A204503 -id:A204503 - cp's OEIS frontend

A028310 Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x.

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

N. J. A. Sloane

1 followed by the natural numbers.
Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).
Engel expansion of e (see A006784 for definition) [when offset by 1]. - Henry Bottomley, Dec 18 2000
Also the denominators of the series expansion of log(1+x). Numerators are A062157. - Robert G. Wilson v, Aug 14 2015
The right-shifted sequence (with a(0)=0) is an autosequence (of the first kind - see definition in links). - Jean-François Alcover, Mar 14 2017

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + 9*x^9  + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Olivia Nabawanda and Fanja Rakotondrajao, The sets of flattened partitions with forbidden patterns, arXiv:2011.07304 [math.CO], 2020.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Oeis Wiki, Autosequence
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for Molien series
Index entries for sequences related to Engel expansions

Cf. A000007, A000027, A000660 (boustrophedon transform).

Cf. A001477, A004001, A005229, A112934, A123110, A212393, A204503.

a028310 n = 0 ^ n + n
a028310_list = 1 : [1..]  -- Reinhard Zumkeller, Nov 06 2012

[n eq 0 select 1 else n: n in [0..75]]; // G. C. Greubel, Jan 05 2024

a:= n-> `if`(n=0, 1, n):
seq(a(n), n=0..60);

Denominator@ CoefficientList[Series[Log[1+x], {x,0,75}], x] (* or *)
CoefficientList[ Series[(1 -x +x^2)/(1-x)^2, {x,0,75}], x] (* Robert G. Wilson v, Aug 14 2015 *)
Join[{1}, Range[75]] (* G. C. Greubel, Jan 05 2024 *)
LinearRecurrence[{2,-1},{1,1,2},80] (* Harvey P. Dale, Jan 29 2025 *)

{a(n) = (n==0) + max(n, 0)} /* Michael Somos, Feb 02 2004 */

A028310(n)=n+!n  \\ M. F. Hasler, Jan 16 2012

def A028310(n): return n|bool(n)^1 # Chai Wah Wu, Jul 13 2023

[n + int(n==0) for n in range(76)] # G. C. Greubel, Jan 05 2024

Binomial transform is A005183. - Paul Barry, Jul 21 2003
G.f.: (1 - x + x^2) / (1 - x)^2 = (1 - x^6) /((1 - x) * (1 - x^2) * (1 - x^3)) = (1 + x^3) / ((1 - x) * (1 - x^2)). a(0) = 1, a(n) = n if n>0.
Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - Michael Somos Jul 30 2006
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x)))). - Michael Somos, Apr 05 2012
G.f. of A112934(x) = 1 / (1 - a(0)*x / (1 - a(1)*x / ...)). - Michael Somos, Apr 05 2012
a(n) = A000027(n) unless n=0.
a(n) = Sum_{k=0..n} A123110(n,k). - Philippe Deléham, Oct 06 2009
E.g.f: 1+x*exp(x). - Wolfdieter Lang, May 03 2010
a(n) = sqrt(floor[A204503(n+3)/9]). - M. F. Hasler, Jan 16 2012
E.g.f.: 1-x + x*E(0), where E(k) = 2 + x/(2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
a(n) = A001477(n) + A000007(n). - Miko Labalan, Dec 12 2015 (See the first comment.)

A023110 Squares which remain squares when the last digit is removed.

Original entry on oeis.org

0, 1, 4, 9, 16, 49, 169, 256, 361, 1444, 3249, 18496, 64009, 237169, 364816, 519841, 2079364, 4678569, 26666896, 92294449, 341991049, 526060096, 749609641, 2998438564, 6746486769, 38453641216, 133088524969, 493150849009, 758578289296, 1080936581761

Offset: 1

David W. Wilson

This A023110 = A031149^2 is the base 10 version of A001541^2 = A055792 (base 2), A001075^2 = A055793 (base 3), A004275^2 = A055808 (base 4), A204520^2 = A055812 (base 5), A204518^2 = A055851 (base 6), A204516^2 = A055859 (base 7), A204514^2 = A055872 (base 8) and A204502^2 = A204503 (base 9). - M. F. Hasler, Sep 28 2014

For the first 4 terms the square has only one digit. It is understood that deleting this digit yields 0. - Colin Barker, Dec 31 2017

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

Jon E. Schoenfield, Table of n, a(n) for n = 1..70 (terms 1..38 from David W. Wilson, terms 39..40 from Robert G. Wilson v, terms 41..67 from Dmitry Petukhov)
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.

