A024352 Numbers which are the difference of two positive squares, c^2 - b^2 with 1 <= b < c.

3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

These are the solutions to the equation x^2 + xy = n where y mod 2 = 0, y is positive and x is any positive integer. - Andrew S. Plewe, Oct 19 2007

Ordered different terms of A120070 = 3, 8, 5, 15, 12, 7, ... (which contains two 15's, two 40's, and two 48's). Complement: A139544. (See A139491.) - Paul Curtz, Sep 01 2009

A024359(a(n)) > 0. - Reinhard Zumkeller, Nov 09 2012

If a(n) mod 6 = 3, n > 1, then a(n) = c^2 - f(a(n))^2 where f(n) = (floor(4*n/3) - 3 - n)/2. For example, 171 = 30^2 - 27^2 and f(171) = 27. - Gary Detlefs, Jul 15 2014

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Ron Knott, Pythagorean Triples and Online Calculators
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Same as A042965 except for initial terms. - Michael Somos, Jun 08 2000

Different from A020884.

Cf. A009005, A020884, A120070, A139544, A139491.

Programs

Haskell

a024352 n = a024352_list !! (n-1)
a024352_list = 3 : drop 4 a042965_list
-- Reinhard Zumkeller, Nov 09 2012

Magma

[3] cat [4 +Floor((4*n-3)/3): n in [2..100]]; // G. C. Greubel, Apr 22 2023

Mathematica

Union[Flatten[Table[Select[Table[b^2 - c^2, {c, b-1}], # < 100 &], {b, 100}]]] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{1,0,1,-1},{3,5,7,8,9},70] (* Harvey P. Dale, Dec 20 2021 *)

PARI

is(n)=(n%4!=2 && n>4) || n==3 \\ Charles R Greathouse IV, May 31 2013

Python

def A024352(n): return 3 if n==1 else 3+(n<<2)//3 # Chai Wah Wu, Feb 10 2025

SageMath

def A024352(n): return 4 + ((4*n-3)//3) - int(n==1)
[A024352(n) for n in range(1,101)] # G. C. Greubel, Apr 22 2023

Formula

Consists of all positive integers except 1, 4 and numbers == 2 (mod 4).

a(n) = a(n-3) + 4, n > 4.

G.f.: (3 + 2*x + 2*x^2 - 2*x^3 - x^4)/(1 - x - x^3 + x^4). - Ralf Stephan, before May 13 2008

a(n) = a(n-1) + a(n-3) - a(n-4), for n > 5. - Ant King, Oct 03 2011

a(n) = 4 + floor((4*n-3)/3), n > 1. - Gary Detlefs, Jul 15 2014

Extensions

Edited by N. J. A. Sloane, Sep 19 2008