0,4

Number of partitions into distinct Fibonacci parts (1 counted as single Fibonacci number)

Inverse Euler transform of sequence has generating function sum_{n>1} x^F(n)-x^{2F(n)} where F() are the Fibonacci numbers.

A065033(n) = a(A000045(n)).

a(n) = 1 if and only if n+1 is a Fibonacci number. The lengths of such quasi-periods (from Fib(i)-1 to Fib(i+1)-1, inclusive) is a Fibonacci number + 1. The maximum value of a(n) within each subsequent quasi-period increases by a Fibonacci number. For example, from n = 143 to n = 232, the maximum is 13. From 232 to 376, the maximum is 16, an increase of 3. From 376 to 609, 21, an increase of 5. From 609 to 986, 26, increasing by 5 again. Each two subsequent maxima seem to increase by the same increment, the next Fibonacci number. [From Kerry Mitchell (lkmitch(AT)gmail.com), Nov 14 2009]

J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60-th anniversary.

M. Bicknell-Johnson, pp. 53-60 in 'Applications of Fibonacci Numbers', volume 8, ed: F T Howard, Kluwer (1999); see Theorem 3.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 54.

D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322.

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).

Paul K. Stockmeyer, "A Smooth Tight Upper Bound for the Fibonacci Representation Function R(N)", Fibonacci Quarterly, Volume 46/47, Number 2, May 2009. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]

T. D. Noe, Table of n, a(n) for n = 0..6765

Jean Berstel, Home Page

Ron Knott Sumthing about Fibonacci Numbers

J. Shallit, Number theory and formal languages, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570. (Eq. 9.2.)

a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), b(k) = Sum_{f} (-1)^(k/f+1)*f, where the last sum is taken over all Fibonacci numbers f dividing k. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 28 2002

a(n)= 1, if n=0, 1, 2; a(n)= a(fib(i-2)+k)+a(k) if n>2 and 0<=k<=fib(i-3); a(n)= 2*a(k) if n>2 and fib(i-3)<=k<=fib(i-2); a(n)= a(fib(i+1)-2-k) otherwise where fib(i) is largest Fibonacci number (A000045) <= n and k=n-fib(i). [Bicknell-Johnson] - Ron Knott (ron(AT)ronknott.com), Dec 06 2004

a(n) = f(n,1,1) with f(x,y,z) = if x<y then 0^x else f(x-y,y+z,y)+f(x,y+z,y). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 11 2009]

with(combinat): p := product((1+x^fibonacci(i)), i=2..25): s := series(p, x, 1000): for k from 0 to 250 do printf(`%d, `, coeff(s, x, k)) od:

CoefficientList[ Normal@Series[ Product[ 1+z^Fibonacci[ k ], {k, 2, 13} ], {z, 0, 233} ], z ]

(PARI) a(n)=local(A, m, f); if(n<0, 0, A=1+x*O(x^n); m=2; while((f=fibonacci(m))<=n, A*=1+x^f; m++); polcoeff(A, n))

Cf. A007000, A003107, A000121. Least inverse is A013583.

Sequence in context: A160696 A152545 A109967 this_sequence A097368 A109699 A029283

Adjacent sequences: A000116 A000117 A000118 this_sequence A000120 A000121 A000122

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms and Maple program from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000

1,2

Every positive integer occurs exactly once in T; thus a is a permutation of the positive integers. Row 1 consists of Fibonacci numbers. To obtain T from the array T' in A094607, add 1 to every number in T', then shift row 1 one place to the right and fill the initial place with 1.

A northwest corner of T:

1 2 3 5 8

4 6 7 10 11

9 12 14 15 19

17 22 27 28 33

Cf. A000045, A000119, A094607.

