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 A161762 Sum of all numbers from and including Fibonacci(n-1) to and including Fibonacci(n). 2
 1, 1, 3, 5, 12, 26, 63, 153, 385, 979, 2520, 6524, 16965, 44225, 115479, 301833, 789412, 2065414, 5405235, 14147705, 37033701, 96946631, 253795248, 664421400, 1739440297, 4553853121, 11922044043, 31212157613, 81714232380, 213930221714 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1). FORMULA From G. C. Greubel, Oct 24 2018: (Start) a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5). G.f.: x*(1 - 2*x - x^2)/(1 - 3*x - x^2 + 5*x^3 + x^4 - x^5). (End) a(n) = (Lucas(2*n - 1) + 5*Fibonacci(n + 1) - 4*(-1)^n)/10. - Greg Dresden, Jan 01 2021 EXAMPLE Fibonacci(0) to Fibonacci(10) are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Hence a(1) = 0+1 = 1, a(2) = 1, a(3) = 1+2 = 3, a(6) = 5+6+7+8 = 26, a(8) = 13+14+15+16+17+18+19+20+21 = 153. MATHEMATICA Table[Sum[k, {k, Fibonacci[n - 1], Fibonacci[n]}], {n, 1, 50}] (* or *) LinearRecurrence[{3, 1, -5, -1, 1}, {1, 1, 3, 5, 12}, 50] (* G. C. Greubel, Oct 24 2018 *) Total[Range[#[[1]], #[[2]]]]&/@Partition[Fibonacci[Range[0, 30]], 2, 1] (* Harvey P. Dale, Feb 13 2023 *) PROG (Magma) [&+[Fibonacci(n-1)..Fibonacci(n)]: n in [1..30]]; // Klaus Brockhaus, Jun 25 2009 (Magma) m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-2*x-x^2)/(1-3*x-x^2+5*x^3+x^4-x^5))); // G. C. Greubel, Oct 24 2018 (PARI) for(n=1, 50, print1(sum(k=fibonacci(n-1), fibonacci(n), k), ", ")) \\ G. C. Greubel, Oct 24 2018 CROSSREFS Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers). Sequence in context: A280000 A241097 A267725 * A245649 A291035 A005913 Adjacent sequences: A161759 A161760 A161761 * A161763 A161764 A161765 KEYWORD nonn,easy AUTHOR Juri-Stepan Gerasimov, Jun 18 2009 EXTENSIONS Edited, corrected (a(2)=2 replaced by 1, a(13)=15310 replaced by 16965) and extended beyond a(13) by Klaus Brockhaus, Jun 25 2009 Definition clarified by Harvey P. Dale, Feb 13 2023 STATUS approved

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Last modified August 12 08:23 EDT 2024. Contains 375085 sequences. (Running on oeis4.)