A087475 -id:A087475 - cp's OEIS frontend

A189833 a(n) = n^2 + 8.

8, 9, 12, 17, 24, 33, 44, 57, 72, 89, 108, 129, 152, 177, 204, 233, 264, 297, 332, 369, 408, 449, 492, 537, 584, 633, 684, 737, 792, 849, 908, 969, 1032, 1097, 1164, 1233, 1304, 1377, 1452, 1529, 1608, 1689, 1772, 1857, 1944, 2033

Offset: 0

Vladimir Joseph Stephan Orlovsky, Apr 28 2011

From César Eliud Lozada, Mar 29 2021: (Start)
Numbers a(n) such that sqrt( a(n) + 4*n*sqrt(2) ) = n + 2*sqrt(2). Examples:
For n=1:  sqrt( 9 +  4*sqrt(2)) = 1 + 2*sqrt(2),
For n=2:  sqrt(12 +  8*sqrt(2)) = 2 + 2*sqrt(2),
For n=3:  sqrt(17 + 12*sqrt(2)) = 3 + 2*sqrt(2). (End)

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..955 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A059100, A117950, A087475.

[n^2+8: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[n^2+8,{n,0,100}]
LinearRecurrence[{3,-3,1},{8,9,12},50] (* Harvey P. Dale, Jun 21 2022 *)

a(n)=n^2+8 \\ Charles R Greathouse IV, Jun 17 2017

From G. C. Greubel, Jan 13 2018: (Start)
G.f.: (8 - 15*x + 9*x^2)/(1 - x)^3.
E.g.f.: (8 + x + x^2)*exp(x). (End)
From Amiram Eldar, Jul 04 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + 2*sqrt(2)*Pi*coth(2*sqrt(2)*Pi))/16.
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*sqrt(2)*Pi*cosech(2*sqrt(2)*Pi))/16. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = (sqrt(7/2)/2)*sinh(sqrt(7)*Pi)/sinh(2*sqrt(2)*Pi).
Product_{n>=0} (1 + 1/a(n)) = (3/(2*sqrt(2)))*sinh(3*Pi)/sinh(2*sqrt(2)*Pi). (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 29 2011

A085554 Greater of twin primes of the form x^2+2, x^2+4.

Original entry on oeis.org

5, 13, 229, 1093, 2029, 3253, 13693, 21613, 59053, 65029, 91813, 140629, 178933, 199813, 205213, 227533, 328333, 567013, 700573, 804613, 815413, 1071229, 2241013, 3629029, 4223029, 4347229, 4809253, 5212093, 5919493, 6185173

Offset: 1

Cino Hilliard, Jul 04 2003

Except for the first term, all a(n)=13 (mod 72) with x=3 (mod 6). The lesser of the twin prime pair is given by A253639, the x-values in A086381. - M. F. Hasler, Jan 18 2015

M. F. Hasler, Table of n, a(n) for n = 1..3044 (all terms below 10^12).

Cf. A085553, A086381, A087475, A059100.

Transpose[Select[Table[x^2+{2,4},{x,5000}],AllTrue[#,PrimeQ]&]][[2]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 15 2015 *)

is_A086381(x)=ispseudoprime(x^2+2)&&ispseudoprime(x^2+4) \\ or is_A067201(x)&&is_A007591(x)
A085554 = apply(A087475,select(is_A086381,vector(9999,n,n))) \\ A087475=x->x^2+4.
write(f="b085554.txt",c=1," 5"); forstep(x=3,1e6,6,is_A086381(x)&&write(f,c++" "x^2+4))
\\ M. F. Hasler, Jan 18 2015

A085554 = A087475 o A086381 = A020725^2 o A253639, i.e., a(n) = A087475(A086381(n)) = A253639(n)+2. - M. F. Hasler, Jan 18 2015

Edited by Don Reble, May 03 2006

Definition corrected by Harvey P. Dale and Franklin T. Adams-Watters, Jan 15 2015

A156798 a(n) = n^4 + 5*n^2 + 4.

