A004275 -id:A004275 - cp's OEIS frontend

A207938 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 0 1 and 1 1 0 vertically.

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 14, 81, 98, 64, 10, 22, 196, 271, 200, 100, 12, 35, 484, 844, 643, 350, 144, 14, 56, 1225, 2706, 2422, 1271, 556, 196, 16, 90, 3136, 8977, 9430, 5594, 2239, 826, 256, 18, 145, 8100, 30168, 38207, 25490, 11256, 3641, 1168, 324, 20, 234

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2...4....6....9....14.....22......35.......56.......90.......145........234
..4..16...36...81...196....484....1225.....3136.....8100.....21025......54756
..6..36...98..271...844...2706....8977....30168...102384....349069....1193648
..8..64..200..643..2422...9430...38207...156792...649758...2703377...11276024
.10.100..350.1271..5594..25490..121313...584386..2841676..13864995...67793828
.12.144..556.2239.11256..58602..319439..1760946..9794226..54631117..305277128
.14.196..826.3641.20568.120276..737575..4570122.28555126.178852957.1121957980
.16.256.1168.5581.34986.226850.1544037.10609482.73474400.509887759.3543126698

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

R. H. Hardin, Table of n, a(n) for n = 1..546

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A207106
Column 4 is A207107
Row 1 is A001611(n+2)
Row 2 is A207436

A208142 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 12, 81, 108, 64, 10, 16, 144, 324, 240, 100, 12, 20, 256, 720, 900, 450, 144, 14, 25, 400, 1600, 2400, 2025, 756, 196, 16, 30, 625, 3000, 6400, 6300, 3969, 1176, 256, 18, 36, 900, 5625, 14000, 19600, 14112, 7056, 1728, 324, 20, 42, 1296

Offset: 1

R. H. Hardin Feb 23 2012

Table starts
..2...4....6.....9....12.....16.....20......25......30.......36.......42
..4..16...36....81...144....256....400.....625.....900.....1296.....1764
..6..36..108...324...720...1600...3000....5625....9450....15876....24696
..8..64..240...900..2400...6400..14000...30625...58800...112896...197568
.10.100..450..2025..6300..19600..49000..122500..264600...571536..1111320
.12.144..756..3969.14112..50176.141120..396900..952560..2286144..4889808
.14.196.1176..7056.28224.112896.352800.1102500.2910600..7683984.17929296
.16.256.1728.11664.51840.230400.792000.2722500.7840800.22581504.57081024

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

R. H. Hardin, Table of n, a(n) for n = 1..4508

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A202195(n-2)
Row 1 is A002620(n+2)
Row 2 is A030179(n+2)
Row 3 is A202093(n-2)

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 3*n^3 + 3*n^2
k=4: a(n) = (9/4)*n^4 + (9/2)*n^3 + (9/4)*n^2
k=5: a(n) = n^5 + 4*n^4 + 5*n^3 + 2*n^2
k=6: a(n) = (4/9)*n^6 + (8/3)*n^5 + (52/9)*n^4 + (16/3)*n^3 + (16/9)*n^2
k=7: a(n) = (5/36)*n^7 + (5/4)*n^6 + (155/36)*n^5 + (85/12)*n^4 + (50/9)*n^3 + (5/3)*n^2

A208379 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 8, 16, 100, 72, 64, 10, 26, 256, 240, 108, 100, 12, 42, 676, 704, 420, 144, 144, 14, 68, 1764, 2080, 1344, 640, 180, 196, 16, 110, 4624, 6216, 4212, 2176, 900, 216, 256, 18, 178, 12100, 18496, 13860, 7072, 3200, 1200, 252, 324, 20, 288

Offset: 1

R. H. Hardin Feb 25 2012

Table starts
..2...4...6...10...16....26....42.....68.....110.....178......288......466
..4..16..36..100..256...676..1764...4624...12100...31684....82944...217156
..6..36..72..240..704..2080..6216..18496...55000..163760...487296..1450192
..8..64.108..420.1344..4212.13860..44880..144540..468852..1517184..4906980
.10.100.144..640.2176..7072.25200..87040..296560.1028128..3545856.12198016
.12.144.180..900.3200.10660.40740.148240..526900.1931300..7015680.25336420
.14.196.216.1200.4416.14976.60984.231744..851400.3276624.12438144.46737936
.16.256.252.1540.5824.20020.86436.340816.1285900.5170900.20393856.79220932

