A067595 -id:A067595 - cp's OEIS frontend

A219496 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 8, 1, 9, 10, 19, 29, 48, ....

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

A274262 Number of positive integers possessing exactly n Fibonacci representations (A000121).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 12, 18, 20, 24, 20, 44, 24, 36, 48, 54, 32, 76, 36, 88, 72, 60, 44, 156, 72, 72, 100, 132, 56, 208, 60, 162, 120, 96, 144, 316, 72, 108, 144, 312, 80, 312, 84, 220, 304, 132, 92, 540, 156, 280, 192, 264, 104, 460, 240, 468, 216, 168, 116, 116, 120, 180, 456, 486, 288, 520, 132, 352, 264, 624, 140

Offset: 1

Steven Finch, Jun 16 2016

nonn

			Let phi denote the Euler totient.
The integer p^2*q has 8 multiplicative compositions:
  (p^2*q), p^2*q, q*p^2, p*(p*q), (p*q)*p, q*p*p, p*q*p, p*p*q
from which
  a(p^2*q) = 2*(3*phi(p^2)*phi(q) + 5*phi(p)^2*phi(q))
follows immediately.

Zai-Qiao Bai and Steven R. Finch, Fibonacci and Lucas Representations, Fibonacci Quart. 54 (2016), no. 4, 319-326.

Cf. A000121, A067595.

Let p, q, r be distinct primes and k be a positive integer.
If n = p^k then a(n) = 2*(p-1)*(2*p-1)^(k-1).
If n = p*q then a(n) = 6*(p-1)*(q-1).
If n = p^2*q then a(n) = 2*(p-1)*(8*p-5)*(q-1).
If n = p^3*q then a(n) = 2*(p-1)*(2*p-1)*(10*p-7)*(q-1).
If n = p^4*q then a(n) = 6*(p-1)*(2*p-1)^2*(4*p-3)*(q-1).
If n = p^2*q^2 then a(n) = 2*(p-1)*(q-1)*(26*p*q-18*p-18*q+13).
If n = p*q*r then a(n) = 26*(p-1)*(q-1)*(r-1).

A331922 Number of compositions (ordered partitions) of n into distinct Lucas numbers (beginning with 1).

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 0, 3, 8, 0, 2, 9, 8, 0, 8, 32, 6, 0, 9, 32, 0, 8, 38, 30, 0, 32, 150, 0, 6, 33, 32, 0, 32, 158, 30, 0, 38, 174, 0, 30, 176, 150, 0, 150, 870, 24, 0, 33, 152, 0, 32, 182, 150, 0, 158, 894, 0, 30, 182, 174, 0, 174, 1014, 144, 0, 176, 990, 0, 150, 1014, 864

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

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

Index entries for sequences related to compositions

Cf. A000204, A003263, A054770 (positions of 0's), A067592, A067595, A218396, A288039.

A241952 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 2,1,3,4,7,11,... and 0,2,2,4,6,10,16,... (A000032 and A118658).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 4, 6, 6, 6, 8, 8, 7, 10, 11, 11, 12, 14, 15, 15, 17, 17, 17, 19, 21, 22, 24, 25, 26, 28, 29, 30, 31, 34, 35, 36, 40, 40, 39, 43, 44, 44, 47, 50, 52, 53, 57, 58, 58, 61, 63, 65, 68, 70, 73, 76, 76, 80, 81, 82, 86, 88, 92, 93, 95, 99, 99, 101, 104, 105, 108, 111, 115, 118, 119, 124, 126, 127, 133, 134, 137, 142, 143, 149

Offset: 1

Casey Mongoven, May 03 2014

nonn

			a(10) = 6 because 10 can be represented in 6 possible ways as a sum of integers in the set {1,2,3,4,6,7,10,11,16,...}: 10, 7+3, 7+2+1, 6+4, 6+3+1, 4+3+2+1.

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. A118658, A000032, A067595.

A241953 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 2,1,3,4,7,11,... and 0,3,3,6,9,15,... (A000032 and A022086).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 5, 6, 6, 7, 8, 8, 9, 11, 10, 13, 13, 14, 16, 17, 16, 19, 21, 19, 24, 24, 25, 27, 30, 28, 32, 34, 33, 38, 37, 39, 42, 45, 42, 49, 48, 48, 55, 54, 55, 59, 63, 60, 68, 66, 68, 74, 74, 76, 81, 82, 81, 91, 86, 89, 97, 96, 97, 105, 104, 104, 114, 110, 113, 120, 120, 123, 130, 128, 131, 140, 137, 141, 149, 146

Offset: 1

Casey Mongoven, May 03 2014

nonn

			a(10) = 6 because 10 can be represented in 6 possible ways as a sum of integers in the set {1,2,3,4,6,7,9,11,15,...}: 9+1, 7+3, 7+2+1, 6+4, 6+3+1, 4+3+2+1.

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. A000032, A022086, A067595.

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.

A357306 Number of compositions (ordered partitions) of n into distinct Lucas numbers (beginning at 2).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 8, 9, 8, 8, 32, 9, 14, 32, 38, 32, 36, 150, 33, 32, 32, 158, 38, 60, 174, 176, 150, 150, 870, 33, 56, 152, 182, 158, 180, 894, 182, 174, 174, 1014, 176, 294, 990, 1014, 870, 888, 5904, 153, 152, 152, 902, 182, 300, 1014, 1022, 894, 894

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

nonn

Index entries for sequences related to compositions

Cf. A000032, A003263, A067593, A067595, A218396, A288039, A331922.

A357427 Expansion of Product_{k>=0} 1 / (1 + x^Lucas(k)).

Original entry on oeis.org

1, -1, 0, -1, 1, 0, 1, -2, 2, -2, 2, -3, 3, -2, 4, -5, 4, -5, 5, -5, 6, -6, 8, -9, 8, -9, 9, -9, 11, -12, 13, -14, 14, -15, 15, -16, 20, -20, 20, -23, 23, -23, 25, -28, 31, -31, 32, -36, 36, -36, 41, -44, 45, -47, 49, -52, 54, -56, 62, -65, 65, -69, 72, -74, 79, -83, 87, -91

Offset: 0

Ilya Gutkovskiy, Sep 28 2022

sign

Convolution inverse of A067595.

Cf. A000032, A067595, A357383.

nmax = 67; CoefficientList[Series[Product[1/(1 + x^LucasL[k]), {k, 0, 20}], {x, 0, nmax}], x]

cp's OEIS Frontend

A219496 Number of representations of n as a sum of distinct elements of the generalized Fibonacci sequence beginning 8, 1, 9, 10, 19, 29, 48, ....

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A274262 Number of positive integers possessing exactly n Fibonacci representations (A000121).

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Formula

A331922 Number of compositions (ordered partitions) of n into distinct Lucas numbers (beginning with 1).

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A241952 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 2,1,3,4,7,11,... and 0,2,2,4,6,10,16,... (A000032 and A118658).

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A241953 Number of possible representations of n as a sum of distinct positive integers from the Fibonacci-type sequences 2,1,3,4,7,11,... and 0,3,3,6,9,15,... (A000032 and A022086).

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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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A357306 Number of compositions (ordered partitions) of n into distinct Lucas numbers (beginning at 2).

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Keywords

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A357427 Expansion of Product_{k>=0} 1 / (1 + x^Lucas(k)).

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Mathematica