A061101 Squares with digital root 7.

16, 25, 169, 196, 484, 529, 961, 1024, 1600, 1681, 2401, 2500, 3364, 3481, 4489, 4624, 5776, 5929, 7225, 7396, 8836, 9025, 10609, 10816, 12544, 12769, 14641, 14884, 16900, 17161, 19321, 19600, 21904, 22201, 24649, 24964, 27556, 27889, 30625

Offset: 1

Author

Amarnath Murthy, Apr 19 2001

Keywords

• nonn
• base
• easy

Examples

			1681=41^2, 1+6+8+1 = 16, 1+6 =7, 4624=68^2, 4+6+2+4 = 16, 1+6 =7.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Amarnath Murthy & Charles Ashbacher, Fabricating a perfect square with a given valid digit sum, in Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences, pp 154-156.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A056991.

Programs

Maple

seq(seq((9*i+j)^2, j=4..5), i=0..100); # Robert Israel, Jan 31 2017

PARI

a(n)=(n\2*9-4*(-1)^n)^2 \\ Charles R Greathouse IV, Sep 21 2012

Formula

Conjecture: a(n)=(9*n-8)^2/4 for n even. a(n)=(9*n-1)^2/4 for n odd. G.f.: x*(16+9*x+112*x^2+9*x^3+16*x^4)/((1-x)^3*(1+x)^2). - Colin Barker, Apr 21 2012

Conjecture is true, because x^2 == 7 (mod 9) if and only if x == 4 or 5 (mod 9). - Robert Israel, Jan 31 2017

Extensions

More terms from Harry J. Smith, Jul 18 2009