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

R. H. Hardin, Table of n, a(n) for n = 1..1463

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A044102(n-1) for n>1
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207840

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 36*n - 36 for n>1
k=4: a(n) = 20*n^2 + 40*n - 60 for n>1
k=5: a(n) = 96*n^2 - 32*n - 64 for n>1
k=6: a(n) = 364*n^2 - 416*n + 52 for n>1
k=7: a(n) = 84*n^3 + 840*n^2 - 1344*n + 420 for n>1
Empirical for row n:
n=1: a(k)=a(k-1)+a(k-2)
n=2: a(k)=2*a(k-1)+2*a(k-2)-a(k-3)
n=3: a(k)=a(k-1)+4*a(k-2)+5*a(k-3)+2*a(k-4)-a(k-5)+a(k-6) for k>8
n=4: a(k)=a(k-1)+4*a(k-2)+9*a(k-3)+5*a(k-4)-2*a(k-5)+4*a(k-6) for k>8
n=5: a(k)=a(k-1)+4*a(k-2)+13*a(k-3)+8*a(k-4)-3*a(k-5)+9*a(k-6) for k>8
n=6: a(k)=a(k-1)+4*a(k-2)+17*a(k-3)+11*a(k-4)-4*a(k-5)+16*a(k-6) for k>8
n=7: a(k)=a(k-1)+4*a(k-2)+21*a(k-3)+14*a(k-4)-5*a(k-5)+25*a(k-6) for k>8

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.

Original entry on oeis.org

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)

A300576 Number of nights required in the worst case to find the princess in a castle with n rooms arranged in a line (Castle and princess puzzle).

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122

Offset: 1

Dmitry Kamenetsky, Mar 09 2018

This is a logic puzzle. There is a castle with n rooms arranged in a line. The princess living in the castle sleeps in a different room each night, but always one adjacent to the one in which she slept on the previous night. She is free to choose any room in which to sleep on the first night. A prince would like to find the princess, but she will not tell him where she is going to sleep each night. Each night the prince can look in a single room. What strategy should he follow in order to guarantee that he finds the princess in a minimum number of nights?
The strategy to find the princess guaranteed within a(n) nights takes on average k(n) nights until the princess is found with lim_{n->oo} k(n) = n-1.5. For n>4, strategies with lower average numbers of trials exist; A386462 provides this strategy for n=8. See there for more information. - Ruediger Jehn, Aug 05 2025
Christian Perfect (see link) considered the case when the rooms are arranged as a general graph. He showed that the set of solvable graphs is exactly the set of trees not containing the "threesy" subgraph, which is A130131. He also showed that for d-level binary trees with 1 <= d <= 4 the number of required nights is 1, 2, 6, 18. Binary trees with d >= 5 are unsolvable as they contain "threesy".

			For n = 1, there is only one room to search, so a(1) = 1.
For n = 2, the prince searches room 1 on the first night. If the princess is not there that means she was in room 2. If the prince searches room 1 again then he is guaranteed to see the princess as she has to move from room 2 to room 1 (she cannot stay in the same room). So a(2) = 2.
For n = 3, the prince searches room 2 on the first night. If the princess is not there that means she was either in room 1 or 3. On the second night she must go to room 2 and this is where the prince will find her. So a(3) = 2.
For n = 4, a solution that guarantees to find the princess in a(4)=4 nights is to search rooms (2,3,3,2).
For n = 5, a solution that guarantees to find the princess in a(5)=6 nights is to search rooms (2,3,4,4,3,2).
In the general case for n >= 3, a solution guaranteeing success in the minimum number of nights is to search rooms (2,3,...,n-1,n-1,...,3,2), so a(n) = 2*n - 4.

