A241850 -id:A241850 - cp's OEIS frontend

A114962 a(n) = n^2 + 14.

14, 15, 18, 23, 30, 39, 50, 63, 78, 95, 114, 135, 158, 183, 210, 239, 270, 303, 338, 375, 414, 455, 498, 543, 590, 639, 690, 743, 798, 855, 914, 975, 1038, 1103, 1170, 1239, 1310, 1383, 1458, 1535, 1614, 1695, 1778, 1863, 1950, 2039, 2130, 2223, 2318, 2415, 2514

Offset: 0

Cino Hilliard, Feb 21 2006

Old name was: "Numbers of the form x^2 + 14".

x^2 + 14 != y^n for all x,y and n > 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. E. Cohn, The diophantine equation x^2 + C = y^n, Acta Arithmetica LXV.4 (1993).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A155136, n^2 - 28; A000290, n^2; A114948, n^2 + 10.

Cf. sequences of the type n^2 + k: A002522 (k=1), A059100 (k=2), A117950 (k=3), A087475 (k=4), A117951 (k=5), A114949 (k=6), A117619 (k=7), A189833 (k=8), A189834 (k=9), A114948 (k=10), A189836 (k=11), A241748 (k=12), A241749 (k=13), this sequence (k=14), A241750 (k=15), A241751 (k=16), A241847 (k=17), A241848 (k=18), A241849 (k=19), A241850 (k=20), A241851 (k=21), A114963 (k=22), A241889 (k=23), A241890 (k=24), A114964 (k=30).

[n^2+14: n in [0..60]]; // Vincenzo Librandi, Apr 30 2014

Table[n^2 + 14, {n, 0, 50}]  (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
CoefficientList[Series[(14 - 27 x + 15 x^2)/(1 - x)^3, {x, 0, 80}], x] (* Vincenzo Librandi, Apr 30 2014 *)

G.f.: (14-27*x+15*x^2)/(1-x)^3. - Colin Barker, Jan 11 2012
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(14)*Pi*coth(sqrt(14)*Pi))/28.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(14)*Pi*cosech(sqrt(14)*Pi))/28. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
E.g.f.: exp(x)*(14 + x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Added 14 from Vincenzo Librandi, Apr 30 2014
Definition changed by Bruno Berselli, Mar 13 2015
Offset corrected by Amiram Eldar, Nov 02 2020

A114964 a(n) = n^2 + 30.

Original entry on oeis.org

30, 31, 34, 39, 46, 55, 66, 79, 94, 111, 130, 151, 174, 199, 226, 255, 286, 319, 354, 391, 430, 471, 514, 559, 606, 655, 706, 759, 814, 871, 930, 991, 1054, 1119, 1186, 1255, 1326, 1399, 1474, 1551, 1630, 1711, 1794, 1879, 1966, 2055, 2146, 2239, 2334, 2431, 2530

Offset: 0

Cino Hilliard, Feb 21 2006

x^2 + 30 != y^n for all x,y and n > 1, so this is a subsequence of A007916.
From Bruno Berselli, May 12 2014: (Start)
This is the case k=5 of the identity n^2 + k*(k+1) = (Sum_{i=-k..k} (n+i)^3)/((2*k+1)*n).
Similar sequences: A059100 (k=1), A114949 (k=2), A241748 (k=3), A241850 (k=4). (End)
The old name of this sequence was: Numbers of the form x^2 + 30. Also numbers that are not a perfect power.

			11*4*a(4) = (-1)^3 + 0^3 + 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = 2024. - _Bruno Berselli_, May 12 2014

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
J. H. E. Cohn, The diophantine equation x^2 + C = y^n, Acta Arithmetica LXV.4 (1993).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007916, A059100, A114949, A241748, A241850.

Range[0,60]^2+30 (* Harvey P. Dale, Oct 17 2022 *)

g(n,p) = for(x=0,n,y=x^2+p;print1(y","));

a(n) = n^2 + 30; \\ Altug Alkan, Apr 30 2018

From Amiram Eldar, Nov 04 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(30)*Pi*coth(sqrt(30)*Pi))/60.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(30)*Pi*cosech(sqrt(30)*Pi))/60. (End)
From Elmo R. Oliveira, Dec 30 2024: (Start)
G.f.: (30 - 59*x + 31*x^2)/(1 - x)^3.
E.g.f.: (30 + x + x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

New name from Shawn A. Broyles and Altug Alkan, Apr 30 2018

A114948 a(n) = n^2 + 10.

Original entry on oeis.org

10, 11, 14, 19, 26, 35, 46, 59, 74, 91, 110, 131, 154, 179, 206, 235, 266, 299, 334, 371, 410, 451, 494, 539, 586, 635, 686, 739, 794, 851, 910, 971, 1034, 1099, 1166, 1235, 1306, 1379, 1454, 1531, 1610, 1691, 1774, 1859, 1946, 2035, 2126, 2219, 2314, 2411, 2510

Offset: 0

Cino Hilliard, Feb 21 2006

Conjecture: n^2 + 10 != x^k for all n,x, and k > 1.

