A083853 -id:A083853 - cp's OEIS frontend

A000121 Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).

1, 2, 2, 3, 3, 3, 4, 3, 4, 5, 4, 5, 4, 4, 6, 5, 6, 6, 5, 6, 4, 5, 7, 6, 8, 7, 6, 8, 6, 7, 8, 6, 7, 5, 5, 8, 7, 9, 9, 8, 10, 7, 8, 10, 8, 10, 8, 7, 10, 8, 9, 9, 7, 8, 5, 6, 9, 8, 11, 10, 9, 12, 9, 11, 13, 10, 12, 9, 8, 12, 10, 12, 12, 10, 12, 8, 9, 12, 10, 13, 11, 9, 12, 9, 10, 11, 8, 9, 6, 6, 10, 9

Offset: 0

N. J. A. Sloane

nonn

Number of partitions into distinct Fibonacci parts (1 counted as two distinct Fibonacci numbers).

Inverse Euler transform of sequence has generating function sum_{n>0} x^F(n)-x^{2F(n)} where F() is Fibonacci.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..6765
Zai-Qiao Bai and Steven R. Finch, Fibonacci and Lucas Representations, Fibonacci Quart. 54 (2016), no. 4, 319-326. See Table 1 p. 324.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 45.

Cf. A000119, A274262, A356874.

Least inverse is A083853.

with(combinat): p := product((1+x^fibonacci(i)), i=1..25): s := series(p,x,1000): for k from 0 to 250 do printf(`%d,`,coeff(s,x,k)) od:

imax = 20; p = Product[1+x^Fibonacci[i], {i, 1, imax}]; CoefficientList[p, x][[1 ;; Fibonacci[imax]+1]] (* Jean-François Alcover, Feb 04 2016, adapted from Maple *)
nmax = 91; s=Total/@Subsets[Select[Table[Fibonacci[i], {i, nmax}], # <= nmax &]];
Table[Count[s, n], {n, 0, nmax}] (* Robert Price, Aug 17 2020 *)

a(n)=local(A,m,f); if(n<0,0,A=1+x*O(x^n); m=1; while((f=fibonacci(m))<=n,A*=1+x^f; m++); polcoeff(A,n))

a(0) = 1; for n >= 1, a(n) = A000119(n) + A000119(n-1). - Peter Munn, Jan 19 2018

More terms from James Sellers, Jun 18 2000

A013583 Smallest positive number that can be written as sum of distinct Fibonacci numbers in n ways.

Original entry on oeis.org

1, 3, 8, 16, 24, 37, 58, 63, 97, 105, 152, 160, 168, 249, 257, 270, 406, 401, 435, 448, 440, 647, 1011, 673, 723, 715, 1066, 1058, 1050, 1092, 1160, 1147, 1694, 1155, 1710, 1702, 2647, 1846, 1765, 1854, 2736, 1867, 2757, 2744, 2841, 2990, 2752, 2854, 2985, 3019, 4511, 3032, 6967, 4456, 3024, 4477, 4616, 4451, 7349, 4629, 7218, 4917, 4621, 4854, 4904, 7179, 7166, 4896, 7200, 7247, 7310, 7213, 7831, 8187, 7488, 7205, 11614, 7480, 7815, 7857, 7925, 11593, 18154, 7912, 11813, 11682, 11653

Offset: 1

Marjorie Bicknell-Johnson (marjohnson(AT)earthlink.net)

Smallest nonnegative number that can be written as sum of distinct Fibonacci numbers in n ways would be the same, except starting with 0.

			1 = 1; 3 = 3 = 2 + 1; 8 = 8 = 5 + 3 = 5 + 2 + 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..5000
Marjorie Bicknell-Johnson and Daniel C. Fielder, The Least Number Having 331 Representations as a Sum of Distinct Fibonacci Numbers, Fibonacci Quarterly 39(2001), pp. 455-461.
Daniel C. Fielder and Marjorie Bicknell-Johnson, The First 330 Terms of Sequence A013583, Fibonacci Quarterly 39 (2001), pp. 75-84.
Petra Kocábová, Zuzana Masáková and Edita Pelantová, Integers with a maximal number of Fibonacci representations, RAIRO-Theor. Inf. Appl., Volume 39, Number 2, April-June 2005.
Paul K. Stockmeyer, A Smooth Tight Upper Bound for the Fibonacci Representation Function R(N), Fibonacci Quarterly, Volume 46/47, Number 2, May 2009.
F. V. Weinstein, Notes on Fibonacci Partitions, arXiv:math/0307150 [math.NT], 2003-2015.

Least inverse of A000119. Cf. A046815, A083853.

A000119(a(n)) = n (for n>1).

Additional terms from Jeffrey Shallit
Extended to 600 terms by Daniel C. Fielder
Entries rechecked by David W. Wilson, Jun 18 2003

A347349 a(n) is the smallest positive integer which can be represented as the sum of distinct Lucas numbers (A000032) in exactly n ways.

Original entry on oeis.org

1, 3, 7, 14, 21, 32, 50, 54, 83, 90, 130, 137, 144, 213, 220, 231, 347, 343, 372, 383, 376, 553, 864, 575, 618, 611, 911, 904, 897, 933, 991, 980, 1447, 987, 1461, 1454, 2261, 1577, 1508, 1584, 2337, 1595, 2355, 2344, 2427, 2554, 2351, 2438, 2550, 2579, 3853, 2590, 5951, 3806, 2583

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Cf. A000032, A013583, A055635, A067595, A083853.

A359391 a(n) is the smallest number which can be represented as the sum of n distinct positive Fibonacci numbers (1 is allowed twice as a part) in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

1, 2, 3, 16, 27, 71, 116, 278, 451, 818, 1305, 2169, 3925, 8119, 13117, 23252, 37858, 62999, 101939, 178088, 298357, 484576, 813710, 1613509, 2610739, 4224275, 6845969, 11280196, 19772533, 32524576, 53157802, 85936132

Offset: 0

Ilya Gutkovskiy, Dec 29 2022

			For n = 3: 16 = Fibonacci(1) + Fibonacci(3) + Fibonacci(7) =
                Fibonacci(2) + Fibonacci(3) + Fibonacci(7) =
                Fibonacci(4) + Fibonacci(5) + Fibonacci(6) =
                1 + 2 + 13 =
                1'+ 2 + 13 =
                3 + 5 + 8.

Cf. A000045, A000121, A013583, A083853.

a(0), a(10)-a(18) from Alois P. Heinz, Dec 29 2022

a(19)-a(31) from David A. Corneth, Dec 30 2022

cp's OEIS Frontend

A000121 Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).

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A013583 Smallest positive number that can be written as sum of distinct Fibonacci numbers in n ways.

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A347349 a(n) is the smallest positive integer which can be represented as the sum of distinct Lucas numbers (A000032) in exactly n ways.

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A359391 a(n) is the smallest number which can be represented as the sum of n distinct positive Fibonacci numbers (1 is allowed twice as a part) in exactly n ways, or -1 if no such number exists.

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