Christian Perfect, Solving the princess on a graph puzzle.
Puzzling Stack Exchange, Why does this solution guarantee that the prince knocks on the right door to find the princess?.
standupmaths, Youtube, The Castle and the Princess Puzzle.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Essentially the same as A005843, A004277 and A004275.

Cf. A130131, A130132, A331693, A386462.

CoefficientList[ Series[(2x^3 - x^2 + 1)/(x - 1)^2, {x, 0, 62}], x] (* Robert G. Wilson v, Mar 12 2018 *)
Join[{1,2},Range[2,200,2]] (* Harvey P. Dale, Jan 25 2019 *)

For n >= 3, a(n) = 2*n - 4.
From Chai Wah Wu, Apr 14 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 4.
G.f.: x*(2*x^3 - x^2 + 1)/(x - 1)^2. (End)
E.g.f.: 4 + 2*exp(x)*(x - 2) + 3*x + x^2. - Stefano Spezia, Aug 15 2025

A251603 Numbers k such that k + 2 divides k^k - 2.

Original entry on oeis.org

3, 4551, 46775, 82503, 106976, 1642796, 4290771, 4492203, 4976427, 21537831, 21549347, 21879936, 51127259, 56786087, 60296571, 80837771, 87761787, 94424463, 96593696, 138644871, 168864999, 221395539, 255881451, 297460451, 305198247, 360306363, 562654203

Offset: 1

Juri-Stepan Gerasimov, Dec 05 2014

nonn

Numbers k such that (k^k - 2)/(k + 2) is an integer.
Since k == -2 (mod k+2), also numbers k such that k + 2 divides (-2)^k - 2. - Robert Israel, Jan 04 2015
Numbers k == 0 (mod 4) such that A066602(k/2+1) = 8, and odd numbers k such that k = 3 or A082493(k+2) = 8. - Robert Israel, Apr 08 2015

			3 is in this sequence because 3 + 2 = 5 divides 3^3 - 2 = 25.

Max Alekseyev, Table of n, a(n) for n = 1..890 (all terms below 10^15)

Cf. A001477, A004273, A004275, A066602, A082493, A081765, A213382, A252606, A357125.

[n: n in [0..10000] | Denominator((n^n-2)/(n+2)) eq 1];

isA251603 := proc(n)
    if modp(n &^ n-2,n+2) = 0 then
        true;
    else
        false;
    end if;
end proc:
A251603 := proc(n)
    option remember;
    local a;
    if n = 1 then
        3;
    else
        for a from procname(n-1)+1 do
            if isA251603(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jan 09 2015

Select[Range[10^6], Mod[PowerMod[#, #, # + 2] - 2, # + 2] == 0 &] (* Michael De Vlieger, Dec 20 2014, based on Robert G. Wilson v at A252041 *)

for(n=1,10^9,if(Mod(n,n+2)^n==+2,print1(n,", "))); \\ Joerg Arndt, Dec 06 2014

A251603_list = [n for n in range(1,10**6) if pow(n, n, n+2) == 2] # Chai Wah Wu, Apr 13 2015

The even terms form A122711, the odd terms are those in A245319 (forming A357125) decreased by 2. - Max Alekseyev, Sep 22 2016

a(6)-a(27) from Joerg Arndt, Dec 06 2014

A084100 Expansion of (1+x-x^2-x^3)/(1+x^2).

Original entry on oeis.org

1, 1, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2

Offset: 0

Paul Barry, May 15 2003

Partial sums are A084099.

The unsigned sequence 1,1,2,2,2,2,.. has g.f. (1+x^2)/(1-x) and a(n)=sum{k=0..n, binomial(1,k/2)(1+(-1)^k)/2}. Its partial sums are A004275(n+1). The sequence 1,-1,2,-2,2,-2,... has g.f. (1+x^2)/(1+x) and a(n)=sum{k=0..n, (-1)^(n-k)binomial(1,k/2)(1+(-1)^k)/2}. - Paul Barry, Oct 15 2004

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

Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A084099, A280560.

