A339623 -id:A339623 - cp's OEIS frontend

A190816 a(n) = 5n^2 - 4n + 1.

1, 2, 13, 34, 65, 106, 157, 218, 289, 370, 461, 562, 673, 794, 925, 1066, 1217, 1378, 1549, 1730, 1921, 2122, 2333, 2554, 2785, 3026, 3277, 3538, 3809, 4090, 4381, 4682, 4993, 5314, 5645, 5986, 6337, 6698, 7069, 7450, 7841, 8242, 8653, 9074

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 20 2011

For n >= 2, hypotenuses of primitive Pythagorean triangles with m = 2*n-1, where the sides of the triangle are a = m^2 - n^2, b = 2*n*m, c = m^2 + n^2; this sequence is the c values, short sides (a) are A045944(n-1), and long sides (b) are A002939(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Short sides (a) A045944(n-1), long sides (b) A002939(n).
Cf. A017281 (first differences), A051624 (a(n)-1), A202141.
Sequences of the form m*n^2 - 4*n + 1: -A131098 (m=0), A028872 (m=1), A056220 (m=2), A045944 (m=3), A016754 (m=4), this sequence (m=5), A126587 (m=6), A339623 (m=7), A080856 (m=8).

[5*n^2 - 4*n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 19 2011

Table[5*n^2 - 4*n + 1, {n, 0, 100}]
LinearRecurrence[{3,-3,1},{1,2,13},100] (* or *) CoefficientList[ Series[ (-10 x^2+x-1)/(x-1)^3,{x,0,100}],x] (* Harvey P. Dale, May 24 2011 *)

a(n)=5*n^2-4*n+1 \\ Charles R Greathouse IV, Oct 16 2015

[5*n^2-4*n+1 for n in range(41)] # G. C. Greubel, Dec 03 2023

From Harvey P. Dale, May 24 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.
G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)
E.g.f.: (1 + x + 5*x^2)*exp(x). - G. C. Greubel, Dec 03 2023

Edited by Franklin T. Adams-Watters, May 20 2011

A339609 Consider a triangle drawn on the perimeter of a triangular lattice with side length n. a(n) is the number of regions inside the triangle after drawing unit circles centered at each lattice point inside the triangle.

Original entry on oeis.org

0, 0, 4, 10, 22, 39, 61, 88, 120, 157, 199, 246, 298, 355, 417, 484, 556, 633, 715, 802, 894, 991, 1093, 1200, 1312, 1429, 1551, 1678, 1810, 1947, 2089, 2236, 2388, 2545, 2707, 2874, 3046, 3223, 3405, 3592, 3784, 3981, 4183, 4390, 4602, 4819, 5041, 5268, 5500, 5737

Offset: 1

Wesley Ivan Hurt, Dec 09 2020

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Peter Kagey, Example for a(4) = 10.

Cf. A005476, A339623 (square version).

Join[{0, 0, 4}, Table[(5 n^2 - 21 n + 24)/2, {n, 4, 60}]]

a(n) = (5*n^2 - 21*n + 24)/2 for n >= 4, with a(1)=a(2)=0, a(3)=4.
a(n) = A005476(n-2)+1 for n >= 4. - Hugo Pfoertner, Dec 10 2020
From Stefano Spezia, Dec 10 2020: (Start)
G.f.: x^3*(4 - 2*x + 4*x^2 - x^3)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)

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A190816 a(n) = 5n^2 - 4n + 1.

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A339609 Consider a triangle drawn on the perimeter of a triangular lattice with side length n. a(n) is the number of regions inside the triangle after drawing unit circles centered at each lattice point inside the triangle.

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cp's OEIS Frontend

A190816 a(n) = 5*n^2 - 4*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A339609 Consider a triangle drawn on the perimeter of a triangular lattice with side length n. a(n) is the number of regions inside the triangle after drawing unit circles centered at each lattice point inside the triangle.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A190816 a(n) = 5n^2 - 4n + 1.