A114949 -id:A114949 - cp's OEIS frontend

A255844 a(n) = 2*n^2 + 6.

6, 8, 14, 24, 38, 56, 78, 104, 134, 168, 206, 248, 294, 344, 398, 456, 518, 584, 654, 728, 806, 888, 974, 1064, 1158, 1256, 1358, 1464, 1574, 1688, 1806, 1928, 2054, 2184, 2318, 2456, 2598, 2744, 2894, 3048, 3206, 3368, 3534, 3704, 3878, 4056, 4238, 4424, 4614

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.
Equivalently, numbers m such that 2*m-12 is a square.
For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).

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

Cf. A016825 (first differences), A087370, A107303, A114949, A117950.
Cf. A152811: nonnegative numbers of the form 2*m^2-6.
Subsequence of A000378.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+6: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 6, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+6)
```
Sage
```
[2*n^2+6 for n in (0..50)]
```

G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117950(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)
E.g.f.: 2*exp(x)*(3 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Corrected and extended by Bruno Berselli, Mar 11 2015

A259054 a(n) = 4n^2 - 4n + 19, n >= 1.

Original entry on oeis.org

19, 27, 43, 67, 99, 139, 187, 243, 307, 379, 459, 547, 643, 747, 859, 979, 1107, 1243, 1387, 1539, 1699, 1867, 2043, 2227, 2419, 2619, 2827, 3043, 3267, 3499, 3739, 3987, 4243, 4507, 4779, 5059, 5347, 5643, 5947, 6259, 6579, 6907, 7243, 7587, 7939, 8299, 8667, 9043, 9427, 9819

Offset: 1

Wolfdieter Lang and Kival Ngaokrajang, Jul 02 2015

a(n) gives twice the inverse radius of the circles touching the large Arbelos (2/3,1/3) circle (radius 1) and the n-th and (n-1)-th circles of the counterclockwise Pappus chain.
For twice the curvatures (inverse radii) of the counterclockwise Pappus chain of the (2/3,1/3) arbelos see A114949, also for the MathWorld link to Pappus chain.
For the small curvatures touching the left circle of the (2/3,1/3) arbelos and the n-th and (n-1)-st circles of the counterclockwise Pappus chain see A259555.
The curvatures of the circles can be computed with Descartes' three (actually 5) circle theorem. See A259555 for links to Descartes' theorem.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A114949, A259555.

Array[4 #^2 - 4 # + 19 &, 40] (* Michael De Vlieger, Jul 02 2015 *)
LinearRecurrence[{3,-3,1},{19,27,43},40] (* Harvey P. Dale, May 17 2016 *)

vector(60, n, 4*n^2 - 4*n + 19) \\ Michel Marcus, Jul 03 2015

a(n) = 4*n^2 - 4*n + 19, n >= 1.
O.g.f.: x*(19 - 30*x + 19*x^2)/(1 - x)^3.
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: exp(x)*(4*x^2 + 19) - 19.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Terms a(37) and beyond from Andrew Howroyd, May 01 2020

A285766 Maximum spillway height for a zero or one bend minimal area lake in a number square.

Original entry on oeis.org

0, 0, 6, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407

Offset: 0

Craig Knecht, May 04 2017

nonn

The water retention model for mathematical surfaces led to definitions for a lake and a pond.  These lakes and ponds divide the square up in interesting ways. This sequence looks at the spillway heights in zero or one bend minimal area lakes.
A lake has dimensions of (n-2) X (n-2) when the square is n X n.  All other water retaining areas are ponds.
A number square contains the numbers 1 to n^2 without repeats.
The larger terms are a(n)= n^2+6 or A114949.

			For the 4 X 4 square a example of a smallest lake is shown. The values 1,2,3 form the lake. The pathway of least resistance off the square is the spillway value 10.
   ( 4  16  15   5)
   (10   1   2  14)
   ( 6  11   3  13)
   ( 7   8  12   9)

Craig Knecht, Minimal lake types in a 7x7 square.
Craig Knecht, Minimal lake area in a square
Wikipedia, Water retention on mathematical surfaces

Cf. A054247, A201126, A268311.

Conjectures from Colin Barker, May 07 2017: (Start)
G.f.: x^2*(6 - 8*x + 3*x^2 + x^3) / (1 - x)^3.
a(n) = 7 - 2*n + n^2 for n>2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.
(End)

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).

A255842 a(n) = 2*n^2 + 12.

Original entry on oeis.org

12, 14, 20, 30, 44, 62, 84, 110, 140, 174, 212, 254, 300, 350, 404, 462, 524, 590, 660, 734, 812, 894, 980, 1070, 1164, 1262, 1364, 1470, 1580, 1694, 1812, 1934, 2060, 2190, 2324, 2462, 2604, 2750, 2900, 3054, 3212, 3374, 3540, 3710, 3884, 4062, 4244, 4430

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.
Equivalently, numbers m such that 2*m - 24 is a square.
For n = 0..10, a(n) - 1 is prime (see A092968).

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

Cf. A016825 (first differences), A092968, A114949.
Subsequence of A047238 and A047406.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+12: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 12, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+12)
```
Sage
```
[2*n^2+12 for n in (0..50)]
```

