A204502 -id:A204502 - cp's OEIS frontend

A220032 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal or diagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..1 nXk array.

2, 2, 3, 3, 4, 4, 4, 6, 6, 5, 5, 9, 10, 8, 6, 6, 12, 19, 15, 10, 7, 7, 15, 30, 34, 21, 12, 8, 8, 18, 42, 61, 55, 28, 14, 9, 9, 21, 55, 95, 111, 83, 36, 16, 10, 10, 24, 69, 137, 192, 187, 119, 45, 18, 11, 11, 27, 84, 187, 302, 358, 297, 164, 55, 20, 12, 12, 30, 100, 246, 442, 613, 626

Offset: 1

R. H. Hardin Dec 03 2012

Table starts
..2..2..3...4....5....6.....7.....8.....9....10....11....12....13....14...15
..3..4..6...9...12...15....18....21....24....27....30....33....36....39...42
..4..6.10..19...30...42....55....69....84...100...117...135...154...174..195
..5..8.15..34...61...95...137...187...246...315...395...487...592...711..845
..6.10.21..55..111..192...302...442...618...838..1111..1447..1857..2353.2948
..7.12.28..83..187..358...613...962..1426..2034..2823..3839..5137..6782.8850
..8.14.36.119..297..626..1165..1963..3088..4630..6711..9492.13175.18010
..9.16.45.164..450.1038..2094..3789..6334..9995.15133.22239.31956
.10.18.55.219..656.1646..3587..6962.12375.20581.32588
.11.20.66.285..926.2513..5893.12243.23132.40583
.12.22.78.363.1272.3714..9335.20705.41537
.13.24.91.454.1707.5337.14323.33819

			Some solutions for n=3 k=4
..0..0..0..0....1..0..0..0....1..0..0..0....0..0..0..0....1..0..0..0
..0..0..0..0....1..1..0..0....1..0..0..0....1..1..0..0....1..1..1..0
..1..1..0..0....1..1..0..0....1..0..0..0....1..1..1..1....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..198

Column 1 is A000027(n+1)
Column 2 is A004275(n+1)
Column 3 is A000217(n+1)
Column 4 is A062748 for n>1
Row 1 is A000027
Row 2 is A204502(n+3)

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)

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

M. F. Hasler, Jan 15 2012

Base-8 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=8;for(n=1,2e9,issquare(n^2\b) & print1((n^2\b)","))

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)

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)

A204575 Squares such that [a(n)/2] is again a square (A055792), written in binary.

Original entry on oeis.org

0, 1, 1001, 100100001, 10011001001001, 1010001010010000001, 101011001001001011001001, 10110111001100110101000100001, 1100001001111011011000010110001001, 110011100111010101001010101001000000001

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).

A204576 Floor[A055792(n-1)/2]=A084703(n-2) (truncated squares), written in binary.

Original entry on oeis.org

0, 0, 100, 10010000, 1001100100100, 101000101001000000, 10101100100100101100100, 1011011100110011010100010000, 110000100111101101100001011000100, 11001110011101010100101010100100000000, 1101101100101100000000000111010111101000100

Offset: 1

M. F. Hasler, Jan 16 2012

A204575 with the last (binary) digit (necessarily = 1, except for a(1)=0) deleted.

Also: Squares, written in binary, such that appending a (binary) digit (necessarily = 1) yields another square (except for a(1)=0 which corresponds to A204575(1)=00, the only square which remains square when multiplied by 2).

David W. Wilson, Table of n, a(n) for n = 1..100

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).

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

M. F. Hasler, Jan 15 2012

Base-7 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=7;for(n=0,200,issquare(n^2\b) & print1(n^2\b,","))

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)

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)

A204577 Sqrt(floor[A204575(n)/2]), written in binary.

Original entry on oeis.org

0, 0, 10, 1100, 1000110, 110011000, 100101001010, 11011000100100, 10011101110001110, 1110010111100110000, 1010011101111110010010, 111101000000111000111100, 101100011100111010111010110

Offset: 1

M. F. Hasler, Jan 16 2012

D. W. Wilson, Table of n, a(n) for n = 1..100

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).

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

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).

cp's OEIS Frontend

A220032 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal or diagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..1 nXk array.

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

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A204575 Squares such that [a(n)/2] is again a square (A055792), written in binary.

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A204576 Floor[A055792(n-1)/2]=A084703(n-2) (truncated squares), written in binary.

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

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A204577 Sqrt(floor[A204575(n)/2]), written in binary.

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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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