A204519 -id:A204519 - cp's OEIS frontend

A204573 A204519(n)^2 = floor(A055851(n)/6): Squares which written in base 6, with some digit appended, yield another square.

0, 0, 0, 1, 4, 16, 121, 400, 1600, 11881, 39204, 156816, 1164241, 3841600, 15366400, 114083761, 376437604, 1505750416, 11179044361, 36887043600, 147548174400, 1095432263641, 3614553835204, 14458215340816, 107341182792481, 354189388806400, 1416757555225600

Offset: 1

M. F. Hasler, Jan 16 2012

Base-6 analog of A202303.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

b=6;for(n=0,1e7,issquare(n^2\b) & print1(n^2\b,","))

Conjecture: a(n) = 99*a(n-3)-99*a(n-6)+a(n-9) for n>10. - Colin Barker, Sep 20 2014

Empirical g.f.: -x^4*(x^6+16*x^5+4*x^4+22*x^3+16*x^2+4*x+1) / ((x-1)*(x^2+x+1)*(x^6-98*x^3+1)). - Colin Barker, Sep 20 2014

A055793 Numbers k such that k and floor[k/3] are both squares; i.e., squares which remain squares when written in base 3 and last digit is removed.

Original entry on oeis.org

0, 1, 4, 49, 676, 9409, 131044, 1825201, 25421764, 354079489, 4931691076, 68689595569, 956722646884, 13325427460801, 185599261804324, 2585064237799729, 36005300067391876, 501489136705686529, 6984842613812219524, 97286307456665386801, 1355023461779503195684, 18873042157456379352769

Offset: 1

Henry Bottomley, Jul 14 2000

Or, squares of the form 3k^2+1.

See A023110, A204503, A204512, A204517, A204519, A055812, A055808 and A055792 for the analog in other bases.

			a(3) = 49 because 49 = 7^2 = 1211 base 3 and 121 base 3 = 16 = 4^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..800
Tom C. Brown and Peter J Shiue, Squares of second-order linear recurrence sequences, Fib. Quart., 33 (1994), 352-356.
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 48, 56.
M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55.
Index to sequences related to truncating digits of squares.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A001075, A023110, A098301.

Cf. also A023110, A204503, A204512, A204517, A204519, A055812, A055808 and A055792 for the analog in other bases.

I:=[0, 1, 4]; [n le 3 select I[n] else 14*Self(n-1) - Self(n-2) - 6: n in [1..30]]; // Vincenzo Librandi, Jan 27 2013

A055793 := proc(n) coeftayl(x*(1-11*x+4*x^2)/((1-x)*(1-14*x+x^2)), x=0, n); end proc: seq(A055793(n), n=0..20); # Wesley Ivan Hurt, Sep 28 2014

CoefficientList[Series[x*(1 - 11*x + 4*x^2)/((1 - x)*(1 - 14*x + x^2)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 28 2014 *)
LinearRecurrence[{15,-15,1},{0,1,4,49},40] (* Harvey P. Dale, Jun 19 2021 *)

sq3nsqplus1(n) = { for(x=1,n, y = 3*x*x+1; \ print1(y" ") if(issquare(y),print1(y" ")) ) }

a(n) = 3*A098301(n-2)+1. - R. J. Mathar, Jun 11 2009
a(n) = 14*a(n-1)-a(n-2)-6, with a(0)=1, a(1)=4. (See Brown and Shiue)
a(n) = (A001075(n-2))^2. - Johannes Boot Dec 16 2011, corrected by M. F. Hasler, Jan 15 2012
G.f.: x*(1 - 11*x + 4*x^2)/((1 - x)*(1 - 14*x + x^2)). - M. F. Hasler, Jan 15 2012

More terms from Cino Hilliard, Mar 01 2003

A031150 Appending a digit to n^2 gives another perfect square.

Original entry on oeis.org

1, 2, 4, 5, 6, 12, 18, 43, 80, 154, 191, 228, 456, 684, 1633, 3038, 5848, 7253, 8658, 17316, 25974, 62011, 115364, 222070, 275423, 328776, 657552, 986328, 2354785, 4380794, 8432812, 10458821, 12484830, 24969660, 37454490

Offset: 1

Patrick De Geest

Square root of 'Squares from A023110 with last digit removed'.

One could include an initial '0', and even list it with multiplicity 3 or 4, since 00, 01, 04 and 09 are all perfect squares: In analogy to corresponding sequences for other bases, this sequence could be defined as sqrt(floor[A023110/10]), see A204512 [base 8], A204517 (base 7), A204519 (base 6), A204521 [base 5], A001353 [base 3], A001542 [base 2]. (For bases 4 and 9, the corresponding sequence contains all integers.) - M. F. Hasler, Jan 16 2012

			5^2 = 25 and 16^2 = 256, so 5 is in the sequence.
115364^2 = 13308852496, 364813^2 = 133088524969.

R. K. Guy, Neg and Reg, preprint, Jan 2012.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012
Joshua Stucky, Pell's Equation and Truncated Squares, Number Theory Seminar, Kansas State University, Feb 19 2018.
Index to sequences related to truncating digits of squares.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,38,0,0,0,0,0,0,-1).

Cf. A023110, A030686, A030687, A053784.

See A202303 for the resulting squares.

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

CoefficientList[Series[(x^10 + 2 x^9 + 4 x^8 + 5 x^7 + 18 x^6 + 12 x^5 + 6 x^4 + 5 x^3 + 4 x^2 + 2 x + 1)/(x^14 - 38 x^7 + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{0,0,0,0,0,0,38,0,0,0,0,0,0,-1},{1,2,4,5,6,12,18,43,80,154,191,228,456,684},40] (* Harvey P. Dale, Jun 09 2017 *)

G.f.: x*(x^10+2*x^9+4*x^8+5*x^7+18*x^6+12*x^5+6*x^4+5*x^3+4*x^2+2*x+1) / (x^14-38*x^7+1). - Colin Barker, Jan 30 2013

A204521 Square root of floor(A055812(n) / 5).

Original entry on oeis.org

0, 0, 0, 1, 3, 4, 8, 21, 55, 72, 144, 377, 987, 1292, 2584, 6765, 17711, 23184, 46368, 121393, 317811, 416020, 832040, 2178309, 5702887, 7465176

Offset: 1

M. F. Hasler, Jan 15 2012

nonn

Or: Numbers whose square yields another square when written in base 5.
(For the first 3 terms, the above "base 5" interpretation is questionable, since they have only 1 digit in base 5. It is understood that dropping this digit yields 0.)
Base-5 analog of A031150 [base 10], A001353 [base 3], A001542 [base 2].
The square roots of A055812 are listed in A204520.

Cf. A055792, A055793, A204519, A204517, A204512, A204503 and A023110.

b=5;for(n=1,2e9,issquare(n^2\b) && print1(sqrtint(n^2\b),","))

Empirical g.f.: x^4*(x^5+3*x^4+8*x^3+4*x^2+3*x+1) / ((x^4-4*x^2-1)*(x^4+4*x^2-1)). - Colin Barker, Sep 15 2014

cp's OEIS Frontend

A204573 A204519(n)^2 = floor(A055851(n)/6): Squares which written in base 6, with some digit appended, yield another square.

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A055793 Numbers k such that k and floor[k/3] are both squares; i.e., squares which remain squares when written in base 3 and last digit is removed.

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A031150 Appending a digit to n^2 gives another perfect square.

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A204521 Square root of floor(A055812(n) / 5).

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