Original entry on oeis.org

4, 10, 40, 130, 340, 754, 1480, 2650, 4420, 6970, 10504, 15250, 21460, 29410, 39400, 51754, 66820, 84970, 106600, 132130, 162004, 196690, 236680, 282490, 334660, 393754, 460360, 535090, 618580, 711490, 814504, 928330, 1053700, 1191370

Offset: 0

Reinhard Zumkeller, Feb 16 2009

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A002522, A059100, A087475.

Magma
```
[n^4+5*n^2+4: n in [0..50]];
```
Mathematica
```
Table[n^4+5n^2+4, {n,0,40}]
```
PARI
```
a(n)=n^4+5*n^2+4
```

[(n^2 +1)*(n^2 +4) for n in (0..50)] # G. C. Greubel, Jun 10 2021

a(n) = A002522(n)*A087475(n) = A000290(n) + A000290(A059100(n)) = A028552(A002522(n)).
a(n) = (n^2 + 1)*(n^2 + 4) = n^2 + (n^2 + 2)^2.
G.f.: 2*(2 -5*x +15*x^2 -5*x^3 +5*x^4)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; corrected by R. J. Mathar, Sep 16 2009
a(0)=4, a(1)=10, a(2)=40, a(3)=130, a(4)=340, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, May 04 2011
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=0} 1/a(n) = (1 + Pi*coth(Pi))/8 - Pi*tanh(Pi)/24.
Sum_{n>=0} (-1)^n/a(n) = 1/8 + Pi*csch(Pi)/6 - Pi*csch(Pi)*sech(Pi)/24. (End)
E.g.f.: (4 + 6*x + 12*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Jun 10 2021

A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 16, 10, 7, 6, 25, 17, 13, 11, 8, 36, 26, 21, 18, 14, 12, 49, 37, 31, 27, 22, 19, 15, 64, 50, 43, 38, 32, 28, 23, 20, 81, 65, 57, 51, 44, 39, 33, 29, 24, 100, 82, 73, 66, 58, 52, 45, 40, 34, 30, 121, 101, 91, 83, 74, 67, 59, 53, 46, 41, 35

Offset: 1

Clark Kimberling, Jul 23 2009

A permutation of the natural numbers.
Row 1 consists of the squares.
Beginning with row 5, the columns obey a 3rd-order recurrence:
c(n)=c(n-1)+c(n-2)-c(n-3)+1; thus disregarding row 1, the nonsquares are partitioned by this recurrence.
Except for initial terms, the first ten rows match A000290, A002522, A002061, A059100, A014209, A117950, A027688, A087475, A027689, A117951, and the first column, A035106.

			Corner:
1....4....9...16...25
2....5...10...17...26
3....7...13...21...31
6...11...18...27...38
The double interspersion A161179 begins thus:
1....4....7...12...17...24
2....3....8...11...18...23
5....6...13...16...25...30
9...10...19...22...33...38
Expel the even-numbered columns, leaving
1....7...17...
2....8...18...
5...13...25...
9...19...33...
Then replace each of those numbers by its rank when all the numbers are jointly ranked.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A000037, A000290, A161179, A163254, A163255, A163256, A163257, A163258.

Let S(n,k) denote the k-th term in the n-th row. Three cases:
S(1,k)=k^2;
if n is even, then S(n,k)=k^2+(n-2)k+(n^2-2*n+4)/4;
if n>=3 is odd, then S(n,k)=k^2+(n-2)k+(n^2-2*n+1)/4.

Edited and augmented by Clark Kimberling, Jul 24 2009

A189836 a(n) = n^2 + 11.

Original entry on oeis.org

11, 12, 15, 20, 27, 36, 47, 60, 75, 92, 111, 132, 155, 180, 207, 236, 267, 300, 335, 372, 411, 452, 495, 540, 587, 636, 687, 740, 795, 852, 911, 972, 1035, 1100, 1167, 1236, 1307, 1380, 1455, 1532, 1611, 1692, 1775, 1860, 1947

Offset: 0

Vladimir Joseph Stephan Orlovsky, Apr 28 2011

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A059100, A117950, A087475.