The conjecture is true: See Cohn. - James Rayman, Feb 14 2023

J. H. E. Cohn, The diophantine equation x^2 + C = y^n, Acta Arithmetica LXV.4 (1993).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A114962, A114963, A114964, A114965, A241850.

a[n_]:=n^2+10; a[Range[200]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011*)

From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(10)*Pi*coth(sqrt(10)*Pi))/20.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(10)*Pi*cosech(sqrt(10)*Pi))/20. (End)
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = (3/sqrt(10))*sinh(3*Pi)/sinh(sqrt(10)*Pi).
Product_{n>=0} (1 + 1/a(n)) = sqrt(11/10)*sinh(sqrt(11)*Pi)/sinh(sqrt(10)*Pi). (End)
From Elmo R. Oliveira, Jan 25 2025: (Start)
G.f.: (10 - 19*x + 11*x^2)/(1 - x)^3.
E.g.f.: (10 + x + x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Edited by Charles R Greathouse IV, Aug 09 2010

a(0) = 10 prepended by Elmo R. Oliveira, Jan 26 2025

A243618 Table read by antidiagonals: T(n,k) is the curvature of a circle in a nested Pappus chain (see Comments for details).

Original entry on oeis.org

2, 6, 3, 12, 7, 6, 20, 13, 10, 11, 30, 21, 16, 15, 18, 42, 31, 24, 21, 22, 27, 56, 43, 34, 29, 28, 31, 38, 72, 57, 46, 39, 36, 37, 42, 51, 90, 73, 60, 51, 46, 45, 48, 55, 66, 110, 91, 76, 65, 58, 55, 56, 61, 70, 83, 132

Offset: 0

Kival Ngaokrajang, Jun 07 2014

Refer to sequential curvatures from Wikipedia. For any integer k > 0, there exists an Apollonian gasket defined by the following curvatures:
  (-k, k+1, k*(k+1), k*(k+1)+1).
For example, the gaskets defined by (-1, 2, 2, 3), (-2, 3, 6, 7), (-3, 4, 12, 13), ..., all follow this pattern (all curvatures are integral). Because every interior circle that is defined by k+1 can become the bounding circle (defined by -k) in another gasket, these gaskets can be nested. When one considers only circles that contact both circles -k and k+1, the pattern will be nested Pappus chains. T(n,k) is the curvature when n = 0 is the circle at the center and n > 0 is in the clockwise direction, k >= 1 for each nested iteration. See illustration in links.

			Table begins:
n/k   1   2   3    4    5    6    7  ...
0     2   6  12   20   30   42   56  ...
1     3   7  13   21   31   43   57  ...
2     6  10  16   24   34   46   60  ...
3    11  15  21   29   39   51   65  ...
4    18  22  28   36   46   58   72  ...
5    27  31  37   45   55   67   80  ...
6    38  42  48   56   66   78   91  ...
7    51  55  61   68   79   91  105  ...
8    66  70  76   83   94  106  120  ...
9    83  87  93  101  111  123  137  ...
..   ..  ..  ..  ...  ...  ...  ...

Kival Ngaokrajang, Illustration of initial terms.
Wikipedia, Apollonian gasket.
Wikipedia, Pappus chain.

Cf. Column 1 = A059100(n), column 2 = A114949(n), column 3 = A241748(n), column 4 = A241850(n), column 5 = A114964(n), row 0 = A002378(k), row 1 = A002061(k+1).

A114965 a(n) = n^2 + 34.

Original entry on oeis.org

34, 35, 38, 43, 50, 59, 70, 83, 98, 115, 134, 155, 178, 203, 230, 259, 290, 323, 358, 395, 434, 475, 518, 563, 610, 659, 710, 763, 818, 875, 934, 995, 1058, 1123, 1190, 1259, 1330, 1403, 1478, 1555, 1634, 1715, 1798, 1883, 1970, 2059, 2150, 2243, 2338, 2435

Offset: 0

Cino Hilliard, Feb 21 2006

Conjecture: n^2 + 34 != x^k for all n,x and k > 1.

The conjecture is true: See Cohn. - James Rayman, Feb 14 2023

J. H. E. Cohn, The diophantine equation x^2 + C = y^n, Acta Arithmetica LXV.4 (1993).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A114948, A114962, A114963, A114964, A241850.

34+Range[50]^2  (* Harvey P. Dale, Jan 28 2011 *)

PARI
```
a(n)=n^2+34
```

From Elmo R. Oliveira, Jan 25 2025: (Start)
G.f.: (34 - 67*x + 35*x^2)/(1 - x)^3.
E.g.f.: (34 + x + x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Edited by Charles R Greathouse IV, Aug 09 2010

a(0) = 34 prepended by Elmo R. Oliveira, Jan 26 2025

cp's OEIS Frontend

A114962 a(n) = n^2 + 14.

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A114964 a(n) = n^2 + 30.

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A114948 a(n) = n^2 + 10.

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A243618 Table read by antidiagonals: T(n,k) is the curvature of a circle in a nested Pappus chain (see Comments for details).

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A114965 a(n) = n^2 + 34.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions