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 1-3 of 3 results.

A090460 Number of essentially different permutations of the numbers 1 to n such that the sum of adjacent numbers is a square.

Original entry on oeis.org

1, 1, 1, 0, 0, 0, 0, 0, 3, 0, 10, 12, 35, 52, 19, 20, 349, 361, 637, 3678, 15237, 11875, 13306, 10964, 27223, 37054, 201408, 510152, 1995949, 4867214, 11255174, 35705858, 63029611, 129860749, 258247089, 190294696, 686125836, 2195910738, 5114909395, 9141343219, 19769529758, 44678128099, 63885400119
Offset: 15

Views

Author

T. D. Noe, Dec 01 2003

Keywords

Comments

For n > 31, some solutions are circular; that is, the first and last numbers also sum to a square. Note that A071983 counts each circular solution n times. This sequence counts each circular solution only once. The Mathematica program uses backtracking to find all solutions, which can be printed by removing the comment symbols.

Examples

			See A071983.

Links

Crossrefs

Cf. A071983, A071984 (number of circular solutions), A090461 (n for which there is a solution).
Cf. A078107 (n for which there is no solution).
Cf. A272259 (row n gives the smallest circular solution, for each n >= 32).

Programs

  • Mathematica
    SquareQ[n_] := IntegerQ[Sqrt[n]]; try[lev_] := Module[{t, j, circular}, If[lev>n, circular=SquareQ[soln[[1]]+soln[[n]]]; If[(!circular&&soln[[1]]

Formula

a(n) = A071983(n) - (n-1)*A071984(n).

Extensions

a(43)-a(45) from Donovan Johnson, Sep 14 2010
a(46)-a(47) from Jud McCranie, Aug 18 2018
a(48) from Jud McCranie, Sep 17 2018
a(49)-a(52) from Bert Dobbelaere, Dec 30 2018
a(47) corrected by Bert Dobbelaere, Jan 12 2019
a(53)-a(54) from Martin Ehrenstein, May 22 2023
a(55)-a(57) from Zhao Hui Du, Apr 26 2024

A071983 Square chains: the number of permutations (reversals not counted as different) of the numbers 1 to n such that the sum of any two consecutive numbers is a square.

Original entry on oeis.org

1, 1, 1, 0, 0, 0, 0, 0, 3, 0, 10, 12, 35, 52, 19, 20, 349, 392, 669, 4041, 17175, 12960, 14026, 11889, 29123, 39550, 219968, 553694, 2178103, 5301127, 12220138, 38838893, 68361609, 140571720, 280217025, 204853870, 738704986, 2368147377, 5511090791, 9802605881, 21164463050, 47746712739, 68092497615, 123092214818
Offset: 15

Views

Author

William Rex Marshall, Jun 16 2002

Keywords

Comments

For n > 31, this sequence counts each circular solution (in which the first and last numbers also sum to a square) n times. Sequence A090460 counts the circular solutions only once, giving the number of essentially different solutions.
The existence of cubic chains in answered affirmatively in Puzzle 311. - T. D. Noe, Jun 16 2005

Examples

			There is only one possible square chain of minimum length, which is: (8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9) so a(15)=1.

References

  • Ruemmler, Ronald E., "Square Loops," Journal of Recreational Mathematics 14:2 (1981-82), page 141; Solution by Chris Crandell and Lance Gay, JRM 15:2 (1982-83), page 155.

Links

Crossrefs

Cf. A071984.
Cf. A090460, A090461.
Cf. A078107 (n for which there is no solution).

Formula

a(n) = A090460(n) + (n-1)*A071984(n). - Martin Ehrenstein, May 16 2023

Extensions

a(43)-a(45) from Donovan Johnson, Sep 14 2010
a(46)-a(47) from Jud McCranie, Aug 18 2018
a(48) from Jud McCranie, Sep 17 2018
a(49)-a(52) from Bert Dobbelaere, Dec 30 2018
a(53)-a(54) from Martin Ehrenstein, May 16 2023
a(55)-a(56) from Zhao Hui Du, Apr 25 2024
a(57)-a(58) from Zhao Hui Du, Apr 26 2024

A090461 Numbers k for which there exists a permutation of the numbers 1 to k such that the sum of adjacent numbers is a square.

Original entry on oeis.org

15, 16, 17, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89
Offset: 1

Views

Author

T. D. Noe, Dec 01 2003

Keywords

Comments

Conjecture: sequence includes all integers k > 24. See A090460 for the number of essentially different solutions.
It is now known that 25..299 are in the sequence, see the Numberphile 2 link. - Jud McCranie, Jan 11 2018
Every 25 <= k <= 2^20 is in the sequence and (71*25^m-1)/2 is also in the sequence for every m, hence this sequence is infinite, see Mersenneforum link for the proof; we give Hamiltonian cycle for these k values if k >= 32. - Robert Gerbicz, Jan 17 2017
The conjecture has been proved: every k >= 25 is in the sequence, moreover for k >= 32 there is a Hamiltonian cycle; see Mersenneforum topic for a code and deterministic algorithm to find a sequence. - Robert Gerbicz, Jan 21 2018

Examples

			See A071983.

Links

Crossrefs

Cf. A071983, A071984 (number of circular solutions), A090460.
Cf. A078107 (k for which there is no solution).

Programs

  • Maple
    F:= proc(n)
uses GraphTheory;
local edg, G;
edg:= select(t -> issqr(t[1]+t[2]),{seq(seq({i,j},i=1..j-1),j=1..n)}) union {seq({i,n+1},i=1..n)};
G:= Graph(n+1,edg);
IsHamiltonian(G)
end proc:
select(F, [$1..50]); # Robert Israel, Jun 05 2015
  • Mathematica
    Join[{15, 16, 17, 23}, Range[25, 100]] (* Paolo Xausa, May 28 2024 *)

Extensions

a(31)-a(69) from Donovan Johnson, Sep 14 2010
Showing 1-3 of 3 results.

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