A002070 -id:A002070 - cp's OEIS frontend

A366346 a(n) = A002070(n) + A036689(n).

0, 5, 21, 40, 111, 160, 270, 342, 505, 812, 937, 1335, 1632, 1800, 2170, 2750, 3427, 3672, 4415, 4967, 5260, 6152, 6800, 7847, 9305, 10102, 10490, 11360, 11782, 12665, 16010, 17012, 18625, 19192, 22042, 22652, 24485, 26410, 27710, 29750, 31847, 32587, 36307

Offset: 1

Mats Granvik, Oct 07 2023

Cf. A366362, A002070, A036689, A001248, A272196.

b[n_] := If[n < 1, 0, With[{m = Prime@n}, SeriesCoefficient[q (Product[(1 - q^(11 k)), {k, Ceiling[m/11]}] Product[1 - q^k, {k, m}])^2, {q, 0, m}]]]; Table[Prime[n] (Prime[n] - 1) + b[n], {n, 1, 43}] (* after Michael Somos in A002070, Jul 04 2011 *)

a(n) = A002070(n) + A036689(n).
a(n) = Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} [GCD(f(x,y), prime(n)) = 1],
a(n) = Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} (1 - [MOD(f(x,y), prime(n)) = 0]) where f(x,y) = x^3 - x^2 - y^2 - y, in the last two formulas.
a(n) = A001248(n) - A272196(n), for n > 1.
a(n) = A366362(prime(n), 1).

A006571 Expansion of qProduct_{k>=1} (1-q^k)^2(1-q^(11*k))^2.

Original entry on oeis.org

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2, 2, -2, -1, 0, -4, -8, 5, -4, 0, 2, 7, 8, -1, 4, -2, -4, 3, 0, -4, 0, -8, -4, -6, 2, -2, 2, 8, 4, -3, 8, 2, 8, -6, -10, 1, 0, 0, 0, 5, -2, 12, -14, 4, -8, 4, 2, -7, -4, 1, 4, -3, 0, 4, -6, 4, 0, -2, 8, -10, -4, 1, 16, -6, 4, -2, 12, 0, 0, 15, 4, -8, -2, -7, -16, 0, -8, -7, 6, -2, -8

Offset: 1

N. J. A. Sloane

Number 23 of the 74 eta-quotients listed in Table I of Martin (1996).
Unique cusp form of weight 2 for congruence group Gamma_1(11). - Michael Somos, Aug 11 2011
For some elliptic curves with p-defects given by this sequence, and for more references, see A272196. See also the Michael Somos formula from May 23 2008 below. - Wolfdieter Lang, Apr 25 2016

			G.f.: q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + ...

Barry Cipra, What's Happening in the Mathematical Sciences, Vol. 5, Amer. Math. Soc., 2002; see p. 5.
M. du Sautoy, Review of "Love and Math: The Heart of Hidden Reality" by Edward Frenkel, Nature, 502 (Oct 03 2013), p. 36.
N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 42.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 412.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Wiles, Modular forms, elliptic curves and Fermat's last theorem, pp. 243-245 of Proc. Intern. Congr. Math. (Zurich), Vol. 1, 1994.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1002 terms from T. D. Noe)
J. Cowles, Some congruence properties of three well-known sequences: Two notes, J. Num. Theory 12(1) (1980) 84.
H. Darmon, A proof of the full Shimura-Taniyama-Weil conjecture is announced, Notices Amer. Math. Soc., Dec. 1999, pp. 1397-1401.
F. Diamond, Congruences between modular forms: raising the level and dropping Euler factors, in Elliptic curves and modular forms (Washington, DC, 1996). Proc. Nat. Acad. Sci. U.S.A. 94 (1997), 11143-11146.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
A. W. Knapp, Review of "Love and Math: The Heart of Hidden Reality" by E. Frenkel, Notices Amer. Math. Soc., 61 (2014), pp. 1056-1060; see p. 1058, but beware typos.
LMFDB, Newform orbit 11.2.a.a
LMFDB, Elliptic curve with LMFDB label 11.a3 (Cremona label 11a3)
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Shimura, Goro, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220.
G. Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220. [Annotated scan of pages 218, 219 only]
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A002070 (terms with prime indices), A032442, A030200.

[ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1),n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 29 2007 */

[ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 31 2007 */

Basis( CuspForms( Gamma1(11), 2), 101) [1]; /* Michael Somos, Jul 14 2014 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ q (Product[ (1 - q^k), {k, 11, n, 11}] Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, May 27 2014 *)

{a(n) = if( n<1, 0, ellak( ellinit( [0, -1, 1, 0, 0], 1), n))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, 1, a0=1; a1 = y = -sum(x=0, p-1, kronecker( 4*x^3 - 4*x^2 + 1, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */

CuspForms( Gamma1(11), 2, prec = 101).0 # Michael Somos, Aug 11 2011

Expansion of (eta(q) * eta(q^11))^2 in powers of q.
a(n) == A000594(n) (mod 11). [Cowles]. - R. J. Mathar, Feb 13 2007
Euler transform of period 11 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...]. - Michael Somos, Feb 12 2006
a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) for p != 11. - Michael Somos, Feb 12 2006
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u*w * (u + 4*v + 4*w) - v^3. - Michael Somos, Mar 21 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11 (t/i)^2 f(t) where q = exp(2 Pi i t).
Convolution square of A030200.
Coefficients of L-series for elliptic curve "11a3": y^2 + y = x^3 - x^2. - Michael Somos, May 23 2008
Convolution inverse is A032442. - Michael Somos, Apr 21 2015
a(prime(n)) = prime(n) - A272196(n), n >= 3.
  a(2) = -2 is not  2 - A272196(1) = 0. Modularity pattern of some elliptic curves. - Wolfdieter Lang, Apr 25 2016

A090305 a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.

Original entry on oeis.org

2, 16, 258, 4144, 66562, 1069136, 17172738, 275832944, 4430499842, 71163830416, 1143051786498, 18359992414384, 294902930416642, 4736806879080656, 76083812995707138, 1222077814810394864, 19629328849962024962

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Lim_{n-> infinity} a(n)/a(n+1) = 0.0622577... = 1/(8+sqrt(65)) = (sqrt(65)-8).

Lim_{n-> infinity} a(n+1)/a(n) = 16.0622577... = (8+sqrt(65)) = 1/(sqrt(65)-8).

			a(4) = 16*a(3) + a(2) = 16*4144 + 258 = (8+sqrt(65))^4 + (8-sqrt(65))^4 = 66561.99998497... + 0.00001502... = 66562.

G. C. Greubel, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (16,1).

Cf. A002070, A015910.
Lucas polynomials: A114525.
Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), this sequence (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25), A087281 (m=29), A087287 (m=76), A089772 (m=199).

m:=16;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 31 2019

m:=16; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 31 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 8*I)), n = 0..20); # G. C. Greubel, Dec 31 2019

LinearRecurrence[{16,1},{2,16},40] (* or *) With[{c=Sqrt[65]}, Simplify/@ Table[(c-8)((8+c)^n-(8-c)^n (129+16c)),{n,20}]] (* Harvey P. Dale, Oct 31 2011 *)
LucasL[Range[20]-1, 16] (* G. C. Greubel, Dec 31 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 8*I) ) \\ G. C. Greubel, Dec 31 2019

[2*(-I)^n*chebyshev_T(n, 8*I) for n in (0..20)] # G. C. Greubel, Dec 31 2019

a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.
a(n) = (8+sqrt(65))^n + (8-sqrt(65))^n.
a(n)^2 = a(2n) - 2 if n = 1, 3, 5, ...;
a(n)^2 = a(2n) + 2 if n = 2, 4, 6, ....
G.f.: (2-16*x)/(1-16*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 16) = 2*(-i)^n * ChebyshevT(n, 8*i). - G. C. Greubel, Dec 31 2019
E.g.f.: 2*exp(8*x)*cosh(sqrt(65)*x). - Stefano Spezia, Jan 01 2020

