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A031149 Numbers n such that n^2 with last digit deleted is still a perfect square. 21
0, 1, 2, 3, 4, 7, 13, 16, 19, 38, 57, 136, 253, 487, 604, 721, 1442, 2163, 5164, 9607, 18493, 22936, 27379, 54758, 82137, 196096, 364813, 702247, 870964, 1039681, 2079362, 3119043, 7446484, 13853287, 26666893, 33073696, 39480499, 78960998, 118441497, 282770296 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Square root of A023110(n).

For the first 4 terms, the square has only one digit, but in analogy to the sequences in other bases (A204502, A204512,  A204514, A204516, A204518, A204520, A004275, A055793, A055792), it is understood that deleting this digit yields 0.

From Robert Israel, Feb 16 2016: (Start)

Solutions x to x^2 = 10*y^2 + j, j in {0,1,4,6,9}, in increasing order of x.

j=0 occurs only for x=0.

Let M be the 2 X 2 matrix [19, 60; 6, 19].

Solutions of x^2 = 10*y^2 + 1 are (x,y)^T = M^k (1,0)^T for k >= 0.

Solutions of x^2 = 10*y^2 + 4 are (x,y)^T = M^k (2,0)^T for k >= 0.

Solutions of x^2 = 10*y^2 + 6 are (x,y)^T = M^k (4,1)^T and M^k (16,5)^T for k >= 0.

Solutions of x^2 = 10*y^2 + 9 are (x,y)^T = M^k (3,0)^T, M^k (7,2)^T, M^k (13,4)^T for k >= 0.

Since (1,0)^T <= (2,0)^T <= (3,0)^T <= (4,1)^T <= (7,2)^T <= (13,4)^T <= (16,5)^T <= (19,6)^T = M (1,0)^T (element-wise) and M has positive entries, we see that the terms always occur in this order, for successive k.

The eigenvalues of M are 19 + 6*sqrt(10) and 19 - 6*sqrt(10).

From this follow my formulas below and the G.f. (End)

REFERENCES

R. K. Guy, Neg and Reg, preprint, Jan 2012. [From N. J. A. Sloane, Jan 12 2012]

LINKS

Dmitry Petukhov and Robert Israel, Table of n, a(n) for n = 1..4400 (n = 1..67 from Dmitry Petukhov)

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012

Index to sequences related to truncating digits of squares.

FORMULA

Appears to satisfy: a(n)=38a(n-7)-a(n-14) which would require a(-k) to look like 3, 2, 1, 4, 7, 13, 16, 57, 38, 19, 136, ... for k>0. - Henry Bottomley, May 08 2001

Empirical g.f.: -x^2*(4*x^13 +7*x^12 +13*x^11 +16*x^10 +57*x^9 +38*x^8 +19*x^7 -16*x^6 -13*x^5 -7*x^4 -4*x^3 -3*x^2 -2*x -1) / (x^14 -38*x^7 +1). - Colin Barker, Jan 17 2014

With e1 = 19 + 6*sqrt(10) and e2 = 19 - 6*sqrt(10),

   a(2+7k) = (e1^k + e2^k)/2,

   a(3+7k) = e1^k + e2^k,

   a(4+7k) = (3/2) (e1^k + e2^k),

   a(5+7k) = (2+sqrt(10)/2) e1^k + (2-sqrt(10)/2) e2^k,

   a(6+7k) = (7/2+sqrt(10)) e1^k + (7/2-sqrt(10)) e2^k,

   a(7+7k) = (13/2+2 sqrt(10)) e1^k + (13/2-2 sqrt(10)) e2^k,

a(8+7k) = (8+5 sqrt(10)/2) e1^k + (8-5 sqrt(10)/2) e2^k. - Robert Israel, Feb 16 2016

EXAMPLE

364813^2 = 133088524969, 115364^2 = 13308852496.

MAPLE

for i from 1 to 150000 do if (floor(sqrt(10 * i^2 + 9)) > floor(sqrt(10 * i^2))) then print(floor(sqrt(10 * i^2 + 9))) end if end do;

MATHEMATICA

fQ[n_] := IntegerQ@ Sqrt@ Quotient[n^2, 10]; Select[ Range[ 0, 40000000], fQ] (* Harvey P. Dale, Jun 15 2011 *) (* modified by Robert G. Wilson v, Jan 16 2012 *)

PROG

(PARI) s=[]; for(n=0, 1e7, if(issquare(n^2\10), s=concat(s, n))); s \\ Colin Barker, Jan 17 2014; typo fixed by Zak Seidov, Jan 31 2014

CROSSREFS

Cf. A023110, A030686, A030687, A031150.

Sequence in context: A057983 A200088 A004783 * A096723 A266498 A137495

Adjacent sequences:  A031146 A031147 A031148 * A031150 A031151 A031152

KEYWORD

nonn,base

AUTHOR

Patrick De Geest

EXTENSIONS

4 initial terms added by M. F. Hasler, Jan 15 2012

a(40) from Robert G. Wilson v, Jan 15 2012

STATUS

approved

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Last modified November 19 11:04 EST 2017. Contains 294936 sequences.