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.

Previous Showing 11-14 of 14 results.

A255842 a(n) = 2*n^2 + 12.

Original entry on oeis.org

12, 14, 20, 30, 44, 62, 84, 110, 140, 174, 212, 254, 300, 350, 404, 462, 524, 590, 660, 734, 812, 894, 980, 1070, 1164, 1262, 1364, 1470, 1580, 1694, 1812, 1934, 2060, 2190, 2324, 2462, 2604, 2750, 2900, 3054, 3212, 3374, 3540, 3710, 3884, 4062, 4244, 4430
Offset: 0

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Author

Avi Friedlich, Mar 08 2015

Keywords

Comments

This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.
Equivalently, numbers m such that 2*m - 24 is a square.
For n = 0..10, a(n) - 1 is prime (see A092968).

Links

Crossrefs

Cf. A016825 (first differences), A092968, A114949.
Subsequence of A047238 and A047406.
Cf. similar sequences listed in A255843.

Programs

  • Magma
    [2*n^2+12: n in [0..50]];
  • Mathematica
    Table[2 n^2 + 12, {n, 0, 50}]
  • PARI
    vector(50, n, n--; 2*n^2+12)
  • Sage
    [2*n^2+12 for n in (0..50)]

Formula

G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A114949(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)
E.g.f.: 2*exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 24 2025

Extensions

Edited by Bruno Berselli, Mar 11 2015

A267068 a(n) = (n+1) / A189733(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 2, 11, 1, 13, 2, 3, 2, 17, 1, 19, 2, 3, 2, 23, 1, 25, 2, 3, 2, 29, 1, 31, 2, 3, 2, 35, 1, 37, 2, 3, 2, 41, 1, 43, 2, 3, 2, 47, 1, 49, 2, 3, 2, 53, 1, 55, 2, 3, 2, 59, 1, 61, 2, 3, 2, 65, 1, 67, 2, 3, 2
Offset: 0

Views

Author

Paul Curtz, Jan 10 2016

Keywords

Comments

A189733(n) is the denominator of an autosequence of the first kind (the main diagonal is A000004).

Crossrefs

Cf. A000004, A047238, A047245, A052901, A130196, A189733.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 2 x + 3 x^2 + 2 x^3 + 5 x^4 + x^5 + 5 x^6 - 2 x^7 - 3 x^8 - 2 x^9 + x^10 - x^11)/((1 - x)^2 (1 + x)^2 (1 - x + x^2)^2 (1 + x + x^2)^2), {x, 0, 69}], x] (* or *)
b[m_, n_] := b[m, n] = Which[m == n, 0, n == m + 1, (-1)^(n + 1)/n, n > m, b[m, n - 1] + b[m + 1, n - 1], n < m, b[m - 1, n + 1] - b[m - 1, n]]; Table[(n + 1)/Denominator@ b[0, n], {n, 0, 69}] (* Michael De Vlieger, Jan 15 2016, Jean-François Alcover at A189733 *)

Formula

a(2n+1) = A130196(n+1).
A052901(n+2) = period 3: 2, 3, 2 is at rank A047245(n+1) = 1, 2, 3, 7, 8, 9, ... .
Conjectures from Colin Barker, Jan 10 2016: (Start)
a(n) = 2*a(n-6) - a(n-12) for n>11.
G.f.: (1+2*x+3*x^2+2*x^3+5*x^4+x^5+5*x^6-2*x^7-3*x^8-2*x^9+x^10-x^11) / ((1-x)^2*(1+x)^2*(1-x+x^2)^2*(1+x+x^2)^2).
(End)
a(3n) + a(3n+1) + a(3n+2) = A047238(n+3).

A295821 Number of coprime pairs (a,b) with -n <= a <= n, -2 <= b <= 2.

Original entry on oeis.org

2, 12, 16, 24, 28, 36, 40, 48, 52, 60, 64, 72, 76, 84, 88, 96, 100, 108, 112, 120, 124, 132, 136, 144, 148, 156, 160, 168, 172, 180, 184, 192, 196, 204, 208, 216, 220, 228, 232, 240, 244, 252, 256, 264, 268, 276, 280, 288, 292, 300, 304, 312, 316, 324, 328
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2017

Keywords

Links

Crossrefs

Column k=2 of A295782.
Cf. A047238, A137243.

Programs

  • PARI
    a(n) = sum(x=-n, n, sum(y=-2, 2, gcd(x,y)==1)); \\ Michel Marcus, Jan 19 2025

Formula

From Alois P. Heinz, Jan 19 2025: (Start)
G.f.: -2*(x^3-x^2-5*x-1)/((x+1)*(x-1)^2).
a(n) = 2 * A047238(n+2) for n>=1. (End)
a(n) = 6*n + 5 - (-1)^n for n>=1. - Keagan Boyce, Jan 19 2025
Sum_{n>=0} (-1)^n/a(n) = 1/4 + Pi/(24*sqrt(3)) + log(3)/8. - Amiram Eldar, Jan 23 2025

A058301 Number of solutions to c(0)F(0) + ... + c(n)F(n) = 0, where c(i) = +-1 for i >= 0, number of (+1)'s >= number of (-1)'s, F(i) = A000045(i) = Fibonacci numbers.

Original entry on oeis.org

1, 0, 2, 3, 0, 6, 4, 0, 8, 11, 0, 22, 16, 0, 32, 42, 0, 84, 64, 0, 128, 165, 0, 330, 256, 0, 512, 654, 0, 1308, 1024, 0, 2048, 2605, 0, 5210, 4096, 0, 8192, 10398, 0, 20796, 16384, 0, 32768, 41550, 0, 83100, 65536, 0, 131072, 166116, 0, 332232, 262144, 0
Offset: 0

Views

Author

Naohiro Nomoto, Dec 08 2000

Keywords

Examples

			a(3) = 3 because +0+1+1-2 = -0+1+1-2 = +0-1-1+2 = 0;
a(5) = 6 because +0+1-1-2-3+5 = +0-1+1-2-3+5 = +0+1-1+2+3-5 = -0+1-1+2+3-5 = +0-1+1+2+3-5 = -0-1+1+2+3-5 = 0.

Crossrefs

Cf. A000045, A002083, A047270, A047238.

Formula

a(3n+1) = 0, a(A047270(n)) = A002083(n+5), a(A047238(n)) = 2^n.

Extensions

More terms from Sean A. Irvine, Aug 02 2022
Previous Showing 11-14 of 14 results.

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