More terms from Ray Chandler, Feb 14 2004

A272196 Number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 4, 4, 9, 10, 9, 19, 19, 24, 29, 24, 34, 49, 49, 39, 59, 54, 49, 74, 74, 69, 89, 89, 74, 104, 99, 119, 89, 99, 104, 119, 149, 144, 129, 159, 149, 164, 159, 179, 179, 194, 174, 174, 189, 199, 199, 199, 204, 209, 214, 209, 269, 249, 274, 259, 249, 259, 299, 279, 299

Offset: 1

Wolfdieter Lang, Apr 22 2016

This elliptic curve is discussed in the second Silverman reference. The present sequence is given in the rows named N_p in Table 45.6, p. 403. In the rows named a_p the p-defects prime(n) - a(n) are shown.
This sequence also gives the number of solutions of the congruences y^2 + y == x^3 - x^2 - 10*x - 20 (mod prime(n)) as well as y^2 + y == x^3 - x^2 (mod prime(n)) for n > 1 (cf. A376073). The first one is given in the Martin and Ono reference in Theorem 2, first row of the table, and the second one is given in the Frenkel reference, p. 84. (Of course, one could change the sign of y in both congruences.)
The modularity pattern for the elliptic curve y^2 = x^3 - 4*x^2 + 16 (and the ones mentioned in the previous comment and a comment below) is exhibited by the modular cusp form of weight 2 and level 11 (eta(z)*eta(11*z))^2, where eta is the Dedekind function, which in the q = exp(2*Pi*i*z), (Im(z) > 0) expansion has coefficients given in A006571 (with A006571(0) = 0). For all odd primes (2 is a bad prime), A002070(n) = A006571(prime(n)) = prime(n) - a(n), n >= 2, the p-defect. A006571(2) = -2, not 2-2 = 0. Note that the discriminant of this elliptic curve is -2^8*11 (sometimes -2^12*11 is used). Prime 11 is also bad for this curve, but A006571(11) = 1 = 11 - a(5) = 11 - 10. The curve y^2 + y = x^3 - x^2 - 10*x - 20 has discriminant -11^5 (see the first Silverman reference, pp. 46-48).
From Wolfdieter Lang, Jan 02 2017: (Start)
The congruence y^2 + y == x^3 - x^2 - 7820*x - 263580 (mod p) as p runs through the odd primes has the same number of solutions. See the Cremona link, N=11.
If b_n(Q) is the number of solutions of the Diophantine equation Q(x1,x2,x3,x4) = n with the quadratic form Q(x1,x2,x3,x4) = x1^2 + 4*(x2^2+x3^2+x4^2) + x1*x3 + 4*x2*x3 + 3*x2*x4 + 7*x3*x4 then the theta series delta(q;Q) = 1 + Sum_{n>=1} b_n(Q)*q^n equals (1/5)*E(q) + (18/5)*f(q) with the expansion coefficients of E(q) given by A185699 and those of f(q) = (eta(z)*eta(11*z))^2 with q = exp(2*Pi*i*z), (Im(z) > 0) given by A006571. See the Moreno-Wagstaff reference, pp. 245-246. b_n(Q), E(q) and f(q) are there denoted by a_n(Q), 12*E_{Chi0}(z) and f(z), respectively, and a missing n in the numerator of E_{Chi0}(z) has to be added (see A185699). (End)

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2  == x^3 - 4*x^2 + 16 (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:  (0, 0), (1, 1)
2,   3,       4:  (0, 1), (0, 2), (1, 1), (1, 2)
3,   5,       4:  (0, 1), (0, 4), (4, 1), (4, 4)
4,   7,       9:  (0, 3), (0, 4), (2, 1), (2, 6),
                  (4, 3), (4, 4), (6, 2), (6, 5)
5,  11,      10:  (0, 4), (0, 7), (4, 4), (4, 7),
                  (6, 0), (7, 3), (7, 8), (9, 5),
                  (9, 6), (10, 0)
...
--------------------------------------------------

Edward Frenkel, Liebe und Mathematik, Springer, Spektrum, 2014, p. 84.
Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247 (corrected).
J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986, pp. 46-48.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Table 45.6, p. 403, Theorem 47.2, p. 413 (4th ed., Pearson 2014, Table 6, p. 369, Theorem 2, p. 383)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. E. Cremona, Algorithms for Modular Elliptic Curves.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247.
J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986, pp. 46-48.