Cf. A023111.
Cf. A031150, A053784, A031149, A055792, A055793, A055808, A055812, A055851, A055859, A055872.
Cf. A001541, A001075, A004275, A204520, A204518, A204516, A204514, A204502, A204503.

count:= 1: A[1]:= 0:
for n from 0 while count < 35 do
  for t in [1,4,6,9] do
    if issqr(10*n^2+t) then
       count:= count+1;
       A[count]:= 10*n^2+t;
    fi
  od
od:
seq(A[i],i=1..count); # Robert Israel, Sep 28 2014

fQ[n_] := IntegerQ@ Sqrt@ Quotient[n^2, 10]; Select[ Range@ 1000000, fQ]^2 (* Robert G. Wilson v, Jan 15 2011 *)

for(n=0,1e7, issquare(n^2\10) & print1(n^2",")) \\  M. F. Hasler, Jan 16 2012

Appears to satisfy a(n)=1444*a(n-7)+a(n-14)-76*sqrt(a(n-7)*a(n-14)) for n >= 16. For n = 15, 14, 13, ... this would require a(1) = 16, a(0) = 49, a(-1) = 169, ... - Henry Bottomley, May 08 2001; edited by Robert Israel, Sep 28 2014
a(n) = A031149(n)^2. - M. F. Hasler, Sep 28 2014
Conjectures from Colin Barker, Dec 31 2017: (Start)
G.f.: x^2*(1 + 4*x + 9*x^2 + 16*x^3 + 49*x^4 + 169*x^5 + 256*x^6 - 1082*x^7 - 4328*x^8 - 9738*x^9 - 4592*x^10 - 6698*x^11 - 6698*x^12 - 4592*x^13 + 361*x^14 + 1444*x^15 + 3249*x^16 + 256*x^17 + 169*x^18 + 49*x^19 + 16*x^20) / ((1 - x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1 - 1442*x^7 + x^14)).
a(n) = 1443*a(n-7) - 1443*a(n-14) + a(n-21) for n>22.
(End)

More terms from M. F. Hasler, Jan 16 2012

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

A055812 a(n) and floor(a(n)/5) are both squares; i.e., squares which remain squares when written in base 5 and last digit is removed.

Original entry on oeis.org

0, 1, 4, 9, 49, 81, 324, 2209, 15129, 25921, 103684, 710649, 4870849, 8346321, 33385284, 228826129, 1568397609, 2687489281, 10749957124, 73681302249, 505019158609, 865363202001, 3461452808004

Offset: 1

Henry Bottomley, Jul 14 2000

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. - M. F. Hasler, Jan 15 2012

			a(4) = 49 because 49 = 7^2 = 144 base 5 and 14 base 5 = 9 = 3^2.

For analogs in other bases see A055792 (base 2), A055793 (base 3), A055808 (base 4), A055851 (base 6), A204517 (base 7), A204512 (base 8), A204503 (base 9) and A023110 (base 10).

Squares of A204520. The square roots of floor[a(n)/5] are given in A204521.

b=5;for(n=1,2e9,issquare(n^2\b) && print1(n^2,","))  \\ M. F. Hasler, Jan 15 2012

Empirical g.f.: -x^2*(9*x^11 +49*x^10 +324*x^9 +81*x^8 -698*x^7 -698*x^6 -968*x^5 -242*x^4 +49*x^3 +9*x^2 +4*x +1) / ((x -1)*(x +1)*(x^2 -4*x -1)*(x^2 +1)*(x^2 +4*x -1)*(x^4 +18*x^2 +1)). - Colin Barker, Sep 15 2014

More terms added and offset changed to 1 by M. F. Hasler, Jan 15 2012

A204502 Numbers such that floor[a(n)^2 / 9] is a square.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177

Offset: 1

M. F. Hasler, Jan 15 2012

nonn

Or, numbers n such that n^2, with its last base-9 digit dropped, is again a square. (Except maybe for the 3 initial terms whose square has only 1 digit in base 9.)

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012
Index to sequences related to truncating digits of squares.

The squares are in A204503, the squares with last base-9 digit dropped in A204504, and the square roots of the latter in A028310.

