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

1,1

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).

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

Example

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

Maple

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

Prog

(PARI) b=0; for (n=1, 1000, until (s==7, b++; s=b^2; s-=9*(s\9)); write("b061101.txt", n, " ", b^2)) \\ Harry J. Smith, Jul 18 2009
(PARI) a(n)=(n\2*9-4*(-1)^n)^2 \\ Charles R Greathouse IV, Sep 21 2012

Crossrefs

Cf. A056991.

Sequence in context: A188826 A152152 A260047 * A166672 A167358 A267764

Adjacent sequences: A061098 A061099 A061100 * A061102 A061103 A061104

Keyword

nonn,base,easy

Author

Amarnath Murthy, Apr 19 2001

Extensions

More terms from Harry J. Smith, Jul 18 2009

Status

approved