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A056991 Numbers with digital root 1, 4, 7 or 9. 16

%I

%S 1,4,7,9,10,13,16,18,19,22,25,27,28,31,34,36,37,40,43,45,46,49,52,54,

%T 55,58,61,63,64,67,70,72,73,76,79,81,82,85,88,90,91,94,97,99,100,103,

%U 106,108,109,112,115,117,118,121,124,126,127,130,133,135,136,139,142

%N Numbers with digital root 1, 4, 7 or 9.

%C All squares are members (see A070433).

%C May also be defined as: possible sums of digits of squares. - _Zak Seidov_, Feb 11 2008

%C First differences are periodic: 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, ... - _Zak Seidov_, Feb 11 2008

%C Minimal n with corresponding sum-of-digits(n^2) are: 1, 2, 4, 3, 8, 7, 13, 24, 17, 43, 67, 63, 134, 83, 167, 264, 314, 313, 707, 1374, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 60663, 41833, 74833, 89437, 94863, 134164, 191833.

%C a(n) is the set of all m such that 9k+m can be a perfect square (quadratic residues of 9 including the trivial case of 0). - _Gary Detlefs_, Mar 19 2010

%H R. J. Mathar, <a href="/A056991/b056991.txt">Table of n, a(n) for n = 1..22222</a>

%H H. I. Okagbue, M. O. Adamu, S. A. Iyase, A. A. Opanuga, <a href="http://www.indjst.org/index.php/indjst/article/view/69912">Sequence of Integers Generated by Summing the Digits of their Squares</a>, Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F From _R. J. Mathar_, Feb 14 2008: (Start)

%F O.g.f.: x*(2x+1)*(x^2+x+1)/((-1+x)^2*(x+1)*(x^2+1)).

%F a(n) = a(n-4) + 9. (End)

%F a(n) = Sum_{k=0..n} (1/8)*(5*(k mod 4) + 5*((k+1) mod 4) + 3*((k+2) mod 4) - ((k+3) mod 4)), with n >= 0. - _Paolo P. Lava_, Feb 15 2008

%F a(n) = 3*(n - floor(n/4)) - (3 - i^n - (-i)^n - (-1)^n)/2, where i = sqrt(-1). - _Gary Detlefs_, Mar 19 2010

%p seq( 3*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/2, n=1..63); # _Gary Detlefs_, Mar 19 2010

%t LinearRecurrence[{1,0,0,1,-1},{1,4,7,9,10},70] (* _Harvey P. Dale_, Aug 29 2015 *)

%o (PARI) forstep(n=1,1e3,[3,3,2,1],print1(n", ")) \\ _Charles R Greathouse IV_, Sep 21 2012

%Y Cf. A000290, A056992, A070433.

%Y For complement see A268226.

%K nonn,base,easy

%O 1,2

%A _Eric W. Weisstein_

%E Edited by _N. J. A. Sloane_, May 16 2008 at the suggestion of _R. J. Mathar_

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Last modified December 6 19:22 EST 2019. Contains 329809 sequences. (Running on oeis4.)