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 A020956 a(n) = Sum_{k>=1} floor(tau^(n-k)) where tau is A001622. 7
 1, 2, 4, 8, 14, 25, 42, 71, 117, 193, 315, 514, 835, 1356, 2198, 3562, 5768, 9339, 15116, 24465, 39591, 64067, 103669, 167748, 271429, 439190, 710632, 1149836, 1860482, 3010333, 4870830, 7881179, 12752025, 20633221, 33385263, 54018502, 87403783, 141422304 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Clark Kimberling, Problem 10520, Amer. Math. Mon. 103 (1996) p. 347. Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1). FORMULA G.f.: x*(1-x^2+x^3)/((1-x-x^2)*(1+x)*(1-x)^2). - Ralf Stephan, Apr 08 2004 a(n) = Lucas(n+1) - floor(n/2) - 1. a(n) = Sum_{k=0..n-1} A014217(k). a(n) = 2^(-2-n)*((-2)^n - 5*2^n + 2*(1-t)^(1+n) + 2*(1+t)^n + 2*t*(1+t)^n - 2^(1+n)*n) where t=sqrt(5). - Colin Barker, Feb 09 2017 From G. C. Greubel, Apr 05 2024: (Start) a(n) = Lucas(n+1) - (1/4)*(2*n + 5 - (-1)^n). E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)) - (1/2)*((x+2)*cosh(x) + (x+3)*sinh(x)). (End) MATHEMATICA LinearRecurrence[{2, 1, -3, 0, 1}, {1, 2, 4, 8, 14}, 40] (* Vincenzo Librandi, Nov 01 2016 *) PROG (Python) prpr = 0 prev = 1 for n in range(2, 100): print(prev, end=", ") curr = prpr+prev + n//2 prpr = prev prev = curr # Alex Ratushnyak, Jul 30 2012 (PARI) Vec(x*(1-x^2+x^3)/((1-x-x^2)*(1+x)*(1-x)^2) + O(x^50)) \\ Michel Marcus, Nov 01 2016 (Magma) I:=[1, 2, 4, 8, 14]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)-3*Self(n-3)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 01 2016 (Magma) [Lucas(n+1)-(2*n+5-(-1)^n)/4: n in [1..40]]; // G. C. Greubel, Apr 05 2024 (SageMath) [lucas_number2(n+1, 1, -1) -(n+2+(n%2))//2 for n in range(1, 41)] # G. C. Greubel, Apr 05 2024 CROSSREFS Cf. A000032, A001622, A014217, A281362. Sequence in context: A340658 A291443 A210145 * A164393 A164391 A164153 Adjacent sequences: A020953 A020954 A020955 * A020957 A020958 A020959 KEYWORD nonn,easy AUTHOR Clark Kimberling EXTENSIONS More terms from Vladeta Jovovic, Apr 04 2002 STATUS approved

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Last modified September 10 21:37 EDT 2024. Contains 375795 sequences. (Running on oeis4.)