A056992 Digital roots of square numbers A000290.

1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9

Offset

1,2

Comments

Cyclic with a period of nine. Note that (7, 9, 4, 1, 9, 1, 4, 9, 7) is palindromic.

a(n) is also the decimal expansion of 499264730/333333333. - Enrique Pérez Herrero, Jul 28 2009

a(n) is also the digital root of the Wonderful Demlo number A002477(n). - Enrique Pérez Herrero, Dec 20 2009

First comment above by Enrique Pérez Herrero and his formula below together give the following identity: 1+Sum_{n>=2}(1+9*((n^2-1)/9-floor((n^2-1)/9)))/10^(n-1) = 499264730/333333333 = 1.49779419149779419149779419... - Alexander R. Povolotsky, Jun 14 201

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Eric Weisstein's World of Mathematics, Square Number.

Formula

a(n) = 1+9*{(n^2-1)/9} , where the symbol {} means fractional part. - Enrique Pérez Herrero, Dec 20 2009

a(n) = (1/108)*{113*(n mod 9)-79*[(n+1) mod 9]+53*[(n+2) mod 9]+77*[(n+3) mod 9]-7*[(n+4) mod 9]+17*[(n+5) mod 9]+41*[(n+6) mod 9]-43*[(n+7) mod 9]-19*[(n+8) mod 9]}, with n>=0. - Paolo P. Lava, Aug 04 2009

a(n) = 3(1 + cos(2n*Pi/3) + cos(4n*Pi/3)) + mod(3n^4+3n^6+4n^8,9). - Ant King, Oct 07 2009

G.f.: x (1+4x+9x^2+7x^3+7x^4+9x^5+4x^6+x^7+9x^8)/((1-x)(1+x+x^2)(1+x^3+x^6)). - Ant King, Oct 20 2009

Also a(n) = A010888(A057147(n)). - Reinhard Zumkeller, Mar 19 2014

Mathematica

DigitalRoot[n_Integer?NonNegative] := 1 + 9*FractionalPart[(n - 1)/9] A056992[n_]:=DigitalRoot[n^2] (* Enrique Pérez Herrero, Dec 20 2009 *)
Table[FixedPoint[Total[IntegerDigits[#]]&, n^2], {n, 90}] (* Zak Seidov, Jun 13 2015 *)
PadRight[{}, 120, {1, 4, 9, 7, 7, 9, 4, 1, 9}] (* Harvey P. Dale, Apr 16 2022 *)

Prog

(Haskell)
a056992 = a010888 . a000290  -- Reinhard Zumkeller, Mar 19 2014

Crossrefs

Cf. A000290, A002477, A010888, A057147, A056991.

Sequence in context: A296425 A335089 A306004 * A339023 A169908 A004159

Adjacent sequences: A056989 A056990 A056991 * A056993 A056994 A056995

Keyword

nonn,base,easy

Author

Eric W. Weisstein

Status

approved