A023110 -id:A023110 - cp's OEIS frontend

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.

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

A055859 a(n) and floor(a(n)/7) are both squares; i.e., squares which remain squares when written in base 7 and last digit is removed.

Original entry on oeis.org

0, 1, 4, 9, 64, 256, 2025, 16129, 64516, 514089, 4096576, 16386304, 130576329, 1040514049, 4162056196, 33165873225, 264286471744, 1057145886976, 8424001222569, 67127723308801, 268510893235204, 2139663144659049, 17050177433963584, 68200709735854336

Offset: 1

Henry Bottomley, Jul 14 2000

Square roots of a(n) are listed in A204516, square roots of floor(a(n)/7) in A204517. - M. F. Hasler, Jan 16 2012

			a(5) = 256 because 256 = 16^2 = 514 base 7 and 51 base 7 = 36 = 6^2.

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

Cf. A023110.

a(n) = A204516(n)^2. - M. F. Hasler, Jan 16 2012

Empirical g.f.: -x^2*(9*x^8+256*x^7+64*x^6-270*x^5-764*x^4-191*x^3+9*x^2+4*x+1) / ((x-1)*(x^2+x+1)*(x^6-254*x^3+1)). - Colin Barker, Sep 15 2014

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

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

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)

A023111 Squares that remain square when the digit 1 is appended.

Original entry on oeis.org

0, 36, 51984, 74960964, 108093658176, 155870980128900, 224765845252215696, 324112192982714904804, 467369557515229640511744, 673946577824768158903030116, 971830497853758169908528915600, 1401378903958541456239939793265156, 2020787407677718926139823273359439424

Offset: 1

David W. Wilson

The terms of the sequence are the squares of the y-values in the solution to the Pellian equation x^2-10*y^2=1. - Colin Barker, Sep 28 2013

After 0, the sequence lists the numbers k for which A055437(k) is a perfect square. - Bruno Berselli, Jan 16 2018

			36 is a term because both 36 and 361 are squares.

Colin Barker, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Cf. A023110.

LinearRecurrence[{1443,-1443,1},{0,36,51984},20] (* Harvey P. Dale, Dec 23 2013 *)

concat(0, Vec(36*x^2*(1 + x) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^15))) \\ Colin Barker, Dec 29 2017

G.f.: 36*x^2*(1 + x) / ((1 - x)*(1 - 1442*x + x^2)). - Colin Barker, Jan 31 2013
a(0)=0, a(1)=36, a(2)=51984, a(n) = 1443*a(n-1)-1443*a(n-2)+a(n-3). - Harvey P. Dale, Dec 23 2013
a(n) = (721 + 228*sqrt(10))^(-n)*(721+228*sqrt(10) - 2*(721+228*sqrt(10))^n + (721-228*sqrt(10))*(721+228*sqrt(10))^(2*n)) / 40. - Colin Barker, Dec 29 2017

A030686 Smallest nontrivial extension of n^2 which is a square.

Original entry on oeis.org

16, 49, 900, 169, 256, 361, 4900, 6400, 8100, 10000, 12100, 1444, 16900, 19600, 22500, 25600, 28900, 3249, 36100, 40000, 44100, 48400, 52900, 57600, 62500, 67600, 72900, 78400, 84100, 90000, 96100, 102400, 108900, 115600, 122500

Offset: 1

Patrick De Geest

Nontrivial extension means appending at least one digit even if the number is already a square.

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

Cf. A030687, A023110, A031149, A031150, A202303.

See also A023110 = A031149^2 and A202303 = A031150^2 for a related concept, and cross-references there (and in links) for the analog in bases other than 10. - M. F. Hasler, Sep 28 2014

a(n) = A030687(n)^2. - M. F. Hasler, Sep 28 2014

a(n) = A030666(n^2). - Alonso del Arte, Apr 01 2020

A204519 Square root of floor(A055851(n)/6).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 11, 20, 40, 109, 198, 396, 1079, 1960, 3920, 10681, 19402, 38804, 105731, 192060, 384120, 1046629, 1901198, 3802396, 10360559, 18819920, 37639840, 102558961, 186298002

Offset: 1

M. F. Hasler, Jan 15 2012

Base-6 analog of A031150 [base 10], A204512 [base 8], A204517 (base 7), A204521 [base 5], A001353 [base 3], A001542 [base 2]. For bases 4 and 9, the corresponding sequence contains all integers.

Cf. A055792, A055793, A055812, A204517, A204512, A204503 and A023110.

Sqrt[Floor[Select[Range[100000],IntegerQ[Sqrt[Quotient[#^2,6]]]&]^2/6]] (* Vaclav Kotesovec, Nov 26 2012 *)

b=6;for(n=1,2e9,issquare(n^2\b) & print1(sqrtint(n^2\b),","))

Conjecture (for n>=8): a(n) = 10*a(n-3) - a(n-6). - Vaclav Kotesovec, Nov 26 2012

Empirical g.f.: x^4*(x^3+4*x^2+2*x+1) / (x^6-10*x^3+1). - Colin Barker, Sep 15 2014

More terms from Vaclav Kotesovec, Nov 26 2012

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

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.

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A055859 a(n) and floor(a(n)/7) are both squares; i.e., squares which remain squares when written in base 7 and last digit is removed.

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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.

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

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

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

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A023111 Squares that remain square when the digit 1 is appended.

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