A189834 -id:A189834 - 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

A098077 a(n) = n^2(n+1)(2*n+1)/3.

Original entry on oeis.org

2, 20, 84, 240, 550, 1092, 1960, 3264, 5130, 7700, 11132, 15600, 21294, 28420, 37200, 47872, 60690, 75924, 93860, 114800, 139062, 166980, 198904, 235200, 276250, 322452, 374220, 431984, 496190, 567300, 645792, 732160, 826914, 930580, 1043700

Offset: 1

Alexander Adamchuk, Oct 24 2004

Sum of all matrix elements M(i,j) = i^2 + j^2 (i,j = 1,...,n).
From Torlach Rush, Jan 05 2020: (Start)
  a(n) = n * A006331(n).
  tr(M(n)) = A006331(n).
The sum of the antidiagonal of M(n) equals tr(M(n)).
  M(n)  = M(n)' (Symmetric).
  M(1,) = M(,1) = A002522(n), n > 0.
  M(2,) = M(,2) = A087475(n), n > 0.
  M(3,) = M(,3) = A189834(n), n > 0.
  M(4,) = M(,4) = A241751(n), n > 0.
(End)
Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, p and p+q. - Wesley Ivan Hurt, Apr 15 2018

			a(2) = (1^2 + 1^2) + (1^2 + 2^2) + (2^2 + 1^2) + (2^2 + 2^2) = 2 + 5 + 5 + 8 = 20.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002522, A006331, A087475, A100431, A108678, A189834, A241751.

[n^2*(n+1)*(2*n+1)/3: n in [1..40]]; // G. C. Greubel, Apr 09 2023

Table[ Sum[i^2 + j^2, {i, n}, {j, n}], {n, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {2, 20, 84, 240, 550}, 40] (* Vincenzo Librandi, Apr 16 2018 *)

a(n)=n^2*(n+1)*(2*n+1)/3 \\ Charles R Greathouse IV, Oct 07 2015

[n^2*(n+1)*(2*n+1)/3 for n in range(1,41)] # G. C. Greubel, Apr 09 2023

a(n) = Sum_{j=1..n} Sum_{i=1..n} (i^2 + j^2).
G.f.: 2*x*(1 + 5*x + 2*x^2)/(1-x)^5. - Colin Barker, May 04 2012
E.g.f.: (1/3)*exp(x)*x*(6 + 24*x + 15*x^2 + 2*x^3) . - Stefano Spezia, Jan 06 2020
a(n) = a(n-1) + (8*n^3 - 3*n^2 + n)/3. - Torlach Rush, Jan 07 2020
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/2 + 24*log(2) - 21.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/4 - 6*Pi - 6*log(2) + 21. (End)
From G. C. Greubel, Apr 09 2023: (Start)
a(n) = (1/4)*A100431(n-1).
a(n) = 2*A108678(n-1). (End)

More terms from Robert G. Wilson v, Nov 01 2004

New definition from Ralf Stephan, Dec 01 2004

A267370 Partial sums of A140091.

Original entry on oeis.org

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510

Offset: 0

Bruno Berselli, Jan 13 2016

After 0, this sequence is the third column of the array in A185874.

Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

			The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005581, A051744, A105163, A140091, A185874.

Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).

Magma
```
[n*(n+1)*(n+5)/2: n in [0..50]];
```

Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* Harvey P. Dale, Jul 18 2019 *)

PARI
```
vector(50, n, n--; n*(n+1)*(n+5)/2)
```
Sage
```
[n*(n+1)*(n+5)/2 for n in (0..50)]
```

O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

A154533 Number of constants of the form a^3u + bc*v, where a, b, c are linear, u of order n-3 and v of order n-2.

Original entry on oeis.org

10, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 109, 130, 153, 178, 205, 234, 265, 298, 333, 370, 409, 450, 493, 538, 585, 634, 685, 738, 793, 850, 909, 970, 1033, 1098, 1165, 1234, 1305, 1378, 1453, 1530, 1609, 1690, 1773, 1858, 1945, 2034, 2125, 2218, 2313

Offset: 0

Reinhard Zumkeller, Jan 11 2009

Emanuel Lasker, Zur Theorie der kanonischen Formen, Mathematische Annalen, 58 (1904), 434-440.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A189834.

Magma
```
[ n^2-2*n+10: n in [0..50] ];
```

LinearRecurrence[{3,-3,1},{10,9,10},60] (* Harvey P. Dale, May 04 2016 *)

a(n)=n^2-2*n+10 \\ Charles R Greathouse IV, Jun 17 2017

[lucas_number1(3,n,-9) for n in range(-1, 49)] # Zerinvary Lajos, May 16 2009

a(n) = n^2 - 2*n + 10.
a(n) = a(n-1) + 2*n-3 (with a(0)=10). - Vincenzo Librandi, Nov 27 2010
G.f.: (-10 + 21*x - 13*x^2) / (x-1)^3 . - R. J. Mathar, Aug 31 2011

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

A373710 Triangle read by rows: T(n,k) is the area of the square whose vertices divide the sides n of a circumscribed square into integer sections k and n - k, 0 <= k <= floor(n/2).

