep( [3, 1; 1, 4], n)[n])}; /* Michael Somos, Nov 20 2006 */
    
  • PARI
    {a(n) = my(A, p, e, f); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==11, 1, f = sum( k=0, p-1, (k^3 - k^2 - k - 1)%p == 0); if( f==0, (e-1)%3-1, if( f==1, (1 + (-1)^e) / 2, e+1)))))}; /* Michael Somos, Nov 20 2006 */
    
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^11 + A), n))}; /* Michael Somos, Nov 20 2006 */
    

Formula

Euler transform of period 11 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, Nov 20 2006
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = 1, b(p^e) = (e-1)%3 - 1 if f=0, b(p^e) = e+1 if f=3, b(p^e) = (1 + (-1)^e) / 2 if f=1 where f = number of zeros of x^3 - x^2 - x - 1 modulo p. - Michael Somos, Nov 20 2006
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(11*k)).
a(n) = sum over all solutions to x^2 + x*y + 3*y^2 = 2*n + 1 with odd integer x>0 of (-1)^y. - Michael Somos, Jan 29 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
Convolution square is A006571.

A191060 Primes that are not squares mod 11.

Original entry on oeis.org

2, 7, 13, 17, 19, 29, 41, 43, 61, 73, 79, 83, 101, 107, 109, 127, 131, 139, 149, 151, 167, 173, 193, 197, 211, 227, 233, 239, 241, 263, 271, 277, 281, 283, 293, 307, 337, 347, 349, 359, 373, 409, 431, 439, 457, 461, 479, 491, 503, 523, 541, 547, 557, 563
Offset: 1

Views

Author

T. D. Noe, May 25 2011

Keywords

Comments

Inert rational primes in the field Q(sqrt(-11)). - N. J. A. Sloane, Dec 25 2017
These are also the primes p for which the polynomial x^3 - x^2 - x - 1 (mod p) has only one integer root. This is important for the Fibonacci and Lucas 3-step recursions, A000073 and A001644. See A106276. - T. D. Noe, Apr 17 2012
It appears that these are the primes p such that the sequence p^(5*n) mod 11 has period length 2, repeating [1, 10]. - Gary Detlefs, May 18 2014

Programs

  • Magma
    [p: p in PrimesUpTo(563) | JacobiSymbol(p, 11) eq -1]; // Vincenzo Librandi, Sep 11 2012
  • Mathematica
    Select[Prime[Range[200]], JacobiSymbol[#, 11] == -1 &]
Showing 1-5 of 5 results.