A249577 -id:A249577 - cp's OEIS frontend

A249576 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

0, 1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 10, 12, 17, 24, 29, 41, 58, 70, 99, 140, 169, 239, 338, 408, 577, 816, 985, 1393, 1970, 2378, 3363, 4756, 5741, 8119, 11482, 13860, 19601, 27720, 33461, 47321, 66922, 80782, 114243, 161564, 195025, 275807, 390050, 470832, 665857, 941664

Offset: 0

Russell Walsmith, Nov 01 2014

Numbers to the left of a(0) are in A249577.
Some identities:
a(3n - 2) + a(3n - 1) = a(3n + 1).
a(3n) + a(3n + 1) = a(3(n + 1)).
a(3n - 2) + a(3n + 1) = a(3n + 2).
a(3n) + a(3n - 1) + a(3(n - 2)) = a(3n + 1).
a(3n - 1)a(3n) + a(3n + 2)a(3(n + 1)) = a(6n + 2).

			M^0 = the 3 X 3 identity matrix = [[1,0,0][0,1,0][0,0,1]]. M[3,1] = 0; M[1,1] = 1; M[1,3] = 0. So the first triple is r = a(0) = 0; s = a(1) = 1; t = a(2) = 0.
M^1 = [[1,4,4][1,3,2][1,2,1]], so r = a(3) = 1; s = a(4) = 1; t = a(5) = 2.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,1).

a(3n) = the n-th term of A000129, the Pell numbers.
a(3n+1) = n-th term of A001333.
a(3n+2) = n-th term of A163271.

LinearRecurrence[{0,0,2,0,0,1},{0,1,0,1,1,2},60] (* Harvey P. Dale, Dec 29 2021 *)

concat(0, Vec(-x*(2*x^4-x^3+x^2+1)/(x^6+2*x^3-1) + O(x^100))) \\ Colin Barker, Nov 02 2014

a(n) = -2*a(n-3)+a(n-6); G.f.: -x*(2*x^4-x^3+x^2+1) / (x^6+2*x^3-1). - Colin Barker, Nov 02 2014

A259579 Number of distinct differences in row n of the reciprocity array of 2.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 4, 5, 4, 3, 6, 3, 4, 5, 6, 3, 6, 3, 6, 7, 6, 3, 10, 3, 6, 7, 8, 3, 12, 3, 8, 9, 6, 5, 12, 3, 6, 9, 10, 3, 12, 3, 10, 9, 6, 3, 16, 5, 8, 9, 10, 3, 10, 5, 10, 9, 6, 3, 20, 3, 6, 9, 10, 5, 14, 3, 10, 9, 12, 3, 16, 3, 6, 11, 10, 9, 14, 3, 14

Offset: 1

Clark Kimberling, Jul 17 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/n] where x is a real number and m and n are positive integers, is proved in Section 3.5 of Concrete Mathematics (see References). See A259572 for a guide to related sequences.

			In the array at A259578, row 6 is (2,5,6,10,12,15,17,20,21,25,27,...), with differences (3,1,4,2,3,2,3,1,4,2,...), and distinct differences {1,2,3,4}, so that a(6) = 4.

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

Cf. A249572, A249577, A259580.

x = 2;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]

cp's OEIS Frontend

A249576 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

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