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 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). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 * A048811 A142155 A092351 Adjacent sequences:  A079286 A079287 A079288 * A079290 A079291 A079292 KEYWORD easy,nonn AUTHOR Paul Barry, Feb 08 2003 STATUS approved

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Last modified September 26 01:25 EDT 2020. Contains 337346 sequences. (Running on oeis4.)