A309376 a(n) appears in the congruences modulo 4 or 32 of Markoff numbers m(n) = A002559(n) for odd or even m(n).

0, 0, 1, 3, 7, 1, 22, 42, 6, 58, 108, 19, 246, 331, 399, 724, 1045, 1435, 202, 1890, 2269, 342, 3675, 7164, 8365, 1177, 10815, 12910, 1944, 18756, 24139, 33784, 48756, 6138, 73671, 106597, 124848, 128557, 20188, 231441, 284172, 39963, 336567, 360472, 421512, 62896, 605881, 730627, 819127, 110143, 1100122, 1656277, 232918, 2099832, 2306866, 2411752, 358911, 3445662

Offset: 1

Author

Wolfdieter Lang, Jul 26 2019

Keywords

• nonn
• easy

Comments

See the Aigner reference, Proposition 3.13., p. 55.

If m(n) is odd then m(n) = 1 + 4*a(n), and if m(n) is even then m(n) = 2 + 32* a(n).

Examples

			a(3) = 1 because m(3) - 1 = 4 = a(3)*4. m(3) is odd.
a(6) = 1 because m(6) - 2 = 32 = a(6)*32. m(6) is even.
		

References

• Martin Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013, p. 55.

Crossrefs

Cf. A002559.

Formula

If m(n) is odd then a(n) = (m(n) - 1)/4, and if m(n) is even then a(n) = (m(n) - 2)/32, for the Markoff numbers m(n) = A002559(n), for n >= 1.