Cf. A002070, A006571, A060457, A185699, A376073.

a(n) gives the number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod prime(n)), n >= 1.

A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Original entry on oeis.org

1, 0, 4, 5, 0, 4, 0, 8, 0, 8, 21, 0, 0, 0, 4, 0, 20, 0, 0, 0, 16, 40, 0, 0, 0, 0, 0, 9, 0, 32, 0, 16, 0, 0, 0, 16, 45, 0, 24, 0, 0, 0, 0, 0, 12, 0, 84, 0, 0, 0, 0, 0, 0, 0, 16, 111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32

Offset: 1

Mats Granvik, Oct 08 2023

Row n appears to have sum n^2. T(prime(m),1) = A366346(m). The number of nonzero terms in row n appears to be A320111(n).

			{
{1}, = 1^2
{0, 4}, = 2^2
{5, 0, 4}, = 3^2
{0, 8, 0, 8}, = 4^2
{21, 0, 0, 0, 4}, = 5^2
{0, 20, 0, 0, 0, 16}, = 6^2
{40, 0, 0, 0, 0, 0, 9}, = 7^2
{0, 32, 0, 16, 0, 0, 0, 16}, = 8^2
{45, 0, 24, 0, 0, 0, 0, 0, 12}, = 9^2
{0, 84, 0, 0, 0, 0, 0, 0, 0, 16}, = 10^2
{111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 11^2
{0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32} = 12^2
}

Cf. A127649, A366346, A000290, A060457, A002070, A036689, A001248, A272196.

f = x^3 - x^2 - y^2 - y; nn = 12; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Conjecture: T(n,n) = A060457(n).

A366450 a(n) = Sum_{k=1..n} A366362(n,k)*A023900(k)/n.

Original entry on oeis.org

1, -2, -1, -4, 1, 2, -2, -8, -3, -2, 1, 4, 4, 4, -1, -16, -2, 6, 0, -4, 2, -2, -1, 8, 5, -8, -9, 8, 0, 2, 7, -32, -1, 4, -2, 12, 3, 0, -4, -8, -8, -4, -6, -4, -3, 2, 8, 16, -14, -10, 2, -16, -6, 18, 1, 16, 0, 0, 5, 4, 12, -14, 6, -64, 4, 2, -7, 8, 1, 4, -3, 24, 4, -6, -5, 0, -2, 8, -10, -16, -27, 16, -6, -8

Offset: 1

Mats Granvik, Oct 10 2023

sign

It appears that: a(A005117(n)) = A006571(A005117(n)), verified up to n = 98. But also a(76) = A006571(76), a(116) = A006571(116) and a(152) = A006571(152). 76 = 19*2^2, 116 = 29*2^2 and 152 = 19*2^3.

Cf. A366362, A023900, A006571, A005117, A002070, A366362.

nn = 84; f = x^3 - x^2 - y^2 - y; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Monitor[Table[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0]*g[k]/n, {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}], n]

a(n) = sum(k=1, n, my(z=sumdivmult(k, d, d*moebius(d))); sum(y=1, n, sum(x=1, n, if (gcd(x^3 - x^2 - y^2 - y, n)==k, z/n)))); \\ Michel Marcus, Oct 10 2023

a(n) = Sum_{k=1..n} A366362(n,k)*A023900(k)/n.

cp's OEIS Frontend

A366346 a(n) = A002070(n) + A036689(n).

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A006571 Expansion of q*Product_{k>=1} (1-q^k)^2*(1-q^(11*k))^2.

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A090305 a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.

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A272196 Number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod p) as p runs through the primes.

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A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

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A366450 a(n) = Sum_{k=1..n} A366362(n,k)*A023900(k)/n.

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A376073 Number of solutions of the congruence y^2 + y == x^3 - x^2 (mod p) as p runs through the primes.

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A006571 Expansion of qProduct_{k>=1} (1-q^k)^2(1-q^(11*k))^2.