A023548 Convolution of natural numbers >= 2 and Fibonacci numbers.

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

Colin Barker, Table of n, a(n) for n = 1..1000

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.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

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

Cf. A000032, A000045, A006327.

Sequence in context: A119970 A326509 A082775 * A144700 A000785 A364552

Adjacent sequences: A023545 A023546 A023547 * A023549 A023550 A023551

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved