A079289 For even n, a(n) = a(n-2) + a(n-1) + 2^(n/2-2), n>2. For odd n, a(n) = a(n-2) + a(n-1).

1, 1, 2, 3, 6, 9, 17, 26, 47, 73, 128, 201, 345, 546, 923, 1469, 2456, 3925, 6509, 10434, 17199, 27633, 45344, 72977, 119345, 192322, 313715, 506037, 823848, 1329885, 2161925, 3491810, 5670119, 9161929, 14864816, 24026745, 38957097, 62983842

Offset

0,3

Comments

Generalized Fibonacci sequence: a(n) = a(n-2) + a(n-1), and for even n a row sum of Pascal's triangle (a power of two) is added.

Call a multiset of nonzero integers good if the sum of the cubes is the square of the sum. The number of ascending chains of good multisets starting from the empty set by adding one element at a time is a(n). - Michael Somos, Apr 14 2005

a(n) is the number of compositions of n which consist of an initial (possibly empty) subsequence of even summands and a remaining (possibly empty) sequence of odd summands.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-2).

Formula

a(n) = a(n-2) + a(n-1) + floor(2^(n/2-2))*(1-(-1)^(n+1))/2 for n>1.

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

a(n) = -A016116(n+1)/2 +A000045(n+2), n>0. - R. J. Mathar, Sep 27 2012

From Gregory L. Simay, Jul 25 2016: (Start)

If n = 2k+1, a(n) = the convolution Sum_{j=0,..k} c(j)*F(n-2j), where c(j) = A011782(j) = 2^(j-1) and f(j)= A000045(j).

If n = 2k, a(n) = c(k) + the convolution Sum_{j=0,..(k-1)} c(j)*F(n-2j), where c(j)=A011782(j)=2^(j-1) and f(j)= A000045(j). (End)

Example

a(4) = 6 from the good multisets {-1,-1,1,1}, {-1,1,1,2}, {-2,-1,1,2}, {-2,1,2,2}, {-3,1,2,3}, {1,2,3,4}.
a(4) = 6 because there are six compositions of four, in which the initial parts are all even and the final parts are all odd: 4, 3+1, 1+3, 2+2, 2+1+1, 1+1+1+1.

Mathematica

CoefficientList[Series[(1-x^2)^2/(1-x-x^2)/(1-2x^2), {x, 0, 37}], x]
LinearRecurrence[{1, 3, -2, -2}, {1, 1, 2, 3, 6}, 25] (* G. C. Greubel, Aug 16 2016; corrected by Georg Fischer, Apr 02 2019 *)
nxt[{n_, a_, b_}]:={n+1, b, If[EvenQ[n], a+b, a+b+2^((n+1)/2-2)]}; Join[{1}, NestList[ nxt, {2, 1, 2}, 40][[All, 2]]] (* Harvey P. Dale, Jul 13 2019 *)

Prog

(PARI) {a(n)=local(A); if(n<3, (n>=0)+(n>1), A=vector(n, i, i); for(i=3, n, A[i]=A[i-1]+A[i-2]+ if(i%2==0, 2^(i/2-2))); A[n])} /* Michael Somos, Apr 14 2005 */
(Magma) I:=[1, 1, 2, 3, 6]; [n le 5 select I[n] else Self(n-1)+3*Self(n-2) -2*Self(n-3)-2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 05 2013

Crossrefs

Cf. A000045, A005674, A007318, A011782, A061667 (bisection).

Sequence in context: A029511 A320271 A056532 * A365377 A048811 A142155

Adjacent sequences: A079286 A079287 A079288 * A079290 A079291 A079292

Keyword

easy,nonn

Author

Paul Barry, Feb 08 2003

Status

approved