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, 10534874, 13136773, 23671647, 29518061

Offset

0,4

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

Paolo Xausa, Table of n, a(n) for n = 0..1000

Floor van Lamoen, Wave sequences

Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-1).

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.

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).

Mathematica

LinearRecurrence[{0, 2, 0, 1, 0, -1}, {1, 1, 1, 2, 3, 5}, 50] (* Paolo Xausa, Apr 19 2026 *)

Prog

(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))

Crossrefs

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

Sequence in context: A276107 A187068 A187070 * A039849 A039896 A367212

Adjacent sequences: A038193 A038194 A038195 * A038197 A038198 A038199

Keyword

easy,nonn

Author

Floor van Lamoen

Extensions

Edited by Floor van Lamoen, Feb 05 2002

More terms from Paolo Xausa, Apr 19 2026

Status

approved