A342757 -id:A342757 - cp's OEIS frontend

A016873 a(n) = 5*n + 2.

2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102, 107, 112, 117, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 172, 177, 182, 187, 192, 197, 202, 207, 212, 217, 222, 227, 232, 237, 242, 247, 252, 257, 262, 267, 272, 277

Offset: 0

N. J. A. Sloane

Numbers ending in 2 or 7. - Lekraj Beedassy, Jul 08 2006
For n > 2, also the number of (not necessarily maximal) cliques in the n-gear graph. - Eric W. Weisstein, Nov 29 2017
Also, positive integers k such that 10*k+5 is equal to the product of two integers ending with 5. Proof: if 10*k+5 = (10*a+5) * (10*b+5), then k = 10*a*b + 5*(a+b) + 2 = 5 * (a + b + 2*a*b) + 2, of the form 5m + 2. So, 262 is a term because 2625 = 35 * 75. - Bernard Schott, May 15 2019
Numbers k such that 2^x + 3^x == 0 mod 31 and 2^x + 3^x == 0 mod 11 with x = 6*k+3. - Pedro Caceres, May 18 2022

G. C. Greubel, Table of n, a(n) for n = 0..1000
Cino Hilliard, solutions to 3^x + 5^x == 2 mod 11. [broken link]
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001622, A008586, A008587, A016861, A342757.
Cf. A053742 (product of two integers ending with 5).
Cf. A324298 (product of two integers ending with 6).

[5*n+2: n in [0..80]]; // G. C. Greubel, Oct 17 2023

a[1]:=2:for n from 2 to 100 do a[n]:=a[n-1]+5 od: seq(a[n], n=1..50); # Zerinvary Lajos, Mar 16 2008

Range[2, 500, 5] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
(* Programs from Eric W. Weisstein, Nov 29 2017 *)
5*Range[0, 70] +2
LinearRecurrence[{2, -1}, {7, 12}, {0, 70}]
CoefficientList[Series[(2+3*x)/(1-x)^2, {x,0,70}], x] (* End *)

a(n)=5*n+2 \\ Charles R Greathouse IV, Jul 10 2016

[i+2 for i in range(300) if gcd(i,5) == 5] # Zerinvary Lajos, May 20 2009

a(n) = 10*n - a(n-1) - 1 (with a(0)=2). - Vincenzo Librandi, Nov 20 2010
G.f.: (2+3*x)/(1-x)^2. - Colin Barker, Jan 08 2012
E.g.f.: exp(x)*(2 + 5*x). - Stefano Spezia, Mar 21 2021
Sum_{n>=0} (-1)^n/a(n) = sqrt(2-2/sqrt(5))*Pi/10 + log(phi)/sqrt(5) - log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

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.

A342758 Array read by ascending antidiagonals: T(k, n) is the maximum value of the magic constant in a perimeter-magic k-gon of order n.

Original entry on oeis.org

12, 15, 23, 19, 30, 37, 22, 37, 48, 54, 26, 44, 60, 71, 74, 29, 51, 71, 88, 97, 97, 33, 58, 83, 105, 121, 128, 123, 36, 65, 94, 122, 144, 159, 162, 152, 40, 72, 106, 139, 168, 190, 202, 201, 184, 43, 79, 117, 156, 191, 221, 241, 250, 243, 219, 47, 86, 129, 173, 215, 252, 281, 299, 303, 290, 257

Offset: 3

Stefano Spezia, Mar 21 2021

			The array begins:
k\n|  3   4   5    6    7 ...
---+---------------------
3  | 12  23  37   54   74 ...
4  | 15  30  48   71   97 ...
5  | 19  37  60   88  121 ...
6  | 22  44  71  105  144 ...
7  | 26  51  83  122  168 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 11 and 13).

Cf. A017005 (n = 4), A135503 (diagonal), A341740 (k = 3), A342719, A342757 (minimum).

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

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

A016873 a(n) = 5*n + 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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A342758 Array read by ascending antidiagonals: T(k, n) is the maximum value of the magic constant in a perimeter-magic k-gon of order n.

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Mathematica

Formula

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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