Original entry on oeis.org

0, 1, 4, 2, 9, 5, 16, 10, 8, 25, 17, 13, 36, 26, 20, 18, 49, 37, 29, 25, 64, 50, 40, 34, 32, 81, 65, 53, 45, 41, 100, 82, 68, 58, 52, 50, 121, 101, 85, 73, 65, 61, 144, 122, 104, 90, 80, 74, 72, 169, 145, 125, 109, 97, 89, 85, 196, 170, 148, 130, 116, 106, 100, 98

Offset: 0

Felix Huber, Jun 17 2024

For a sketch see linked illustration "Square in square".

			Triangle T(n,k) begins:
   n\k   0     1     2     3     4     5     6     7   ...
   0     0
   1     1
   2     4     2
   3     9     5
   4    16    10     8
   5    25    17    13
   6    36    26    20    18
   7    49    37    29    25
   8    64    50    40    34    32
   9    81    65    53    45    41
  10   100    82    68    58    52    50
  11   121   101    85    73    65    61
  12   144   122   104    90    80    74    72
  13   169   145   125   109    97    89    85
  14   196   170   148   130   116   106   100    98
  ...

Felix Huber, Table of n, a(n) for n = 0..100000
Felix Huber, Square in square

Cf. A000290(first column), A005563 (second column), A048147 (rows: first half of each diagonal there), A087475 (third column), A189834 (fourth column), A241751 (fifth column).

A373710:=(n,k)->n^2+2*k^2-2*n*k;
seq(seq(A373710(n,k),k=0..floor(n/2)),n=0..14);

T(n,k) = n^2 + 2*k^2 - 2*n*k, 0 <= k <= floor(n/2).
Sequence of row n = r: a(i) = 2*i^2 - 4*i - 2*r*i + r^2 + 2*r + 2, 1 <= i <= floor(r/2 + 1).
Sequence of column k = c: a(j) = j^2 - 2*j + 2*c*j + 2*c^2 - 2*c + 1, j >= 1.

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).

A255848 a(n) = 2*n^2 + 18.

Original entry on oeis.org

18, 20, 26, 36, 50, 68, 90, 116, 146, 180, 218, 260, 306, 356, 410, 468, 530, 596, 666, 740, 818, 900, 986, 1076, 1170, 1268, 1370, 1476, 1586, 1700, 1818, 1940, 2066, 2196, 2330, 2468, 2610, 2756, 2906, 3060, 3218, 3380, 3546, 3716, 3890, 4068, 4250, 4436

Offset: 0

Avi Friedlich, Mar 08 2015

For n>3, the sequence gives the 6th diagonal of triangle in A055096.
Also, this is the case k=9 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. It is noted that a(n)*n = (n + sqrt(3))^3 + (n - sqrt(3))^3.
Equivalently, numbers m such that 2*m-36 is a square.

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

Cf. A016825 (first differences), A055096, A189834.
Subsequence of A047463.
Cf. similar sequences listed in A255843.

[2*n^2+18: n in [0..50]]; // Vincenzo Librandi, Mar 08 2015

f[n_] := 2 n^2 + 18; Array[f, 50, 0] (* Robert G. Wilson v, Mar 08 2015 *)
CoefficientList[Series[(18 - 34 x + 20 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 08 2015 *)
LinearRecurrence[{3,-3,1},{18,20,26},50] (* Harvey P. Dale, Aug 20 2021 *)

vector(50, n, 2*n^2+18) \\ Derek Orr, Mar 09 2015

[2*n^2+18 for n in (0..50)] # Bruno Berselli, Mar 11 2015

a(n) = 2*A189834(n).
From Vincenzo Librandi, Mar 08 2015: (Start)
G.f.: 2*(9 - 17*x + 10*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + 3*Pi*coth(3*Pi))/36.
Sum_{n>=0} (-1)^n/a(n) = (1 + 3*Pi*cosech(3*Pi))/36. (End)
E.g.f.: 2*exp(x)*(9 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Edited by Bruno Berselli, Mar 11 2015

Showing 1-9 of 9 results.

cp's OEIS Frontend

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

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A098077 a(n) = n^2*(n+1)*(2*n+1)/3.

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A267370 Partial sums of A140091.

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A154533 Number of constants of the form a^3*u + b*c*v, where a, b, c are linear, u of order n-3 and v of order n-2.

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Sage

Formula

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

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Python

Extensions

A373710 Triangle read by rows: T(n,k) is the area of the square whose vertices divide the sides n of a circumscribed square into integer sections k and n - k, 0 <= k <= floor(n/2).

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Maple

Formula

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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A098077 a(n) = n^2(n+1)(2*n+1)/3.

A154533 Number of constants of the form a^3u + bc*v, where a, b, c are linear, u of order n-3 and v of order n-2.