A004275 -id:A004275 - cp's OEIS frontend

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

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

M. F. Hasler, Jan 16 2012

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

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

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

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

A204574 Numbers such that floor[a(n)^2/2] is a square (A001541), written in binary.

Original entry on oeis.org

0, 1, 11, 10001, 1100011, 1001000001, 110100100011, 100110010010001, 11011111001000011, 10100010100100000001, 1110110011011111000011, 1010110010010010110010001, 111110110111010100110100011, 101101110011001101010001000001

Offset: 1

M. F. Hasler, Jan 16 2012

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).

A373744 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)(1+sqrt(1-4x))), (1-2x-sqrt(1-4x))/(2*x) ).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 9, 13, 6, 1, 1, 23, 41, 26, 8, 1, 1, 65, 131, 101, 43, 10, 1, 1, 197, 428, 376, 197, 64, 12, 1, 1, 626, 1429, 1377, 834, 337, 89, 14, 1, 1, 2056, 4861, 5017, 3382, 1597, 529, 118, 16, 1, 1, 6918, 16795, 18277, 13378, 7105, 2773, 781, 151, 18, 1

Offset: 0

Stefano Spezia, Jun 16 2024

			The triangle begins as:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,   9,  13,   6,   1;
  1,  23,  41,  26,   8,  1;
  1,  65, 131, 101,  43, 10,  1;
  1, 197, 428, 376, 197, 64, 12, 1;
  ...

Tian-Xiao He and Roksana Słowik, Total Positivity of Almost-Riordan Arrays, arXiv:2406.03774 [math.CO], 2024. See pp. 16-17.

Cf. A000012 (k=0 and n=k), A001453 (k=2), A004275 (subdiagonal), A014137, A091823, A143955 (k=3).

T[n_, 0]:=1; T[n_, k_]:=SeriesCoefficient[2/((1-x)(1+Sqrt[1-4x]))((1-2x-Sqrt[1-4x])/(2x))^(k-1), {x, 0, n-1}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

T(n,1) = A014137(n-1).

T(n,n-2) = A091823(n-1) for n > 2.

cp's OEIS Frontend

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

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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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A204574 Numbers such that floor[a(n)^2/2] is a square (A001541), written in binary.

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A373744 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)(1+sqrt(1-4x))), (1-2x-sqrt(1-4x))/(2*x) ).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A204574 Numbers such that floor[a(n)^2/2] is a square (A001541), written in binary.

Views

Author

Keywords

Crossrefs

A373744 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+sqrt(1-4*x))), (1-2*x-sqrt(1-4*x))/(2*x) ).

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A373744 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)(1+sqrt(1-4x))), (1-2x-sqrt(1-4x))/(2*x) ).