[n^2 + 11: n in [0..50]]; // G. C. Greubel, Jan 13 2018

Table[n^2+11,{n,0,100}]
LinearRecurrence[{3,-3,1},{11,12,15},60] (* Harvey P. Dale, Aug 24 2020 *)

a(n)=n^2+11 \\ Charles R Greathouse IV, Jun 17 2017

From G. C. Greubel, Jan 13 2018: (Start)
G.f.: (11 - 21*x + 12*x^2)/(1 - x)^3.
E.g.f.: (11 + x + x^2)*exp(x). (End)
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(11)*Pi*coth(sqrt(11)*Pi))/22.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(11)*Pi*cosech(sqrt(11)*Pi))/22. (End)
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = sqrt(10/11)*sinh(sqrt(10)*Pi)/sinh(sqrt(11)*Pi).
Product_{n>=0} (1 + 1/a(n)) = 2*sqrt(3/11)*sinh(2*sqrt(3)*Pi)/sinh(sqrt(11)*Pi). (End)

A155965 a(n) = n*(n^2+4).

Original entry on oeis.org

0, 5, 16, 39, 80, 145, 240, 371, 544, 765, 1040, 1375, 1776, 2249, 2800, 3435, 4160, 4981, 5904, 6935, 8080, 9345, 10736, 12259, 13920, 15725, 17680, 19791, 22064, 24505, 27120, 29915, 32896, 36069, 39440, 43015, 46800, 50801, 55024, 59475, 64160

Offset: 0

Vincenzo Librandi, Jan 31 2009

The identity (n^3+4*n)^2 + (2*n^2+8)^2 = (n^2+4)^3 can be written as a(n)^2 + A155966(n)^2 = A087475(n)^3.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006003, A087475, A155966.

Table[n (n^2 + 4), {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)
CoefficientList[Series[x (5 - 4 x + 5 x^2)/(1 - x)^4, {x, 0, 60}], x] (* Vincenzo Librandi, May 03 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,5,16,39},50] (* Harvey P. Dale, Jan 23 2019 *)

a(n)=n*(n^2+4) \\ Charles R Greathouse IV, Jan 11 2012

[lucas_number1(4,n,-2) for n in range(0, 41)] # Zerinvary Lajos, May 16 2009

G.f.: x*(5 - 4*x + 5*x^2)/(1 - x)^4. - Vincenzo Librandi, May 03 2014
a(n) = 4*a(n-1) - 6*a(n-2) +4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, May 03 2014
a(n) = A006003(n-1) + A006003(n+1). - Lechoslaw Ratajczak, Oct 31 2021

A155966 a(n) = 2*n^2 + 8.

Original entry on oeis.org

8, 10, 16, 26, 40, 58, 80, 106, 136, 170, 208, 250, 296, 346, 400, 458, 520, 586, 656, 730, 808, 890, 976, 1066, 1160, 1258, 1360, 1466, 1576, 1690, 1808, 1930, 2056, 2186, 2320, 2458, 2600, 2746, 2896, 3050, 3208, 3370, 3536, 3706, 3880, 4058, 4240, 4426, 4616

Offset: 0

Vincenzo Librandi, Jan 31 2009

The identity (n^3 + 4*n)^2 + (2*n^2 + 8)^2 = (n^2 + 4)^3 can be written as A155965(n)^2 + a(n)^2 = A087475(n)^3.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A087475, A155965.

Cf. similar sequences listed in A255843.

[2*n^2+8: n in [0..50]]; // Vincenzo Librandi, Feb 22 2012

LinearRecurrence[{3, -3, 1}, {8, 10, 16}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
2*Range[0,50]^2+8 (* Harvey P. Dale, Mar 01 2018 *)

a(n)=2*n^2+8 \\ Charles R Greathouse IV, Jan 11 2012

G.f.: 2*(4 - 7*x + 5*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A087475(n). - Bruno Berselli, Mar 13 2015
From Amiram Eldar, Feb 25 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/16 + coth(2*Pi)*Pi/8.
Sum_{n>=0} (-1)^n/a(n) =  1/16 + cosech(2*Pi)*Pi/8. (End)
E.g.f.: 2*exp(x)*(4 + x + x^2). - Elmo R. Oliveira, Jan 17 2025

Offset changed from 1 to 0 and added a(0)=8 by Bruno Berselli, Mar 13 2015

A213921 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 clockwise. Table T(n,k) read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 05 2013

A 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) is 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(2,n), T(4,n), ... T(n,4), T(n,2);
...

