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.

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A130554 Denominators of partial sums for a series for 2*Pi*sqrt(3)/9.

Original entry on oeis.org

1, 6, 5, 140, 1260, 110, 60060, 72072, 680680, 12932920, 38798760, 11440660, 185910725, 5736673800, 4621209450, 3438179830800, 10314539492400, 140744203600, 59365905078480, 127212653739600, 4056670180362800
Offset: 1

Views

Author

Wolfdieter Lang, Jul 13 2007

Keywords

Comments

Numerators are given in A130553.
For the rationals r(n) := 2*Sum_{j=1..n} 1/(j*binomial(2*j,j)), n >= 1, the A. Comtet reference and a W. Lang link see A130553.

Crossrefs

Cf. A130553 (numerators), A248897 (2*Pi*sqrt(3)/9).

Programs

  • PARI
    a(n) = denominator(2*sum(j=1, n, 1/(j*binomial(2*j, j)))); \\ Michel Marcus, Nov 09 2019

Formula

a(n) = denominator(r(n)), n >= 1, with the rationals r(n) defined above.

A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

0, 1, 1, -2, -1, 0, -2, 2, 1, -2, 0, 0, 2, 2, 2, 0, -1, 0, -2, 2, 0, -4, 0, 0, -2, 1, 2, -2, -2, 0, 0, 2, 1, 0, 0, 0, 2, 2, 2, -4, 0, 0, -4, 2, 0, 0, 0, 0, 2, 3, 1, 0, -2, 0, -2, 0, 2, -4, 0, 0, 0, 2, 2, -4, -1, 0, 0, 2, 0, 0, 0, 0, -2, 2, 2, -2, -2, 0, -4, 2
Offset: 0

Views

Author

Michael Somos, Aug 04 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = x + x^2 - 2*x^3 - x^4 - 2*x^6 + 2*x^7 + x^8 - 2*x^9 + 2*x^12 + ...

Links

Crossrefs

Cf. A033687, A033762, A097195, A093829, A112848, A113447, A123530, A123863, A227696, A246838, A248897, A253243, A260941, A260942, A260943, A260944.
Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(36), 1), 80); A[2] + A[3] - 2*A[4] - A[5] - 2*A[7] + 2*A[8] + A[9] - 2*A[10] + 2*A[13] + 2*A[14] + 2*A[15] - A[17] - 2*A[19] - 4*A[20];
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ {1, 1, 0, -1, -1, 0}[[Mod[ d, 6, 1]]] {1, 0, -2, 0, 1, 0}[[Mod[ n/d, 6, 1]]], {d, Divisors @ n}]]
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -(-1)^#2, # == 3, -2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] EllipticTheta[ 3, 0, q] / (2 q^(1/4) QPochhammer[ q^6]), {q, 0, n}];
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 1, 0, -1, -1][d%6 + 1] * [0, 1, 0, -2, 0, 1][n\d%6 + 1]))};
  • PARI
    {a(n) = if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, -2, p%6==5, 1-e%2, e+1)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

Formula

Expansion of (a(q) + a(q^2) - 3*a(q^3) - 2*a(q^4) - 3*a(q^6) + 6*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q * phi(q) * psi(-q) * psi(-q^9) / f(-q^6) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^4 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, -3, 1, -2, 1, -2, 1, -2, 0, -3, 1, -1, 1, -3, 1, -2, 1, -2, 1, -2, 1, -3, 1, -1, 1, -3, 0, -2, 1, -2, 1, -2, 1, -3, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, -3, -2, -1, 0, 1, 2, 0, 0, -1, 6, 1, 0, 3, -2, -1, 0, 1, 2, -3, 0, -1, -6, 1, 0, 0, -2, -1, 0, 1, 2, 3, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = -2, if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123863.
a(2*n) = A112848(n). a(2*n + 1) = A123530(n). a(3*n) = -2 * A113447(n). a(3*n + 1) = A227696(n).
a(4*n) = - A112848(n). a(4*n + 1) = A253243(n). a(4*n + 2) = A123530(n). a(4*n + 3) = -2 * A246838(n).
a(6*n) = -2 * A093829(n). a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n). a(6*n + 3) = -2 * A033762(n). a(6*n + 5) = 0.
a(8*n + 1) = A260941(n). a(8*n + 2) = A253243(n). a(8*n + 3) = -2 * A260943(n). a(8*n + 4) = - A123530(n). a(8*n + 5) = 2 * A260942(n). a(8*n + 6) = -2 * A246838(n). a(8*n + 7) = 2 * A260944(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A371858 Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^7) dx.