Cf. A031149=sqrt(A023110) (base 10), 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).

Select[Range[0,200],IntegerQ[Sqrt[Floor[#^2/9]]]&] (* Harvey P. Dale, May 05 2018 *)

b=9;for(n=0,200,issquare(n^2\b) & print1(n","))

Conjecture: a(n) = 3*n-12 for n>5. G.f.: x^2*(x^2+x+1)*(x^3-x+1)/(x-1)^2. [Colin Barker, Nov 23 2012]

A204514 Numbers such that floor(a(n)^2 / 8) is again a square.

Original entry on oeis.org

0, 1, 2, 3, 6, 17, 34, 99, 198, 577, 1154, 3363, 6726, 19601, 39202, 114243, 228486, 665857, 1331714, 3880899, 7761798, 22619537, 45239074, 131836323, 263672646, 768398401, 1536796802, 4478554083, 8957108166, 26102926097, 52205852194, 152139002499, 304278004998, 886731088897

Offset: 1

M. F. Hasler, Jan 15 2012

Or: Numbers whose square, with its last base-8 digit dropped, is again a square. (Except maybe for the 3 initial terms whose square has only 1 digit in base 8.)

See A204504 for the squares resulting from truncation of a(n)^2, and A204512 for their square roots. - M. F. Hasler, Sep 28 2014

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012
Index to sequences related to truncating digits of squares.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), 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).

Cf. A003499, A055872, A204504, A204512.

A204514 := proc(n) coeftayl((x^2+2*x^3-3*x^4-6*x^5)/(1-6*x^2+x^4), x=0, n); end proc: seq(A204514(n), n=1..30); # Wesley Ivan Hurt, Sep 28 2014

CoefficientList[Series[(x^2 + 2*x^3 - 3*x^4 - 6*x^5)/(x (1 - 6*x^2 + x^4)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 28 2014 *)
LinearRecurrence[{0,6,0,-1},{0,1,2,3,6},40] (* Harvey P. Dale, Nov 23 2022 *)

b=8;for(n=0,1e7,issquare(n^2\b) & print1(n","))

A204514(n)=polcoeff((x + 2*x^2 - 3*x^3 - 6*x^4)/(1 - 6*x^2 + x^4+O(x^(n+!n))),n-1,x)

G.f. = (x^2 + 2*x^3 - 3*x^4 - 6*x^5)/(1 - 6*x^2 + x^4).
a(n) = sqrt(A055872(n)). - M. F. Hasler, Sep 28 2014
a(2n) = A001541(n-1). a(2n+1) = A003499(n-1). - R. J. Mathar, Feb 05 2020

A055851 a(n) and floor(a(n)/6) are both squares; i.e., squares that remain squares when written in base 6 and last digit is removed.

Original entry on oeis.org

0, 1, 4, 9, 25, 100, 729, 2401, 9604, 71289, 235225, 940900, 6985449, 23049601, 92198404, 684502569, 2258625625, 9034502500, 67074266169, 221322261601, 885289046404, 6572593581849, 21687323011225, 86749292044900

Offset: 1

Henry Bottomley, Jul 14 2000

For the first 3 terms, the above "base 6" interpretation is questionable, since they have only 1 digit in base 6. It is understood that dropping this digit yields 0. - M. F. Hasler, Jan 15 2012

Base-6 analog of A055792 (base 2), A055793 (base 3), A055808 (base 4), A055812 (base 5), A204517 (base 7), A204503 (base 9) and A023110 (base 10). - M. F. Hasler, Jan 15 2012

			a(5) = 100 because 100 = 10^2 = 244 base 6 and 24 base 6 = 16 = 4^2.

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012
Index to sequences related to truncating digits of squares.

Cf. A023110.

b=6;for(n=1,2e9,issquare(n^2\b) & print1(n^2,",")) \\ M. F. Hasler, Jan 15 2012

a(n) = A204518(n)^2. - M. F. Hasler, Jan 15 2012

Empirical g.f.: -x^2*(9*x^8+100*x^7+25*x^6-162*x^5-296*x^4-74*x^3+9*x^2+4*x+1) / ((x-1)*(x^2+x+1)*(x^6-98*x^3+1)). - Colin Barker, Sep 15 2014

More terms added and offset changed to 1 by M. F. Hasler, Jan 16 2012

A055872 a(n) and floor(a(n)/8) are both squares; i.e., squares that remain squares when written in base 8 and last digit is removed.