Sequence in context: A077529 A143516 A120620 this_sequence A163280 A056537 A083221

Adjacent sequences: A094605 A094606 A094607 this_sequence A094609 A094610 A094611

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), May 14 2004

0,3

The partitions allow repeated items but the order of items is immaterial (1+2=2+1) - Ron Knott (ron(AT)ronknott.com), Oct 22 2003

A098641(n) = a(A000045(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 24 2005

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..1000

G. Almkvist, Partitions with Parts in a Finite Set and with Parts Outside a Finite Set, Exper. Math. vol 11 no 4 (2002) p 449-456

a(n)=(1/n)*Sum_{k=1..n} A005092(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 21 2002

G.f.: Product(1/(1-x^fibonacci(i)), i=2..infinity). - Ron Knott (ron(AT)ronknott.com), Oct 22 2003

a(n) = f(n,1,1) with f(x,y,z) = if x<y then 0^x else f(x-y,y,z)+f(x,y+z,y). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 11 2009]

a(4)=4 since the 4 partitions of 4 using only Fibonacci numbers, reptitions allowed, are 1+1+1+1, 2+2, 2+1+1, 3+1

CoefficientList[ Series[1/ Product[1 - x^Fibonacci[i], {i, 2, 21}], {x, 0, 53}], x] (from Robert G. Wilson v (rgwv(at)rgwv.com), Mar 28 2006)

Cf. A007000, A005092, A003107, A028290 (where the only Fibonacci numbers allowed are 1, 2, 3, 5 and 8).

Cf. A102848.

A000119. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 11 2009]

Sequence in context: A027589 A039851 A028290 this_sequence A014977 A008583 A053253

Adjacent sequences: A003104 A003105 A003106 this_sequence A003108 A003109 A003110

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Herman P. Robinson

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 21 2002

0,2

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.

D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322.

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

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:

(PARI) 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))

Cf. A000119. Least inverse is A083853.

Sequence in context: A072789 A126302 A134674 this_sequence A049846 A086712 A125842

Adjacent sequences: A000118 A000119 A000120 this_sequence A000122 A000123 A000124

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 18 2000

1,2

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

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.

Paul K. Stockmeyer, "A Smooth Tight Upper Bound for the Fibonacci Representation Function R(N)", Fibonacci Quarterly, Volume 46/47, Number 2, May 2009. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]

Charles R Greathouse IV, Table of n, a(n) for n = 1..5000

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

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

Least inverse of A000119. Cf. A046815, A083853.

Sequence in context: A136532 A030417 A123979 this_sequence A122794 A115006 A122796

Adjacent sequences: A013580 A013581 A013582 this_sequence A013584 A013585 A013586

nonn

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

Additional terms from Jeffrey Shallit (shallit(AT)graceland.uwaterloo.ca); extended to 600 terms by Daniel C. Fielder

Entries rechecked by David W. Wilson, Jun 18 2003

0,4

Gives number of terms in n-th row of many common tables.

Number of partitions of the (n+1)-th Fibonacci number into distinct Fibonacci numbers: a(n) = A000119(A000045(n)), see also A098641. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 24 2005

Harry J. Smith, Table of n, a(n) for n=0,...,1000

a(0)=a(1)=a(2)= 1, a(3)=2, a(n)=a(n-1)+a(n-2)-a(n-3) for n>3 . G.f. (1-x^2+x^3)/(1-x-x^2+x^3) - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 28 2006

a(n)=(3/4)+(1/4)*(-1)^(n-1)+(1/2)*(n-1)+[C(2*n,n) mod 2], with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]

(PARI) { for (n=0, 1000, if (n<3, a=1, if (n%2, a++)); write("b065033.txt", n, " ", a) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Oct 03 2009]

Cf. A004526, A008619.

