A061099 Squares with digital root 1.

1, 64, 100, 289, 361, 676, 784, 1225, 1369, 1936, 2116, 2809, 3025, 3844, 4096, 5041, 5329, 6400, 6724, 7921, 8281, 9604, 10000, 11449, 11881, 13456, 13924, 15625, 16129, 17956, 18496, 20449, 21025, 23104, 23716, 25921, 26569, 28900, 29584

Offset

1,2

Links

Harry J. Smith, Table of n, a(n) for n = 1..1001

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+2)^2/4 for n even. a(n)=(9*n+7)^2/4 for n odd. G.f.: (1+63*x+34*x^2+63*x^3+x^4)/((1-x)^3*(1+x)^2). [Colin Barker, Apr 21 2012]

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). - Wesley Ivan Hurt, Apr 21 2021

Example

289 = 17^2, 2+8+9 = 19, 1+9 = 1, 1369 = 37^2, 1+3+6+9 = 19, 1+9 = 1.

Prog

(PARI) { b=0; for (n=0, 1000, until (s==1, b++; s=b^2; s-=9*(s\9)); write("b061099.txt", n, " ", b^2) ) } [Harry J. Smith, Jul 19 2009]
(PARI) SumD(x)= { s=0; while (x>9, s=s+x-10*(x\10); x=x\10); return(s + x) } { b=0; for (n=0, 1000, s=2; while (s!= 1, b++; s=b^2; while (s>9, s=SumD(s))); write("b061099.txt", n, " ", b^2) ) } [Harry J. Smith, Jul 17 2009]
(PARI) a(n)=(n\2*9-(-1)^n)^2 \\ Charles R Greathouse IV, Sep 20 2012

Crossrefs

Squares of A056020.

Cf. A056991.

Sequence in context: A111730 A255990 A104022 * A118488 A088033 A303960

Adjacent sequences: A061096 A061097 A061098 * A061100 A061101 A061102

Keyword

nonn,base,easy

Author

Amarnath Murthy, Apr 19 2001

Extensions

More terms from Harry J. Smith, Jul 17 2009

Status

approved