A114962 -id:A114962 - cp's OEIS frontend

A241850 a(n) = n^2 + 20.

20, 21, 24, 29, 36, 45, 56, 69, 84, 101, 120, 141, 164, 189, 216, 245, 276, 309, 344, 381, 420, 461, 504, 549, 596, 645, 696, 749, 804, 861, 920, 981, 1044, 1109, 1176, 1245, 1316, 1389, 1464, 1541, 1620, 1701, 1784, 1869, 1956, 2045, 2136, 2229, 2324, 2421, 2520

Offset: 0

Vincenzo Librandi, May 01 2014

The only solution for x at the Diophantine equation x^2 + 20 = y^m (with m > 2) is 14: 14^2 + 20 = a(14) = 6^3. - Bruno Berselli, May 01 2014

Vincenzo Librandi, 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, p. 379.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Magma
```
[n^2+20: n in [0..60]];
```
Mathematica
```
Table[n^2 + 20, {n, 0, 60}]
```

a(n)=n^2+20 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (20 - 39*x + 21*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 03 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(20)*Pi*coth(sqrt(20)*Pi))/40.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(20)*Pi*cosech(sqrt(20)*Pi))/40. (End)
E.g.f.: exp(x)*(20 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

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

A241751 a(n) = n^2 + 16.

Original entry on oeis.org

16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 116, 137, 160, 185, 212, 241, 272, 305, 340, 377, 416, 457, 500, 545, 592, 641, 692, 745, 800, 857, 916, 977, 1040, 1105, 1172, 1241, 1312, 1385, 1460, 1537, 1616, 1697, 1780, 1865, 1952, 2041, 2132, 2225, 2320, 2417, 2516

Offset: 0

Vincenzo Librandi, May 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Cf. A000290, A243451 (primes).

a241751 = (+ 16) . (^ 2)  -- Reinhard Zumkeller, Apr 11 2015

Magma
```
[n^2+16: n in [0..60]];
```
Mathematica
```
Table[n^2 + 16, {n, 0, 60}]
```

a(n)=n^2+16 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (16 - 31*x + 17*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 03 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + 4*Pi*coth(4*Pi))/32.
Sum_{n>=0} (-1)^n/a(n) = (1 + 4*Pi*cosech(4*Pi))/32. (End)
E.g.f.: exp(x)*(16 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A114963 a(n) = n^2 + 22.

Original entry on oeis.org

22, 23, 26, 31, 38, 47, 58, 71, 86, 103, 122, 143, 166, 191, 218, 247, 278, 311, 346, 383, 422, 463, 506, 551, 598, 647, 698, 751, 806, 863, 922, 983, 1046, 1111, 1178, 1247, 1318, 1391, 1466, 1543, 1622, 1703, 1786, 1871, 1958, 2047, 2138, 2231, 2326, 2423, 2522, 2623

Offset: 0

Cino Hilliard, Feb 21 2006

Old name was: "Numbers of the form x^2 + 22".

x^2 + 22 != y^n for all x,y and n > 1.

Vincenzo Librandi, 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. similar sequences listed in A114962.

[n^2+22: n in [0..60]]; // Vincenzo Librandi, Apr 30 2014

Table[n^2 + 22, {n, 0, 60}] (* Vincenzo Librandi, Apr 30 2014 *)

a(n)=n^2+22 \\ Amiram Eldar, Nov 04 2020

G.f.: (22 - 43*x + 23*x^2)/(1 - x)^3. - Vincenzo Librandi, Apr 30 2014
From Amiram Eldar, Nov 04 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(22)*Pi*coth(sqrt(22)*Pi))/44.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(22)*Pi*cosech(sqrt(22)*Pi))/44. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
E.g.f.: exp(x)*(22 + x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

a(0)=22 from Vincenzo Librandi, Apr 30 2014
Definition changed by Bruno Berselli, Mar 13 2015
Offset corrected by Amiram Eldar, Nov 04 2020

A241748 a(n) = n^2 + 12.

Original entry on oeis.org

12, 13, 16, 21, 28, 37, 48, 61, 76, 93, 112, 133, 156, 181, 208, 237, 268, 301, 336, 373, 412, 453, 496, 541, 588, 637, 688, 741, 796, 853, 912, 973, 1036, 1101, 1168, 1237, 1308, 1381, 1456, 1533, 1612, 1693, 1776, 1861, 1948, 2037, 2128, 2221, 2316, 2413, 2512

Offset: 0

Vincenzo Librandi, Apr 30 2014

3/a(n) = R(n)/r, n >= 0, with R(n) the n-th radius of the counterclockwise Pappus chain of the arbelos with semicircle radii r, r1 = 3*r/4, r2 = r - r1 = r/4. See a comment on A114949 also for the MathWorld Pappus chain link. - Wolfdieter Lang, Jun 29 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of Pappus chain (3/4, 1/4).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Cf. A114964 (see comment), A114949.

Magma
```
[n^2+12: n in [0..60]];
```
Mathematica
```
Table[n^2 + 12, {n, 0, 60}]
```

a(n)=n^2+12 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (12-23*x+13*x^2)/(1-x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
(2*n)*a(n) = (n+2)^3 + (n-2)^3; also, 2*a(n) = (n+sqrt(12))^2 + (n-sqrt(12))^2. - Bruno Berselli, Mar 13 2015
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(12)*Pi*coth(sqrt(12)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(12)*Pi*cosech(sqrt(12)*Pi))/24. (End)
E.g.f.: exp(x)*(12 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A241847 a(n) = n^2 + 17.