CoefficientList[Series[(1+x-x^2-x^3)/(1+x^2),{x,0,100}],x]  (* Harvey P. Dale, Apr 20 2011 *)
a[ n_] := (-1)^Quotient[n, 2] If[ Quotient[n, 2] != 0, 2, 1]; (* Michael Somos, Jan 05 2017 *)

{a(n) = (-1)^(n\2) * if( n\2, 2, 1)}; /* Michael Somos, Jan 05 2017 */

Euler transform of length 4 sequence [1, -3, 0, 1]. - Michael Somos, Jan 05 2017
G.f.: (1 + x) * (1 - x^2) / (1 + x^2). - Michael Somos, Jan 05 2017
a(n) = a(1-n) for all n in Z. - Michael Somos, Jan 05 2017
a(2*n) = a(2*n + 1) = A280560(n) for all n in Z. - Michael Somos, Jan 05 2017

A252041 Numbers m such that m - 3 divides m^m + 3.

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 85, 105, 136, 186, 262, 820, 1161, 2626, 2926, 4924, 10396, 11656, 19689, 27637, 33736, 36046, 42886, 42901, 53866, 55189, 82741, 95266, 103762, 106822, 127401, 135460, 251506, 366796, 375220, 413326, 466966, 531445, 553456, 568876

Offset: 1

Juri-Stepan Gerasimov, Dec 12 2014

nonn

Numbers m such that (m^m + 3)/(m - 3) is an integer.
Most but not all terms are congruent to 4 modulo 6. - Robert G. Wilson v, Dec 19 2014
Note that m^m == 3^m (mod m-3). - Robert Israel, Dec 19 2014

			2 is in this sequence because (2^2 + 3)/(2 - 3) = -7 is an integer.
4 is in this sequence because (4^4 + 3)/(4 - 3) = 259 is an integer.
7 is not in the sequence because (7^7 + 3)/4 = 411773/2, which is not an integer.

Robert G. Wilson v, Table of n, a(n) for n = 1..331

Cf. ...............Numbers n such that x divides y, where:
...x......y....k = 0.....k = 1.....k = 2......k = 3.......
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751..this sequence..
..n+k..n^n-k..A000027...A004275...A251603....A251862......
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Dec 28 2014)

[n: n in [4..50000] | Denominator((n^n+3)/(n-3)) eq 1];

select(t -> 3 &^t + 3 mod (t-3) = 0, [1,2,$4..10^6]); # Robert Israel, Dec 19 2014

fQ[n_] := Mod[PowerMod[n, n, n - 3] + 3, n - 3] == 0; Select[Range@ 1000000, fQ] (* Michael De Vlieger, Dec 13 2014; modified by Robert G. Wilson v, Dec 19 2014 *)

isok(n) = (n != 3) && (Mod(n, n-3)^n  == -3); \\ Michel Marcus, Dec 13 2014

More terms from Michel Marcus, Dec 13 2014

A103829 Sum of even digits less than sum of odd digits.

Original entry on oeis.org

0, 2, 4, 6, 8, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 34, 36, 38, 40, 41, 42, 43, 44, 46, 48, 56, 58, 60, 61, 62, 63, 64, 65, 66, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 102, 104, 106, 108, 114, 116, 118, 120, 122, 124, 126, 128, 136, 138, 140, 141, 142, 144, 146

Offset: 1

Zak Seidov, Feb 17 2005

0 is assumed as even digit: A005843, A004275, A007928. Sum of even digits equals sum of odd digits A036301.

Cf. A005843, A004275, A007928, A036301.

Select[Range[300], Plus@@Select[IntegerDigits[ # ], OddQ]

cp's OEIS Frontend

A207938 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 0 1 and 1 1 0 vertically.

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A208142 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

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A208379 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

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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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Mathematica

PARI

PARI

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A300576 Number of nights required in the worst case to find the princess in a castle with n rooms arranged in a line (Castle and princess puzzle).

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A251603 Numbers k such that k + 2 divides k^k - 2.

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Magma

Maple

Mathematica

PARI

Python

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A084100 Expansion of (1+x-x^2-x^3)/(1+x^2).

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