Original entry on oeis.org

1, 0, 3, 4, 3, 7, 6, 0, 5, 5, 2, 6, 6, 7, 9, 6, 4, 8, 2, 9, 4, 5, 3, 0, 6, 4, 0, 6, 5, 1, 2, 4, 8, 8, 7, 4, 8, 3, 6, 4, 2, 5, 6, 7, 2, 6, 4, 2, 7, 3, 3, 7, 5, 8, 1, 0, 2, 8, 3, 3, 2, 6, 8, 8, 1, 5, 2, 5, 9, 3, 1, 0, 0, 7, 4, 8, 6, 2, 5, 4, 8, 5, 5, 5, 2, 0, 7, 5, 8, 9, 3, 8, 1, 8, 2, 0, 0, 0, 5, 9, 6, 0
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 09 2024

Keywords

Examples

			1.0343760552667964829453064065124887483642567...

Links

Crossrefs

Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^k) dx: A019669 (k=2), A248897 (k=3), A093954 (k=4), A352324 (k=5), A019670 (k=6), this sequence (k=7), A352125 (k=8).
Cf. A019674, A047336, A121598.

Programs

  • Mathematica
    RealDigits[(1/7) Pi Csc[Pi/7], 10, 102][[1]]

Formula

Equals (1/7) * Pi * csc(Pi/7).
Equals A019674 * A121598.
Equals Product_{k>=2} (1 + (-1)^k/A047336(k)). - Amiram Eldar, Nov 22 2024

A267803 Number of unit hexagonal lattice cells (hexagon in the origin) enclosed by origin centered circle of radius n.

Original entry on oeis.org

1, 1, 7, 13, 19, 31, 43, 61, 85, 97, 121, 151, 175, 211, 241, 271, 313, 349, 385, 439, 499, 535, 583, 649, 691, 757, 823, 877, 955, 1027, 1087, 1159, 1249, 1321, 1401, 1483, 1561, 1663, 1765, 1827, 1945, 2053, 2133, 2233, 2347, 2443, 2563, 2677, 2779, 2905
Offset: 1

Views

Author

Luca Petrone, Jan 20 2016

Keywords

Links

Crossrefs

Cf. A053458, A267710.
Cf. A248897 (2*Pi/(3*sqrt(3))).

Formula

Limit_{n->oo} a(n)/(n^2) = 2*Pi/(3*sqrt(3)).

A370562 Decimal expansion of (2*Pi - 3*sqrt(3))/2.

Original entry on oeis.org

5, 4, 3, 5, 1, 6, 4, 4, 2, 2, 3, 6, 4, 7, 7, 2, 9, 8, 1, 7, 1, 4, 7, 3, 8, 7, 1, 0, 2, 0, 6, 9, 4, 3, 3, 3, 7, 8, 2, 9, 6, 1, 5, 1, 8, 6, 5, 9, 5, 3, 4, 8, 7, 8, 8, 9, 1, 2, 3, 4, 1, 2, 3, 1, 2, 9, 9, 1, 6, 8, 8, 0, 9, 2, 3, 0, 0, 8, 9, 4, 3, 0
Offset: 0

Views

Author

Wolfdieter Lang, Mar 15 2024

Keywords

Comments

This constant is the difference of the area of a disk with radius 1 (length unit) and the inscribed regular hexagon.

Examples

			0.5435164422364772981714738710206943337829615186595348788912...

Crossrefs

Cf. A000796, A019692, A010482, A104956, A248897.

Programs

  • Mathematica
    RealDigits[Pi - 3*Sqrt[3]/2, 10, 120][[1]] (* Amiram Eldar, Mar 15 2024 *)

Formula

Equals (A019692 - A010482)/2.
Equals Pi - 3*sqrt(3)/2 = A000796 - A104956.
Previous Showing 21-25 of 25 results.

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