A067595 -id:A067595 - cp's OEIS frontend

A103344 Number of representations of n as a sum of distinct elements of the Fibonacci-type sequence beginning 1, 4, 5, 9, 14, 23, 37, 60, ....

1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 0, 3, 3, 0, 0, 2, 4, 2, 0, 0, 3, 3, 0, 0, 1, 4, 3, 0, 0, 3, 5, 2, 0, 0, 4, 4, 0, 0, 2, 5, 3, 0, 0, 3, 4, 1, 0, 0, 4, 4, 0, 0, 3, 6, 3, 0, 0, 5, 5, 0, 0, 2, 6, 4, 0, 0, 4, 6, 2, 0, 0, 5, 5, 0, 0, 3, 6, 3, 0, 0, 4, 4, 0, 0, 1, 5, 4, 0, 0

Offset: 0

Casey Mongoven, Feb 01 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..19308
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 60th anniversary.
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.
Ron Knott, Ron Knott's Sequence Visualiser.
Casey Mongoven, U(n) Rep Sequence no. 1; electronic music created with this sequence.
Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.

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

imax = 10;
f[1] = 1; f[2] = 4; f[n_] := f[n] = f[n-1] + f[n-2];
p = Product[1+x^f[i], {i, 1, imax}];
CoefficientList[p, x][[1;;f[imax]+1]] (* Jean-François Alcover, May 08 2019 *)

a(0)=1 corrected by Alois P. Heinz, Sep 16 2015

A103343 Number of representations of n as a sum of distinct elements of the Fibonacci-type sequence beginning 3, 1, 4, 5, 9, 14, 23, 37, ....

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 2, 0, 2, 3, 3, 2, 1, 3, 4, 3, 1, 2, 3, 4, 3, 0, 3, 5, 4, 2, 2, 4, 5, 3, 0, 3, 4, 4, 3, 1, 4, 6, 5, 2, 3, 5, 6, 4, 0, 4, 6, 5, 3, 2, 5, 6, 4, 1, 3, 4, 5, 4, 0, 4, 7, 6, 3, 3, 6, 8, 5, 0, 5, 7, 6, 4, 2, 6, 8, 6, 2, 4, 6, 7, 5, 0, 5, 8, 6, 3, 3, 6, 7, 4, 0, 4, 5, 5, 4, 1, 5

Offset: 0

Casey Mongoven, Feb 01 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..11933
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 60th anniversary.
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.
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.

a(0)=1 corrected by Alois P. Heinz, Sep 16 2015

A294204 Number of partitions of n into distinct Lucas parts (A000032) greater than 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 1, 2, 1, 3, 0, 3, 2, 2, 2, 2, 3, 0, 3, 1, 3, 1, 3, 3, 2, 3, 2, 4, 0, 4, 2, 3, 2, 3, 3, 1, 3, 1, 4, 0, 4, 3, 3, 3, 3, 5, 0, 5, 2, 4, 2, 4, 4, 2, 4, 2, 5, 0, 5, 3, 3, 3, 3, 4, 0, 4, 1, 4, 1, 4, 4, 3, 4, 3, 6, 0, 6, 3, 5, 3, 5, 5, 2, 5, 2, 6, 0, 6, 4, 4, 4, 4

Offset: 0

Ilya Gutkovskiy, Oct 24 2017

nonn

Convolution of the sequences A067595 and A033999.

			a(9) = 2 because we have [7, 2] and [4, 3, 2].

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences related to partitions

Cf. A000032, A033999, A067593, A067595, A239002, A294203.

CoefficientList[Series[(1 + x^2) Product[1 + x^LucasL[k], {k, 2, 15}], {x, 0, 100}], x]

G.f.: (1 + x^2)*Product_{k>=2} (1 + x^Lucas(k)).

A219479 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 6, 7, 13, 20, 33, 53, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 2, 5, 3, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 3, 6, 3, 0, 0, 0, 0, 5, 5

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..17101
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322.
C. Mongoven, Musical Composition Based on This Sequence.

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

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219480 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 7, 8, 15, 23, 38, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 2, 5, 3, 0, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 0, 4, 4, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..12166
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 60th anniversary.
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.

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

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219481 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 8, 9, 17, 26, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 2, 5, 3, 0, 0, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..13763
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 60th anniversary.
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.

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

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219482 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 9, 10, 19, 29, 48, ....

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15360
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 60th anniversary.
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.

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

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219483 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 7, 9, 16, 25, 41, ....

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 3, 0, 2, 0, 0, 0, 0, 2, 0, 3, 0, 1, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 1, 0, 4, 0, 3, 0, 0, 0, 0, 3, 0, 5, 0, 2, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 2, 0, 5, 0, 3, 0, 0, 0, 0, 3, 0, 4, 0, 1, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..13153
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 60th anniversary.
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.

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

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219484 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 9, 11, 20, 31, 51, ....

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 3, 0, 0, 0, 0, 0, 0, 3, 0, 5, 0, 2, 0, 0, 0, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16347
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 60th anniversary.
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.

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

a(0)=1 from Alois P. Heinz, Sep 16 2015

A219485 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 3, 7, 10, 17, 27, 44, ....

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 0, 3, 0, 0, 5, 0, 0, 2, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 2, 0, 0, 5, 0, 0

Offset: 0

Casey Mongoven, Nov 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..14140
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 60th anniversary.
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.

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

a(0)=1 from Alois P. Heinz, Sep 16 2015

cp's OEIS Frontend

A103344 Number of representations of n as a sum of distinct elements of the Fibonacci-type sequence beginning 1, 4, 5, 9, 14, 23, 37, 60, ....

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A103343 Number of representations of n as a sum of distinct elements of the Fibonacci-type sequence beginning 3, 1, 4, 5, 9, 14, 23, 37, ....

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A294204 Number of partitions of n into distinct Lucas parts (A000032) greater than 1.

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A219479 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 6, 7, 13, 20, 33, 53, ....

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A219480 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 7, 8, 15, 23, 38, ....

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A219481 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 8, 9, 17, 26, ....

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A219482 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 1, 9, 10, 19, 29, 48, ....

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A219483 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 7, 9, 16, 25, 41, ....

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A219484 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 2, 9, 11, 20, 31, 51, ....

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A219485 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 3, 7, 10, 17, 27, 44, ....

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