Original entry on oeis.org

0, 1, 4, 9, 36, 289, 1156, 9801, 39204, 332929, 1331716, 11309769, 45239076, 384199201, 1536796804, 13051463049, 52205852196, 443365544449, 1773462177796, 15061377048201, 60245508192804

Offset: 1

Henry Bottomley, Jul 14 2000

For the first 3 terms which have only 1 digit in base 8, removing this digit is meant to yield 0.

Base-8 analog of A055792 (base 2), A055793 (base 3), A055808 (base 4), A055812 (base 5), A055851 (base 6), A055859 (base 7), A204503 (base 9) and A023110 (base 10). - M. F. Hasler, Jan 15 2012

			a(5) = 289 because 289 = 17^2 = 441 base 8 and 44 base 8 = 36 = 6^2.

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012.
Index to sequences related to truncating digits of squares.
Index entries for linear recurrences with constant coefficients, signature (0,35,0,-35,0,1)

Cf. A023110, A055792 (bisection).

Select[Range[0,8*10^6]^2,IntegerQ[Sqrt[FromDigits[Most[ IntegerDigits[ #,8]], 8]]]&] (* Harvey P. Dale, Aug 02 2016 *)

b=8;for(n=1,200,issquare(n^2\b) && print1(n^2,",")) \\ M. F. Hasler, Jan 15 2012

a(n) = A204514(n)^2. - M. F. Hasler, Jan 15 2012

Empirical g.f.: -x^2*(4*x+1)*(9*x^4-26*x^2+1) / ((x-1)*(x+1)*(x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Sep 15 2014

More terms added and offset changed to 1 by M. F. Hasler, Jan 15 2012

A204517 Square root of floor[A055859(n)/7].

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 17, 48, 96, 271, 765, 1530, 4319, 12192, 24384, 68833, 194307, 388614, 1097009, 3096720, 6193440, 17483311, 49353213, 98706426, 278635967, 786554688, 1573109376, 4440692161, 12535521795, 25071043590, 70772438609, 199781794032, 399563588064

Offset: 1

M. F. Hasler, Jan 15 2012

nonn

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012
Index to sequences related to truncating digits of squares.

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=7;for(n=1,2e9,issquare(n^2\b) & print1(sqrtint(n^2\b),","))

A204517(n)=polcoeff((x^4 + 3*x^5 + 6*x^6 + x^7)/(1 - 16*x^3 + x^6+O(x^n)),n)

A204517(n) = sqrt(floor(A204516(n)^2/7)).

G.f. = (x^4 + 3*x^5 + 6*x^6 + x^7)/(1 - 16*x^3 + x^6)

A204512 Square roots of [A055872/8]: Their square written in base 8, with some digit appended, is again a square.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 12, 35, 70, 204, 408, 1189, 2378, 6930, 13860, 40391, 80782, 235416, 470832, 1372105, 2744210, 7997214, 15994428, 46611179, 93222358, 271669860, 543339720, 1583407981, 3166815962, 9228778026, 18457556052, 53789260175, 107578520350

Offset: 1

M. F. Hasler, Jan 15 2012

Base-8 analog of A031150. The square of the terms (= truncated squares A055872) are listed in A204504.

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).

CoefficientList[Series[(x^4 (1+2x))/(1-6x^2+x^4),{x,0,40}],x] (* Harvey P. Dale, Nov 30 2020 *)

b=8;for(n=1,1e7,issquare(n^2\b) & print1(sqrtint(n^2\b)","))

a(n)=polcoeff((2*x^5 + x^4)/(x^4 - 6*x^2 + 1+O(x^n)),n)

G.f. = x^4*(1 + 2*x)/(1 - 6*x^2 + x^4)

cp's OEIS Frontend

A028310 Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x.

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A023110 Squares which remain squares when the last digit is removed.

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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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A055812 a(n) and floor(a(n)/5) are both squares; i.e., squares which remain squares when written in base 5 and last digit is removed.

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A204502 Numbers such that floor[a(n)^2 / 9] is a square.

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A204514 Numbers such that floor(a(n)^2 / 8) is again a square.

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Maple

Mathematica