A342758 -id:A342758 - cp's OEIS frontend

A017005 a(n) = 7n + 2.

2, 9, 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93, 100, 107, 114, 121, 128, 135, 142, 149, 156, 163, 170, 177, 184, 191, 198, 205, 212, 219, 226, 233, 240, 247, 254, 261, 268, 275, 282, 289, 296, 303, 310, 317, 324, 331, 338, 345, 352, 359, 366, 373, 380

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1120
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A342758.

Magma
```
[ 7*n+2: n in [0..60] ];
```

Range[2, 1000, 7] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
LinearRecurrence[{2,-1},{2,9},60] (* Harvey P. Dale, Sep 07 2015 *)

G.f.: (2+5*x)/(1-x)^2.
E.g.f.: exp(x)*(2 + 7*x). - Stefano Spezia, Mar 21 2021
a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Mar 22 2021

A135503 a(n) = n*(n^2 - 1)/2.

Original entry on oeis.org

0, 0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

Offset: 0

Cino Hilliard, Feb 09 2008

Previous name was: Integer values of sqrt(b) solving sqrt(d) + sqrt(b) = sqrt(c) with d^2 + b = c.
Squaring the first equation and setting the result equal to the second, we need d + b + 2*sqrt(d*b) = d^2+b -> d + 2*sqrt(d*b) = d^2 -> d^2 - d = 2*sqrt(d*b)
-> d^2*(d-1)^2 = 4*d*b -> b = d*(d-1)^2/4 -> sqrt(b) = (d-1)*sqrt(d)/2. Setting d = (n+1)^2 yields sqrt(b) = A027480(n).
This is the case k = 2 for FLTR, Fermat's Last Theorem with rational exponents 1/k: Consider x + y = x + y. Then (x^k)^(1/k) + (y^k)^(1/k) = ((x+y)^k)^(1/k).
For k > 2, there are infinitely many solutions to d^(1/k) + b^(1/k) = c^(1/k). E.g., 8^(1/3) + 27^(1/3) = 125^(1/3) at k = 3. However, in conjunction with d^2 + b = c, I could not find any nontrivial solutions.
A shifted version of A027480. - R. J. Mathar, Apr 07 2009
For n > 2, a(n) is the maximum value of the magic constant in a perimeter-magic n-gon of order n (see A342758). - Stefano Spezia, Mar 21 2021
a(n) is equal to the total number of P_3 edge-disjoint subgraphs of the complete graph on n vertices. - Samuel J. Bevins, May 09 2023

			For d = 9, b = 144, c = 225, 9^(1/2) + 144^(1/2) = 225^(1/2) and 9^2 + 144 = 225. So b^(1/2) = 12 is the 4th entry in the sequence.

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. D. Bennet, A. M. W. Glass and G. J. Székely, Fermat's Last Theorem for Rational Exponents, Am. Math. Monthly 111 (2004), 322-329. - _R. J. Mathar_, Apr 21 2009
Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20.

Cf. A000217, A000578, A002411, A004188, A006003, A007531, A007588, A027480, A342758.

Array[# (#^2 - 1)/2 &, 42, 0] (* Michael De Vlieger, Feb 20 2018 *)

flt2(n,p) = { local(a,b); for(a=0,n, b = (a^3-a)/2; print1(b", ") ) }

a(n) = 3*A000292(n-1).
From R. J. Mathar Feb 20 2008: (Start)
O.g.f.: 3*x^2/(-1+x)^4.
a(n) = n*(n^2 - 1)/2 = A007531(n+1)/2. (End)
G.f.: 3*x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
a(n) = A006003(n+1) - A000326(n+1). - J. M. Bergot, Dec 04 2014
E.g.f.: (1/2)* x^2 *(3 + x)*exp(x). - G. C. Greubel, Oct 15 2016
From Miquel Cerda, Dec 25 2016: (Start)
a(n) = A000578(n) - A006003(n).
a(n) = A004188(n) - A000578(n).
a(n) = A007588(n) - A004188(n). (End)
a(n) = A002411(n) - A000217(n). - Justin Gaetano, Feb 20 2018
From Amiram Eldar, Jan 09 2021: (Start)
Sum_{n>=2} 1/a(n) = 1/2.
Sum_{n>=2} (-1)^n/a(n) = 4*log(2) - 5/2. (End)

Edited by R. J. Mathar, Apr 21 2009

New name using R. J. Mathar's formula, Joerg Arndt, Dec 05 2014

A342819 Table read by ascending antidiagonals: T(k, n) is the number of distinct values of the magic constant in a perimeter-magic k-gon of order n.