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   8  14  22  32 ...
   7   9   6  11  18  27 ...
  13  16  12  15  23  33 ...
  21  25  20  24  19  28 ...
  31  36  30  35  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  8,  9, 13;
  17, 14,  6, 16, 21;
  26, 22, 11, 12, 25, 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 W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A002522, A014206, A059100, A027688, A117950, A027689, A087475, A027690, A117951, A027691, A114949, A027692, A117619; in columns A002061, A000290, A002378, A005563, A028387, A008865, A028552, A028872, A014209, A028347, A028875.

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-(j%2)*i+2-int((j+2)/2)
else:
   result=j*j-((i%2)+1)*j + int((i+3)/2)

As a table:
T(n,k) = n*n - (k mod 2)*n + 2 - floor((k+2)/2), if n>k;
T(n,k) = k*k - ((n mod 2)+1)*k + floor((n+3)/2), if n<=k.
As a linear sequence:
a(n) = i*i - (j mod 2)*i + 2 - floor((j+2)/2), if i>j;
a(n) = j*j - ((i mod 2)+1)*j + floor((i+3)/2), 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).

A132354 Integers m such that 7*m + 1 is a square.

Original entry on oeis.org

0, 5, 9, 24, 32, 57, 69, 104, 120, 165, 185, 240, 264, 329, 357, 432, 464, 549, 585, 680, 720, 825, 869, 984, 1032, 1157, 1209, 1344, 1400, 1545, 1605, 1760, 1824, 1989, 2057, 2232, 2304, 2489, 2565, 2760, 2840, 3045, 3129, 3344, 3432, 3657, 3749, 3984, 4080

Offset: 0

Mohamed Bouhamida, Nov 08 2007

Numbers of the form m*(7*m + 2) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A054000, A056220, A000217, A087475, A028347, A062717, A036666, A002378, A001082, A046092, A005563.

[n^2+n+3*Ceiling(n/2)^2: n in [0..50]]; // Wesley Ivan Hurt, Jul 07 2014

seq(n^2+n+3*ceil(n/2)^2, n=0..48); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+7*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

a(2*k) = k*(7*k + 2), a(2*k + 1) = 7*k^2 + 12*k + 5.
a(n) = n^2 + n + 3*ceiling(n/2)^2. - Gary Detlefs, Feb 23 2010
G.f.: -x*(5*x^2 + 4*x + 5)/((x - 1)^3*(x + 1)^2). - Colin Barker, Oct 24 2012
Sum_{n>=1} 1/a(n) = 7/4 - cot(2*Pi/7)*Pi/2. - Amiram Eldar, Mar 15 2022

More terms from Max Alekseyev, Nov 13 2009

Better definition from Max Alekseyev, Oct 24 2012

A156701 a(n) = 4n^4 + 17n^2 + 4.

Original entry on oeis.org

4, 25, 136, 481, 1300, 2929, 5800, 10441, 17476, 27625, 41704, 60625, 85396, 117121, 157000, 206329, 266500, 339001, 425416, 527425, 646804, 785425, 945256, 1128361, 1336900, 1573129, 1839400, 2138161, 2471956, 2843425, 3255304, 3710425

Offset: 0

Reinhard Zumkeller, Feb 13 2009

a(n) = A087475(n)*A053755(n).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016850, A053755, A087475, A099761.

[4*n^4+17*n^2+4: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

Table[4n^4+17n^2+4,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{4,25,136,481,1300},50] (* Harvey P. Dale, Nov 08 2017 *)

a(n)=4*n^4+17*n^2+4 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = (2*(n^2 - 1))^2 + (5*n)^2.
G.f.: (-4-25*x^4-11*x^3-51*x^2-5*x)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
E.g.f.: exp(x)*(4 + 21*x + 45*x^2 + 24*x^3 + 4*x^4). - Stefano Spezia, Jul 08 2023

cp's OEIS Frontend

A189833 a(n) = n^2 + 8.

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A085554 Greater of twin primes of the form x^2+2, x^2+4.

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A156798 a(n) = n^4 + 5*n^2 + 4.

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A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

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A189836 a(n) = n^2 + 11.

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A155965 a(n) = n*(n^2+4).

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A155966 a(n) = 2*n^2 + 8.

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A156701 a(n) = 4n^4 + 17n^2 + 4.