A272259 -id:A272259 - cp's OEIS frontend

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

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

T. D. Noe, Dec 01 2003

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.

			See A071983.

Zhao Hui Du, Table of n, a(n) for n = 15..59

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).

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

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

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

A112663 Smallest circular sequence of period 32 such that any two adjacent numbers sum to a square number.

Original entry on oeis.org

1, 8, 28, 21, 4, 32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 31, 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15, 1, 8, 28, 21, 4, 32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 31, 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15, 1, 8, 28, 21, 4, 32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 31, 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15

Offset: 0

Federico Ramondino, Dec 29 2005

nonn

The terms of this sequence are given in A071984. An algorithm for computing circular chains of squares is given in A090460. - T. D. Noe, Dec 30 2005

			1+8=9
8+28=36
28+21=49
...
26+10=36
10+15=25
15+1=16

Cf. A000217, A000290.

Cf. A272259 (has terms a(0..31) in row 32).

apply( {A112663(n)=my(r=1);foreach(digits(403079653644429064719159, 6)[1..n%32],s,r=(s+2)^2-r); r}, [0..77]) \\ M. F. Hasler, Jun 23 2025

a(n) = A272259(32, (n-1) mod 32) for all n, where "mod" is the (nonnegative) remainder operator. - M. F. Hasler, Jun 23 2025

A336749 Number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 1 or 4.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 3, 6, 5, 10, 12, 14, 25, 27, 40, 57, 68, 104, 133, 177, 255, 324, 454, 617, 811, 1136, 1507, 2042, 2803, 3729, 5109, 6904, 9290, 12692, 17070, 23152, 31430, 42361, 57567, 77842, 105279, 142865, 193040, 261589, 354316, 479189, 649498, 878905

Offset: 5

Ethan Patrick White, Aug 02 2020

Permutations in which adjacent terms sum to a particular value is a property central to the sequences A090460, A071984, A108658, A272259, and A107929.

Ethan P. White, Richard K. Guy, Renate Scheidler, Difference Necklaces, arXiv:2006.15250 [math.CO], 2020. See Table A.1 p. 31.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1,2,2,1,1,1).

See A079977 or A017899 for other sequences counting similar circular arrangements of positive integers.

CoefficientList[ Series[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1 + x - x^3 - x^4 - 2*x^5 - 2*x^6 - x^7 - x^8 - x^9), {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 07 2020 *)

a(n) = -a(n-1) + a(n-3) + a(n-4) + 2*a(n-5) + 2*a(n-6) + a(n-7) + a(n-8) + a(n-9) for n > 13.

G.f.: x^5*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1 + x - x^3 - x^4 - 2*x^5 - 2*x^6 - x^7 - x^8 - x^9). - Stefano Spezia, Aug 03 2020

cp's OEIS Frontend

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

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A112663 Smallest circular sequence of period 32 such that any two adjacent numbers sum to a square number.

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A336749 Number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 1 or 4.

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