Original entry on oeis.org

4, 4, 7, 6, 9, 10, 6, 11, 12, 13, 8, 13, 16, 17, 16, 8, 15, 18, 21, 20, 19, 10, 17, 22, 25, 26, 25, 22, 10, 19, 24, 29, 30, 31, 28, 25, 12, 21, 28, 33, 36, 37, 36, 33, 28, 12, 23, 30, 37, 40, 43, 42, 41, 36, 31, 14, 25, 34, 41, 46, 49, 50, 49, 46, 41, 34, 14, 27, 36, 45, 50, 55, 56, 57, 54, 51, 44, 37

Offset: 3

Stefano Spezia, Mar 22 2021

			The table begins:
k\n|  3   4   5   6   7 ...
---+-------------------
3  |  4   7  10  13  16 ...
4  |  4   9  12  17  20 ...
5  |  6  11  16  21  26 ...
6  |  6  13  18  25  30 ...
7  |  8  15  22  29  36 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).

Cf. A005408 (n = 4), A016813 (n = 6), A016921 (n = 8), A017077 (n = 10), A146951 (n = 7), A238290 (n = 9), A342757, A342758.

T[k_,n_]:=k(n-2)+(Mod[k,2]-1)Mod[n,2]+1;Table[T[k+3-n,n],{k,3,14},{n,3,k}]//Flatten

O.g.f.: (1 - y + 2*x*(y^2 + y - 1) + x^2*(4*y^2 + y - 3))/((1 - x)^2*(1 + x)*(1 - y)^2*(1 + y)).
E.g.f.: (1 + x*(y - 2))*cosh(x + y) + cosh(y)*sinh(x) + x*(y - 2)*sinh(x + y).
T(k, n) = k*(n - 2) + ((k mod 2) - 1)*(n mod 2) + 1.
T(k, n) = A342758(k, n) - A342757(k, n) + 1.

A342757 Array read by ascending antidiagonals: T(k, n) is the minimum value of the magic constant in a perimeter-magic k-gon of order n.

Original entry on oeis.org

9, 12, 17, 14, 22, 28, 17, 27, 37, 42, 19, 32, 45, 55, 59, 22, 37, 54, 68, 78, 79, 24, 42, 62, 81, 96, 104, 102, 27, 47, 71, 94, 115, 129, 135, 128, 29, 52, 79, 107, 133, 154, 167, 169, 157, 32, 57, 88, 120, 152, 179, 200, 210, 208, 189, 34, 62, 96, 133, 170, 204, 232, 251, 258, 250, 224

Offset: 3

Stefano Spezia, Mar 21 2021

			The array begins:
k\n|  3   4   5   6    7 ...
---+------------------------
3  |  9  17  28  42   59 ...
4  | 12  22  37  55   78 ...
5  | 14  27  45  68   96 ...
6  | 17  32  54  81  115 ...
7  | 19  37  62  94  133 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10 and 12).

Cf. A016873 (n = 4), A285009 (k = 3), A342719, A342758 (maximum).

T[k_,n_]:= ((1-Mod[k,2])Mod[n,2]+k*(n^2-2*n+2)+n)/2; Table[T[k+3-n,n],{k,3,13},{n,3,k}]//Flatten

G.f.: (x^2*(-3*y^3 + 2*y - 1) - x*(2*y^3 + y^2 - 2*y + 1) + (y - 1)*y)/((x - 1)^2*(x + 1)*(y - 1)^3*(y + 1)).
T(k, n) = (n^2/2 - n + 1)*k + n/2 if n is even or both n and k are odd.
T(k, n) = (n^2/2 - n + 1)*k + (n + 1)/2 if n is odd and k is even.
T(k, n) = ((1 - (k mod 2))*(n mod 2) + k*(n^2 - 2*n + 2) + n)/2.

A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

Original entry on oeis.org

21, 36, 45, 55, 78, 78, 78, 120, 136, 120, 105, 171, 210, 210, 171, 136, 231, 300, 325, 300, 231, 171, 300, 406, 465, 465, 406, 300, 210, 378, 528, 630, 666, 630, 528, 378, 253, 465, 666, 820, 903, 903, 820, 666, 465, 300, 561, 820, 1035, 1176, 1225, 1176, 1035, 820, 561

Offset: 3

Stefano Spezia, Mar 19 2021

			The array begins:
k\n|   3    4    5    6    7 ...
---+------------------------
3  |  21   45   78  120  171 ...
4  |  36   78  136  210  300 ...
5  |  55  120  210  325  465 ...
6  |  78  171  300  465  666 ...
7  | 105  231  406  630  903 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equation 3).

Cf. A014105 (n = 3), A033585 (n = 5), A037270 (1st superdiagonal), A081266 (n = 4), A083374 (1st subdiagonal), A110450 (diagonal), A144312 (n = 6), A144314 (n = 7), A342757, A342758.

T[k_,n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n,n],{k,3,12},{n,3,k}]//Flatten

O.g.f.: (x^2 - 3*x^2*y + x*y^2 + 3*x^2*y^2)/((1 - x)^3*(1 - y)^3).
E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.
T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.

cp's OEIS Frontend

A017005 a(n) = 7n + 2.

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A135503 a(n) = n*(n^2 - 1)/2.

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A342819 Table read by ascending antidiagonals: T(k, n) is the number of distinct values of the magic constant in a perimeter-magic k-gon of order n.

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A342757 Array read by ascending antidiagonals: T(k, n) is the minimum value of the magic constant in a perimeter-magic k-gon of order n.

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A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

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