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A280523 a(n) = Fibonacci(2n + 1) - n.

%I A280523 #72 Feb 06 2024 17:47:05
%S A280523 1,3,10,30,84,227,603,1589,4172,10936,28646,75013,196405,514215,
%T A280523 1346254,3524562,9227448,24157799,63245967,165580121,433494416,
%U A280523 1134903148,2971215050,7778742025,20365011049,53316291147,139583862418
%N A280523 a(n) = Fibonacci(2n + 1) - n.
%C A280523 Old (and equivalent) definition: these are the indices of records in the Fibonachos sequence A280521: the least k such that A280521(k) = n.
%C A280523 Conjecture: a(n) = A215004(2n - 2). - _Peter Kagey_. The conjecture is true. See link for analysis of connections with A215004 and A280521. - _Nathan Fox_, Jan 27 2017
%C A280523 Define the n-th Fibonacci compositions CF(n) by CF(1)={(1)}, CF(2)={(2)}, and CF(n) is 1 adjoined at the end of each composition in CF(n-1) union 2 adjoined to the end of each composition in CF(n-2). The union is disjoint, so |CF(n)| is the n-th Fibonacci number. Define the weight of a composition c by 2^(number of singletons in c). For example, 2122 has 1 singleton and weight 2. Let s(n) be the sum of the weights of CF(n). Conjecture: a(n)= s(2n+4)-s(2n+3). - _George Beck_, Jan 31 2020
%H A280523 Nathan Fox, <a href="/A280523/a280523.pdf">Proof that a(n) = Fibonacci(2n + 1) - n</a>.
%H A280523 Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2
%H A280523 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,5,-1).
%F A280523 G.f.: -x*(-1+2*x-3*x^2+x^3) / ( (x^2-3*x+1)*(x-1)^2 ). - _R. J. Mathar_, Mar 11 2017
%F A280523 a(n) = 5*a(n-1)-8*a(n-2)+5*a(n-3)-a(n-4). - _Wesley Ivan Hurt_, Apr 26 2021
%e A280523 a(1) = 1    because A280521(1)    = 1;
%e A280523 a(2) = 3    because A280521(3)    = 2;
%e A280523 a(3) = 10   because A280521(10)   = 3;
%e A280523 a(4) = 30   because A280521(30)   = 4;
%e A280523 a(5) = 84   because A280521(84)   = 5;
%e A280523 a(6) = 227  because A280521(227)  = 6;
%e A280523 a(7) = 603  because A280521(603)  = 7;
%e A280523 a(8) = 1589 because A280521(1589) = 8.
%t A280523 Table[Fibonacci[2n + 1] - n, {n, 30}] (* _Alonso del Arte_, Jan 29 2017 *)
%t A280523 LinearRecurrence[{5,-8,5,-1},{1,3,10,30},30] (* _Harvey P. Dale_, Feb 06 2024 *)
%o A280523 (PARI) F=vector(64,n,fibonacci(n+2)-1); \\ Resize as needed
%o A280523 A280521(n)=my(s); while(n, s++; t=setsearch(F,n,1); if(t, n-=F[t-1], return(s))); s
%o A280523 first(n)=my(v=vector(n),k,t,mn=1,gaps=n); while(gaps, t=A280521(k++); if(t>=mn && t<=n && v[t]==0, v[t]=k; while(mn<=n && v[mn], mn++); print("a("t") = "k); gaps--)); v \\ _Charles R Greathouse IV_, Jan 04 2017
%Y A280523 Cf. A001519, A215004, A280521.
%K A280523 nonn,easy
%O A280523 1,2
%A A280523 _Peter Kagey_, Jan 04 2017
%E A280523 Corrected and extended by _Charles R Greathouse IV_, Jan 04 2017
%E A280523 a(26) from _Charles R Greathouse IV_, Jan 09 2017
%E A280523 a(27) from _Charles R Greathouse IV_, Jan 16 2017
%E A280523 Replaced old definition by simple formula established by _Nathan Fox_. - _N. J. A. Sloane_, Jan 30 2017