Sequence in context: A111660 A127365 A168050 this_sequence A001057 A130472 A004526

Adjacent sequences: A065030 A065031 A065032 this_sequence A065034 A065035 A065036

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Nov 04, 2001

1,1

M. Bicknell-Johnson & D.C. Fielder, 'The number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas' Fibonacci Quart. 37.1 (1999) pgs 47 ff

Ron Knott Sumthing about Fibonacci Numbers

GF: (8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(x^9-x^8+x^5-x^4-x+1). {a(n)}={ series starting [8,8,11,11,13,13,18,19] then a(n)=a(n-4)+a(n-8)+1,n>7} a(0)=8, a(1)=11, a(2)=13, a(3)=18, then: a(4n)=A022318(n+3) = 2 A00045(n+5) + A000045(n+3) - 1,n>=1 a(4n+1)=A022406(n+2) = 4 A000045(n+4) - 1,n>=1 a(4n+2)=A022308(n+4)= 2 A000045(n+4) + A000045(n+6) - 1,n>=1, a(4n+3) = 3 A000045(n+4) - 1,n>=1

8 is the sum of only 3 sets of Fibonacci numbers: {8}, {3,5} and {1,2,5}

11 is the sum of only {3,8}, {1,2,8}, {1,2,3,5}

first N terms: series((8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(x^9-x^8+x^5-x^4-x+1), x, N);

Cf. A000045, A000071, A013583, A000119, A122194.

Sequence in context: A077060 A123939 A134787 this_sequence A064153 A106670 A153039

Adjacent sequences: A122192 A122193 A122194 this_sequence A122196 A122197 A122198

nonn

Ron Knott (enquiry(AT)ronknott.com), Aug 25 2006, corrected Aug 29 2006

0,5

J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498.

D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322.

Ron Knott, Ron Knott's Sequence Visualiser.

Casey Mongoven, V(n) Rep Sequence no. 1; electronic music created with this sequence.

Cf. A000121, A000119, A067595, A003263, A103344.

Sequence in context: A122915 A030298 A098281 this_sequence A085263 A115092 A169623

Adjacent sequences: A103340 A103341 A103342 this_sequence A103344 A103345 A103346

nonn

Casey Mongoven (cm(AT)caseymongoven.com), Feb 01 2005

0,6

J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498.

D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322.

Ron Knott, Ron Knott's Sequence Visualiser.

Casey Mongoven, U(n) Rep Sequence no. 1; electronic music created with this sequence.

Cf. A000121, A000119, A067595, A003263, A103343.

Sequence in context: A125071 A161528 A136176 this_sequence A123484 A008626 A058626

Adjacent sequences: A103341 A103342 A103343 this_sequence A103345 A103346 A103347

nonn

Casey Mongoven (cm(AT)caseymongoven.com), Feb 01 2005

0,1

Number of partitions of n with an even number of distinct Fibonacci parts minus the number of partitions of n with an odd number of distinct Fibonacci parts.

Every term is -1, 0 or 1.

N. Robbins, "Fibonacci Partitions", The Fibonacci Quarterly, 34.4 (1996), pp. 306-313

Yufei Zhao, The coefficients of a truncated Fibonacci power series, Fib. Q., 46/47 (2008/2009), 53-55. - from N. J. A. Sloane, May 30 2009

F. Ardila, The Coefficients of a Fibonacci power series

Ardila gives a fast recurrence.

1 - x - x^2 + x^4 + x^7 - x^8 + x^11 - x^12 - x^13 + x^14 + x^18 - x^19 - x^20 + x^22 + x^23 - x^24 + x^29 - x^30 - x^31 + x^33 + x^36 - x^38 - x^39 + x^40 + x^47 - ... - N. J. A. Sloane, May 30 2009

Take[ CoefficientList[ Expand[ Product[1 - x^Fibonacci[k], {k, 2, 13}]], x], 105] (from Robert G. Wilson v May 29 2004)

Cf. A000045, A000119, A093997, A093998, A151661.

Sequence in context: A079295 A088025 A082416 this_sequence A083187 A080339 A167752

Adjacent sequences: A093993 A093994 A093995 this_sequence A093997 A093998 A093999

easy,sign

Naoki Sato (nsato7(AT)yahoo.ca), May 24 2004

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), May 29 2004