A114962 -id:A114962 - cp's OEIS frontend

A241849 a(n) = n^2 + 19.

19, 20, 23, 28, 35, 44, 55, 68, 83, 100, 119, 140, 163, 188, 215, 244, 275, 308, 343, 380, 419, 460, 503, 548, 595, 644, 695, 748, 803, 860, 919, 980, 1043, 1108, 1175, 1244, 1315, 1388, 1463, 1540, 1619, 1700, 1783, 1868, 1955, 2044, 2135, 2228, 2323, 2420, 2519

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+19: n in [0..60]];
```

A241849:=n->n^2+19: seq(A241849(n), n=0..100); # Wesley Ivan Hurt, Jan 16 2017

Table[n^2 + 19, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{19,20,23},50] (* Harvey P. Dale, Dec 05 2018 *)

a(n)=n^2+19 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (19 - 37*x + 20*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(19)*Pi*coth(sqrt(19)*Pi))/38.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(19)*Pi*cosech(sqrt(19)*Pi))/38. (End)
E.g.f.: exp(x)*(19 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A241889 a(n) = n^2 + 23.

Original entry on oeis.org

23, 24, 27, 32, 39, 48, 59, 72, 87, 104, 123, 144, 167, 192, 219, 248, 279, 312, 347, 384, 423, 464, 507, 552, 599, 648, 699, 752, 807, 864, 923, 984, 1047, 1112, 1179, 1248, 1319, 1392, 1467, 1544, 1623, 1704, 1787, 1872, 1959, 2048, 2139, 2232, 2327, 2424, 2523

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+23: n in [0..60]];
```

CoefficientList[Series[(23 - 45 x + 24 x^2)/(1 - x)^3,{x, 0, 60}], x]
Range[0, 50]^2 + 23 (* or *) LinearRecurrence[{3, -3, 1}, {23, 24, 27}, 50] (* Harvey P. Dale, May 27 2014 *)

a(n)=n^2+23 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (23 - 45*x + 24*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(23)*Pi*coth(sqrt(23)*Pi))/46.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(23)*Pi*cosech(sqrt(23)*Pi))/46. (End)
E.g.f.: exp(x)*(23 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A241890 a(n) = n^2 + 24.

Original entry on oeis.org

24, 25, 28, 33, 40, 49, 60, 73, 88, 105, 124, 145, 168, 193, 220, 249, 280, 313, 348, 385, 424, 465, 508, 553, 600, 649, 700, 753, 808, 865, 924, 985, 1048, 1113, 1180, 1249, 1320, 1393, 1468, 1545, 1624, 1705, 1788, 1873, 1960, 2049, 2140, 2233, 2328, 2425, 2524

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+24: n in [0..60]];
```

Table[n^2 + 24, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{24,25,28},60] (* Harvey P. Dale, Jun 24 2025 *)

a(n)=n^2+24 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (24 - 47*x + 25*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(24)*Pi*coth(sqrt(24)*Pi))/48.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(24)*Pi*cosech(sqrt(24)*Pi))/48. (End)
E.g.f.: exp(x)*(24 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

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

A214871 Natural numbers placed in table T(n,k) layer by layer. The order of placement - T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1). Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 2, 7, 11, 8, 9, 12, 18, 13, 5, 14, 19, 27, 20, 15, 16, 21, 28, 38, 29, 22, 10, 23, 30, 39, 51, 40, 31, 24, 25, 32, 41, 52, 66, 53, 42, 33, 17, 34, 43, 54, 67, 83, 68, 55, 44, 35, 36, 45, 56, 69, 84, 102, 85, 70, 57, 46, 26, 47, 58, 71, 86, 103, 123

Offset: 1

Boris Putievskiy, Mar 11 2013

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(2,2), T(1,2), T(2,1);
  . . .
  T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1);
  . . .

			The start of the sequence as table:
  1....3...6..11..18..27...
  4....2...8..13..20..29...
  7....9...5..15..22..31...
  12..14..16..10..24..33...
  19..21..23..25..17..35...
  28..30..32..34..36..26...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  3,4;
  6,2,7;
  11,8,9,12;
  18,13,5,14,19;
  27,20,15,16,21,28;
  . . .

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A059100, A087475, A114949, A189833, A114948, A114962; in columns A117950, A117951, A117619, A189834, A189836; the main diagonal is A002522.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i == j:
   result=(i-1)**2+1
if i > j:
   result=(i-1)**2+2*j+1
if i < j:
   result=(j-1)**2+2*i

As table
T(n,k) = (n-1)^2+1,     if n=k;
T(n,k) = (n-1)^2+2*k+1, if n>k;
T(n,k) = (k-1)^2+2*n,   if nAs linear sequence
a(n) = (i-1)^2+1,     if i=j;
a(n) = (i-1)^2+2*j+1, if i>j;
a(n) = (j-1)^2+2*i,   if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

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A241849 a(n) = n^2 + 19.

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A241889 a(n) = n^2 + 23.

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A241890 a(n) = n^2 + 24.

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

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A214871 Natural numbers placed in table T(n,k) layer by layer. The order of placement - T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1). Table T(n,k) read by antidiagonals.

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