Original entry on oeis.org

17, 18, 21, 26, 33, 42, 53, 66, 81, 98, 117, 138, 161, 186, 213, 242, 273, 306, 341, 378, 417, 458, 501, 546, 593, 642, 693, 746, 801, 858, 917, 978, 1041, 1106, 1173, 1242, 1313, 1386, 1461, 1538, 1617, 1698, 1781, 1866, 1953, 2042, 2133, 2226, 2321, 2418, 2517

Offset: 0

Vincenzo Librandi, May 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Magma
```
[n^2+17: n in [0..60]];
```
Mathematica
```
Table[n^2 + 17, {n, 0, 60}]
```

a(n)=n^2+17 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (17 - 33*x + 18*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 03 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(17)*Pi*coth(sqrt(17)*Pi))/34.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(17)*Pi*cosech(sqrt(17)*Pi))/34. (End)
E.g.f.: exp(x)*(17 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A241749 a(n) = n^2 + 13.

Original entry on oeis.org

13, 14, 17, 22, 29, 38, 49, 62, 77, 94, 113, 134, 157, 182, 209, 238, 269, 302, 337, 374, 413, 454, 497, 542, 589, 638, 689, 742, 797, 854, 913, 974, 1037, 1102, 1169, 1238, 1309, 1382, 1457, 1534, 1613, 1694, 1777, 1862, 1949, 2038, 2129, 2222, 2317, 2414, 2513

Offset: 0

Vincenzo Librandi, Apr 30 2014

For i=0..28, 2*a(i) + 3 is prime. - Vincenzo Librandi, Jun 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Magma
```
[n^2+13: n in [0..60]];
```
Mathematica
```
Table[n^2 + 13, {n, 0, 60}]
```

a(n)=n^2+13 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (13 - 25*x + 14*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(13)*Pi*coth(sqrt(13)*Pi))/26.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(13)*Pi*cosech(sqrt(13)*Pi))/26. (End)
E.g.f.: exp(x)*(13 + x + x^2). - Elmo R. Oliveira, Apr 20 2025

A241848 a(n) = n^2 + 18.

Original entry on oeis.org

18, 19, 22, 27, 34, 43, 54, 67, 82, 99, 118, 139, 162, 187, 214, 243, 274, 307, 342, 379, 418, 459, 502, 547, 594, 643, 694, 747, 802, 859, 918, 979, 1042, 1107, 1174, 1243, 1314, 1387, 1462, 1539, 1618, 1699, 1782, 1867, 1954, 2043, 2134, 2227, 2322, 2419, 2518

Offset: 0

Vincenzo Librandi, May 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Magma
```
[n^2+18: n in [0..60]];
```

Table[n^2 + 18, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{18,19,22},60] (* Harvey P. Dale, Jan 18 2025 *)

a(n)=n^2+18 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (18 - 35*x + 19*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 03 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(18)*Pi*coth(sqrt(18)*Pi))/36.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(18)*Pi*cosech(sqrt(18)*Pi))/36. (End)
E.g.f.: exp(x)*(18 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A241851 a(n) = n^2 + 21.

Original entry on oeis.org

21, 22, 25, 30, 37, 46, 57, 70, 85, 102, 121, 142, 165, 190, 217, 246, 277, 310, 345, 382, 421, 462, 505, 550, 597, 646, 697, 750, 805, 862, 921, 982, 1045, 1110, 1177, 1246, 1317, 1390, 1465, 1542, 1621, 1702, 1785, 1870, 1957, 2046, 2137, 2230, 2325, 2422, 2521

Offset: 0

Vincenzo Librandi, May 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequence listed in A114962.

Magma
```
[n^2+21: n in [0..60]];
```
Mathematica
```
Table[n^2 + 21, {n, 0, 60}]
```

a(n)=n^2+21 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (21 - 41*x + 22*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 04 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(21)*Pi*coth(sqrt(21)*Pi))/42.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(21)*Pi*cosech(sqrt(21)*Pi))/42. (End)
E.g.f.: exp(x)*(21 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A241750 a(n) = n^2 + 15.

Original entry on oeis.org

15, 16, 19, 24, 31, 40, 51, 64, 79, 96, 115, 136, 159, 184, 211, 240, 271, 304, 339, 376, 415, 456, 499, 544, 591, 640, 691, 744, 799, 856, 915, 976, 1039, 1104, 1171, 1240, 1311, 1384, 1459, 1536, 1615, 1696, 1779, 1864, 1951, 2040, 2131, 2224, 2319, 2416, 2515

Offset: 0

Vincenzo Librandi, May 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Magma
```
[n^2+15: n in [0..60]];
```

Table[n^2 + 15, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{15,16,19},60] (* Harvey P. Dale, Jul 04 2025 *)

a(n)=n^2+15 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (15 - 29*x + 16*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 03 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(15)*Pi*coth(sqrt(15)*Pi))/30.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(15)*Pi*cosech(sqrt(15)*Pi))/30. (End)
E.g.f.: exp(x)*(15 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

cp's OEIS Frontend

A241850 a(n) = n^2 + 20.

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

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A241751 a(n) = n^2 + 16.

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A114963 a(n) = n^2 + 22.

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A241748 a(n) = n^2 + 12.

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A241847 a(n) = n^2 + 17.

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A241749 a(n) = n^2 + 13.

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