cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Patrick De Geest

Keywords

Comments

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

Examples

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

References

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

Links

Crossrefs

Cf. A023110, A030686, A030687, A053784.
See A202303 for the resulting squares.

Programs

  • Maple
    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;
  • Mathematica
    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 *)

Formula

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

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

Views

Author

Patrick De Geest

Keywords

Comments

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

Links

Crossrefs

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

Formula

a(n) = A030687(n)^2. - M. F. Hasler, Sep 28 2014
a(n) = A030666(n^2). - Alonso del Arte, Apr 01 2020

A204504 A204512(n)^2 = floor[A055872(n)/8]: Squares such that appending some digit in base 8 yields another square.

Original entry on oeis.org

0, 0, 0, 1, 4, 36, 144, 1225, 4900, 41616, 166464, 1413721, 5654884, 48024900, 192099600, 1631432881, 6525731524, 55420693056, 221682772224, 1882672131025, 7530688524100, 63955431761796, 255821727047184, 2172602007770041, 8690408031080164, 73804512832419600
Offset: 1

Views

Author

M. F. Hasler, Jan 15 2012

Keywords

Comments

Base-8 analog of A202303.

Crossrefs

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).

Programs

  • PARI
    b=8;for(n=1,2e9,issquare(n^2\b) & print1((n^2\b)","))
  • PARI
    a(n)=polcoeff(x^4*(1 + 4*x + x^2 + 4*x^3)/(1 - 35*x^2 + 35*x^4 - x^6+O(x^n)), n)

Formula

a(n)=A204512(n)^2.
G.f. = x^4*(1 + 4*x + x^2 + 4*x^3)/(1 - 35*x^2 + 35*x^4 - x^6)

A204513 A204517(n)^2 = floor[A055859(n)/7]: Squares which written in base 7, with some digit appended, yield another square.

Original entry on oeis.org

0, 0, 0, 1, 9, 36, 289, 2304, 9216, 73441, 585225, 2340900, 18653761, 148644864, 594579456, 4737981889, 37755210249, 151020840996, 1203428746081, 9589674758400, 38358699033600, 305666163522721, 2435739633423369, 9742958533693476, 77638002106025089, 618668277214777344, 2474673108859109376, 19719746868766849921, 157139306672920022025, 628557226691680088100, 5008738066664673854881, 39912765226644470817024, 159651060906577883268096
Offset: 1

Views

Author

M. F. Hasler, Jan 15 2012

Keywords

Comments

Base-7 analog of A202303.

Crossrefs

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).

Programs

  • PARI
    b=7;for(n=0,200,issquare(n^2\b) & print1(n^2\b,","))
  • PARI
    A204513(n)=polcoeff((x^4 + 9*x^5 + 36*x^6 + 34*x^7 + 9*x^8 + 36*x^9 + x^10)/(1 - 255*x^3 + 255*x^6 - x^9+O(x^n)),n)

Formula

G.f. = (x^4 + 9*x^5 + 36*x^6 + 34*x^7 + 9*x^8 + 36*x^9 + x^10)/(1 - 255*x^3 + 255*x^6 - x^9)

A203719 A204521(n)^2 = floor[A055812(n)/5]: Squares which written in base 5, with some digit appended, yield another square.

Original entry on oeis.org

0, 0, 0, 1, 9, 16, 64, 441, 3025, 5184, 20736, 142129, 974169, 1669264, 6677056, 45765225, 313679521, 537497856, 2149991424, 14736260449, 101003831721, 173072640400, 692290561600, 4745030099481, 32522920134769, 55728852710976, 222915410843904
Offset: 1

Views

Author

M. F. Hasler, Jan 16 2012

Keywords

Comments

Base-5 analog of A202303.

Crossrefs

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).

Programs

  • PARI
    b=5;for(n=0,1e7,issquare(n^2\b) & print1(n^2\b,","))

Formula

Conjecture: a(n) = 323*a(n-4)-323*a(n-8)+a(n-12) for n>13. - Colin Barker, Sep 20 2014
Empirical g.f.: -x^4*(x^9 +9*x^8 +64*x^7 +16*x^6 +118*x^5 +118*x^4 +64*x^3 +16*x^2 +9*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 20 2014

Extensions

More terms from Colin Barker, Sep 20 2014

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Jan 16 2012

Keywords

Comments

Base-6 analog of A202303.

Crossrefs

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).

Programs

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

Formula

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
Showing 1-6 of 6 results.

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