G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A114949(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)
E.g.f.: 2*exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 24 2025

Edited by Bruno Berselli, Mar 11 2015

A259055 a(n) = 9n^2 + 18n + 7.

Original entry on oeis.org

7, 34, 79, 142, 223, 322, 439, 574, 727, 898, 1087, 1294, 1519, 1762, 2023, 2302, 2599, 2914, 3247, 3598, 3967, 4354, 4759, 5182, 5623, 6082, 6559, 7054, 7567, 8098, 8647, 9214, 9799, 10402, 11023, 11662, 12319, 12994, 13687, 14398, 15127

Offset: 0

Wolfdieter Lang and Kival Ngaokrajang, Jul 03 2015

a(n) gives twice the curvature of the n-th circle touching the two semicircles of the (2/3,1/3) arbelos and the (n-1)-th circle, with input circle of twice the curvature a(0) = A114949(1) = 7 (referring to the second circle of the counterclockwise Pappus chain).

Kival Ngaokrajang, Illustration of one half of the initial terms.
Eric Weisstein's World of Mathematics, Descartes Circle theorem.
Eric Weisstein's World of Mathematics, Pappus chain.
Wikipedia, Descartes' Theorem.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A114949, A259054, A259555.

A259055:=n->9*n^2+18*n+7: seq(A259055(n), n=0..100); # Wesley Ivan Hurt, Feb 04 2017

Table[9 n^2 + 18 n + 7, {n, 0, 40}] (* Michael De Vlieger, Jul 03 2015 *)
LinearRecurrence[{3,-3,1},{7,34,79},50] (* Harvey P. Dale, Sep 05 2018 *)

a(n)=9*n^2+18*n+7 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 9*(n+1)^2 - 2, n >= 0.
O.g.f.: (-2*x^2+13*x+7)/(1-x)^3.
Recurrence: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3, with a(0)=7, a(1)=34, and a(2)=79.
Descartes' three (actually five) circle theorem (see links) leads to a nonlinear recurrence for twice the curvatures: a(n) = 2*(3 + 3/2) + a(n-1) + 4*sqrt((3 + 3/2)*a(n-1)/2 + 9/2) = 9 + a(n-1) + 6*sqrt(a(n-1) + 2), with input a(0) = 7 = 2*A114949(1). This leads to a quadratic equation with the relevant solution a(n) = 9*n^2 + 18*n + 7.
E.g.f.: exp(x)*(9*x*(x + 3) + 7). - Elmo R. Oliveira, Oct 20 2024

Previous Showing 11-18 of 18 results.

cp's OEIS Frontend

A255844 a(n) = 2*n^2 + 6.

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A259054 a(n) = 4*n^2 - 4*n + 19, n >= 1.

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A285766 Maximum spillway height for a zero or one bend minimal area lake in a number square.

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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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A255842 a(n) = 2*n^2 + 12.

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Magma

Mathematica

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Sage

Formula

A259054 a(n) = 4n^2 - 4n + 19, n >= 1.

A259055 a(n) = 9n^2 + 18n + 7.