A241751 -id:A241751 - cp's OEIS frontend

A243451 Primes of the form n^2 + 16.

17, 41, 97, 137, 241, 457, 641, 857, 977, 1697, 2417, 2617, 3041, 4241, 5641, 6257, 6577, 7937, 8297, 9041, 9817, 11897, 13241, 14177, 14657, 15641, 16657, 22817, 27241, 32057, 36497, 44537, 47977, 48857, 52457, 53377, 60041, 62017, 70241, 75641, 78977, 83537

Offset: 1

Vincenzo Librandi, Jun 05 2014

nonn

Intersection of A241751 and A028916; conjecture: sequence is infinite. - Reinhard Zumkeller, Apr 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A122062 (associated n).
Cf. similar sequences listed in A243449.
Cf. A010051, A241751; subsequence of A028916.
Primes of form n^2+b^4, b fixed: A002496 (b=1),  A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a243451 n = a243451_list !! (n-1)
a243451_list = [x | x <- a241751_list, a010051' x == 1]
-- Reinhard Zumkeller, Apr 11 2015

[a: n in [0..1000] | IsPrime(a) where a is n^2+16];

Select[Table[n^2 + 16, {n, 0, 1000}], PrimeQ]
Select[Range[1,301,2]^2+16,PrimeQ] (* Harvey P. Dale, Nov 05 2015 *)

list(lim)=if(lim<17,return([])); my(v=List(),t); forstep(n=1,sqrtint(lim\1-16),2, if(isprime(t=n^2+16), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Aug 18 2017

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

Original entry on oeis.org

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

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.

cp's OEIS Frontend

A243451 Primes of the form n^2 + 16.

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