A189833 -id:A189833 - 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

A255847 a(n) = 2*n^2 + 16.

Original entry on oeis.org

16, 18, 24, 34, 48, 66, 88, 114, 144, 178, 216, 258, 304, 354, 408, 466, 528, 594, 664, 738, 816, 898, 984, 1074, 1168, 1266, 1368, 1474, 1584, 1698, 1816, 1938, 2064, 2194, 2328, 2466, 2608, 2754, 2904, 3058, 3216, 3378, 3544, 3714, 3888, 4066, 4248, 4434, 4624

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=8 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 32 is a square.

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

Cf. A189833.
Subsequence of A047467.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+16: n in [0..50]];
```

Table[2 n^2 + 16, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{16,18,24},50] (* Harvey P. Dale, Nov 11 2017 *)

PARI
```
vector(50, n, n--; 2*n^2+16)
```
Sage
```
[2*n^2+16 for n in (0..50)]
```

G.f.: 2*(8 - 15*x + 9*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A189833(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + 2*sqrt(2)*Pi*coth(2*sqrt(2)*Pi))/32.
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*sqrt(2)*Pi*cosech(2*sqrt(2)*Pi))/32. (End)
E.g.f.: 2*exp(x)*(8 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Edited by Bruno Berselli, Mar 13 2015

A338432 Triangle read by rows: T(n, k) = (n - k + 1)^2 + 2*k^2, for n >= 1, and k = 1, 2, ..., n.

Original entry on oeis.org

3, 6, 9, 11, 12, 19, 18, 17, 22, 33, 27, 24, 27, 36, 51, 38, 33, 34, 41, 54, 73, 51, 44, 43, 48, 59, 76, 99, 66, 57, 54, 57, 66, 81, 102, 129, 83, 72, 67, 68, 75, 88, 107, 132, 163, 102, 89, 82, 81, 86, 97, 114, 137, 166, 201

Offset: 1

Wolfdieter Lang, Dec 09 2020

This triangle is obtained from the array A(m, k) = m^2 + 2*k^2, for k and m >= 1, read by upwards antidiagonals. This array A is of interest for representing numbers as a sum of three non-vanishing squares with two squares coinciding.
For the numbers represented this way, see A154777. To find the actual values for m and k (taken positive), given a representable number from A154777, one can also use the number triangle T(n, k) = A(n-k+1, k).
To find the number of representations of value N (from A154777), it is sufficient to consider the rows n >= 1 not exceeding n_{max} = Floor(N, Min), where the sequence Min gives the minima of the numbers in each row: Min = {min(n)}_{n>=1} with min(n) = min(T(n, 1), T(n, 2), ..., T(n, n)) and Floor(N, Min) is the greatest member of Min not exceeding N.
Conjecture: min(n) = T(n, ceiling(n/3)), n >= 1. This is the sequence (n+1)^2 - ceiling(n/3)*(2*(n+1) - 3*ceiling(n/3)) = A071619(n+1) = ceiling((2/3)*(n+1)^2) = (n+1)^2 - floor((1/3)*(n+1)^2) = 3, 6, 11, 17, 24, 33, 43, .... (Proof of these identities by considering the three n (mod 3) cases.)
For the multiplicities of the representable values A154777(n), see A339047.
The author met this representation problem in connection with special triples of integer curvatures in the Descartes-Steiner five circle problem.

			The triangle T(n, k) begins:
n \ k  1   2   3   4   5   6   7   8   9  10  11  12 ...
1:     3
2:     6   9
3:    11  12  19
4:    18  17  22  33
5:    27  24  27  36  51
6:    38  33  34  41  54  73
7:    51  44  43  48  59  76  99
8:    66  57  54  57  66  81 102 129
9:    83  72  67  68  75  88 107 132 163
10:  102  89  82  81  86  97 114 137 166 201
11:  123 108  99  96  99 108 123 144 171 204 243
12:  146 129 118 113 114 121 134 153 178 209 246 289
...
----------------------------------------------------
T(5, 1) = 5^2 + 2*1^2 = 27 = T(5, 3) = 3^2 + 2*3^2. A338433(11) = 2 for A154777(11) = 27.
T(4, 4) = 1^2 + 2*4^2 = 33 = T(6, 2) = 5^2 + 2*2^2. A338433(12) = 2 for A154777(12) = 33.
T(5, 5) = 1^2 + 2*5^2 = 51 = T(7, 1) = 7^2 + 2*1^2. A338433(20) = 2 for A154777(20) = 51.
T(7, 7) = 1^1 - 2*7^2 = 99 = T(11, 3) = 9^2 + 2*3^2 = 99 = T(11, 5) = 7^2 + 2*5^2. A338433(39) = 3 for A154777(39) = 99.
The first multiplicity 4 appears for 297.

Cf. A154777, A339047.
Cf. Columns k = 1..3: A059100, A189833, A241848.
Cf. Diagonals m = 1..4: A058331, A255843, A339048, A255847.

T(n, k) = A(n - k + 1, k), with the array A(m, k) = m^2 + 2*k^2, for n >= 1 and k = 1, 2, ..., n, and 0 otherwise.
G.f. of T and A column k (offset 0): G(k, x) = (1 + x + 2*(1 - x)^2*k^2)/(1-x)^3, for k >= 1.
G.f. of T diagonal m (A row m) (offset 0): D(m, x) = ((2*(1+x) + (1-x)^2*m^2)/(1-x)^3), for m >= 1.
G.f. of row polynomials in x (that is, g.f. of the triangle): G(z,x) = (3 - 3*z + (2 - 6*x + x^2)*z^2 + (2 + x)*x*z^3)*x*z / ((1 - z)*(1 - x*z))^3.

A357995 Frobenius number for A = (n, n+1^2, n+2^2, n+3^2, ...) for n>=2.

Original entry on oeis.org

1, 5, 11, 13, 11, 20, 31, 24, 27, 29, 43, 37, 49, 52, 63, 58, 69, 53, 75, 61, 65, 84, 95, 98, 85, 96, 107, 115, 88, 121, 127, 122, 130, 136, 139, 134, 145, 148, 159, 151, 154, 157, 171, 174, 169, 180, 191, 194, 178, 181, 203, 198, 201, 212, 223, 210, 221, 232, 235, 214

Offset: 2

Michel Marcus, Oct 23 2022

nonn

Feihu Liu and Guoce Xin, On Frobenius Formulas of Power Sequences, arXiv:2210.02722 [math.CO], 2022. See Table 1 p. 26.

Cf. A059100, A117950, A087475, A117951, A114949, A117619, A189833, A189834,

def A357995(n):
    a, b = set([0]), set(range(1,n**2))
    for m in [n+k**2 for k in range(n+1)]:
        d=m
        while d < n**2:
            c2 = set([x for x in b if x-d in a])
            a |= c2 ; b -= c2 ; d*=2
    return max(b) # Bert Dobbelaere, Oct 30 2022

More terms from Bert Dobbelaere, Oct 30 2022

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
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(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

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

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 A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

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*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), 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).

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

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A255847 a(n) = 2*n^2 + 16.

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A338432 Triangle read by rows: T(n, k) = (n - k + 1)^2 + 2*k^2, for n >= 1, and k = 1, 2, ..., n.

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A357995 Frobenius number for A = (n, n+1^2, n+2^2, n+3^2, ...) for n>=2.

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A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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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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