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A023548
Convolution of natural numbers >= 2 and Fibonacci numbers.
15
2, 5, 11, 21, 38, 66, 112, 187, 309, 507, 828, 1348, 2190, 3553, 5759, 9329, 15106, 24454, 39580, 64055, 103657, 167735, 271416, 439176, 710618, 1149821, 1860467, 3010317, 4870814, 7881162, 12752008, 20633203, 33385245, 54018483, 87403764, 141422284
OFFSET
1,1
COMMENTS
Minimal cost of maximum height Huffman tree of size n for strictly "worst case height" sequences. (A strictly "worst case height" sequence generates only maximum height Huffman trees; a non-strictly "worst case height" sequence can generate also non-maximum height Huffman trees.) - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004
Record-positions for A107910: A107910(a(n+2)) = A005578(n), A107910(m) < A005578(n) for m < a(n+2). - Reinhard Zumkeller, May 28 2005
LINKS
N.-N. Cao, F.-Z. Zhao, Some Properties of Hyperfibonacci and Hyperlucas Numbers, J. Int. Seq. 13 (2010) # 10.8.8
Ligia L. Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, Journal of Integer Sequences, Volume 19, 2016, Issue 7, #16.7.6.
A. B. Vinokur, Huffman trees and Fibonacci numbers, Kibernetika Issue 6 (1986) 9-12 (in Russian); English translation in Cybernetics 21, Issue 6 (1986), 692-696.
Alex Vinokur, Fibonacci connection between Huffman codes and Wythoff array, arXiv:cs/0410013 [cs.DM], 2004-2005.
FORMULA
From Wolfdieter Lang: (Start)
Convolution of natural numbers n >= 1 with Lucas numbers (A000032).
a(n) = 4*(F(n+1) - 1) + 3*F(n) - n, F(n)=A000045 (Fibonacci).
G.f.: x*(2-x)/((1-x-x^2)*(1-x)^2). (End)
For n >= 1, a(n) = L(n+3) - (n+4), where L(n) are Lucas numbers. - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
a(n) = Fib(n+3) + F(n+1) - (n+3) for n > 1. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004
a(n) = (-4 + (2^(-n)*((1-sqrt(5))^n*(-5+2*sqrt(5)) + (1+sqrt(5))^n*(5+2*sqrt(5)))) / sqrt(5) - n). - Colin Barker, Mar 11 2017
a(n) = Sum_{i=1..n} C(n-i+2,i+1) + C(n-i+1,i). - Wesley Ivan Hurt, Sep 13 2017
MATHEMATICA
Table[4(Fibonacci[n+1] -1) +3Fibonacci[n] -n, {n, 40}] (* Vincenzo Librandi, Sep 16 2017 *)
PROG
(PARI) a(n) = 4*fibonacci(n+1) + 3*fibonacci(n) - n - 4; \\ Michel Marcus, Sep 08 2016
(PARI) Vec(x*(2-x) / ((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017
(Magma) [4*(Fibonacci(n+1)-1)+3*Fibonacci(n)-n: n in [1..40]]; // Vincenzo Librandi, Sep 16 2017
(Sage) [lucas_number2(n+3, 1, -1) -n-4 for n in (1..40)] # G. C. Greubel, Jul 08 2019
(GAP) List([1..40], n-> Lucas(1, -1, n+3)[2] -n-4) # G. C. Greubel, Jul 08 2019
CROSSREFS
KEYWORD
nonn,easy
STATUS
approved