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A144115 Total number of Fibonacci parts in all partitions of n. 9
1, 3, 6, 11, 19, 32, 49, 77, 114, 169, 241, 345, 480, 667, 910, 1237, 1656, 2213, 2918, 3840, 5003, 6497, 8368, 10751, 13711, 17441, 22052, 27806, 34879, 43645, 54355, 67535, 83571, 103171, 126907, 155766, 190554, 232629, 283158, 343969, 416716, 503900, 607807 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is also the sum of the differences between the sum of f-th largest and the sum of (f+1)-st largest elements in all partitions of n for all Fibonacci parts f. - Omar E. Pol, Oct 27 2012

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..8000

FORMULA

G.f.: Sum_{i>=2} x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Jan 24 2017

EXAMPLE

From Omar E. Pol, Nov 20 2011 (Start):

For n = 6 we have:

--------------------------------------

.                        Number of

Partitions            Fibonacci parts

--------------------------------------

6 .......................... 0

3 + 3 ...................... 2

4 + 2 ...................... 1

2 + 2 + 2 .................. 3

5 + 1 ...................... 2

3 + 2 + 1 .................. 3

4 + 1 + 1 .................. 2

2 + 2 + 1 + 1 .............. 4

3 + 1 + 1 + 1 .............. 4

2 + 1 + 1 + 1 + 1 .......... 5

1 + 1 + 1 + 1 + 1 + 1 ...... 6

------------------------------------

Total ..................... 32

So a(6) = 32. (End)

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

      b(n, i-1)+ (p-> p+`if`((t-> issqr(t+4) or issqr(t-4)

      )(5*i^2), [0, p[1]], 0))(b(n-i, min(n-i, i)))))

    end:

a:= n-> b(n$2)[2]:

seq(a(n), n=1..60); # Alois P. Heinz, Jun 24 2009, revised Aug 06 2019

MATHEMATICA

Clear[b]; b[_] = False; l = {0, 1}; For[k=1, k <= 100, k++, b[l[[1]]] = True; l = {l[[2]], l[[1]] + l[[2]]}]; aa[n_, i_] := aa[n, i] = Module[{g, h}, If[n==0, {1, 0}, If[i==0 || n<0, {0, 0}, g = aa[n, i-1]; h = aa[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}]]]; a[n_] := aa[n, n][[2]]; Table[a[n], {n, 1, 60}] (* Jean-Fran├žois Alcover, Jul 30 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000045, A006128, A037032, A144116, A144117, A144118, A199936, A309537.

Sequence in context: A050228 A114089 A001976 * A183088 A326957 A116557

Adjacent sequences:  A144112 A144113 A144114 * A144116 A144117 A144118

KEYWORD

nonn

AUTHOR

Omar E. Pol, Sep 11 2008

EXTENSIONS

More terms from Alois P. Heinz, Jun 24 2009

STATUS

approved

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Last modified September 26 16:43 EDT 2020. Contains 337374 sequences. (Running on oeis4.)