A038196 3-wave sequence starting with 1, 1, 1.

1, 1, 1, 2, 3, 5, 6, 11, 14, 25, 31, 56, 70, 126, 157, 283, 353, 636, 793, 1429, 1782, 3211, 4004, 7215, 8997, 16212, 20216, 36428, 45425, 81853, 102069, 183922, 229347, 413269, 515338, 928607, 1157954, 2086561, 2601899, 4688460, 5846414

Offset: 0

Author

Floor van Lamoen

Keywords

• easy
• nonn

Comments

The 3-wave sequence with initial values a, b, c is formed by the following construction:

a.......a+b+c............3a+5b+6c...

..b...b+c...a+2b+2c..2a+4b+5c...

....c..........a+2b+3c...

References

• J. Kappraff, Beyond Measure, World Scientific, Inc. 2002, p. 497.

Links

• F. v. Lamoen, Wave sequences
• Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, 1, 0, -1).

Crossrefs

a(2n) forms A006356, a(2n+1) ("the middle row") forms A006054. Cf. A038197, A038201, A187070.

Programs

PARI

a(n)=if(n>-1,polcoeff((1+x-x^2)/(1-2*x^2-x^4+x^6)+x*O(x^n),n),if(n<-3,polcoeff((1-x-x^2)/(1-x^2-2*x^4+x^6)+O(x^(-3-n)),-4-n),0))

Formula

a(n) = a(n-1) + a(n-2) if n is odd,

a(n) = a(n-1) + a(n-4) if n is even.

Also: a(n) = 2*a(n-2) + a(n-4) - a(n-6).

G.f.: (1 + x - x^2)/(1 - 2*x^2 - x^4 + x^6).

Extensions

Edited by Floor van Lamoen, Feb 05 2002