A054770 -id:A054770 - cp's OEIS frontend

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

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.

A332529 Rectangular array by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

Original entry on oeis.org

0, 2, 1, 5, 3, 2, 7, 6, 4, 3, 10, 8, 7, 5, 4, 13, 11, 9, 8, 6, 5, 15, 14, 12, 10, 9, 7, 6, 18, 16, 15, 13, 11, 10, 8, 7, 20, 19, 17, 16, 14, 12, 11, 9, 8, 23, 21, 20, 18, 17, 15, 13, 12, 10, 9, 26, 24, 22, 21, 19, 18, 16, 14, 13, 11, 10, 28, 27, 25, 23, 22

Offset: 0

Clark Kimberling, Jun 15 2020

Column 0: Nonnegative integers.
Row 0: Upper Wythoff sequence, A001950, with 0 prepended.
Main Diagonal: A003231, with 0 prepended.
Diagonal (2,6,9,13,...) = A054770.
Diagonal (1,4,8,11,...) = A214971.
Diagonal (2,5,9,12,...) = A284624.

			Northwest corner:
  0   2    5    7   10   13   15
  1   3    6    8   11   14   16
  2   4    7    9   12   15   17
  3   5    8   10   13   16   18
  4   6    9   11   14   17   19
  5   7   10   12   15   18   20
  6   8   11   13   16   19   21
As a triangle (antidiagonals):
  0
  1   2
  2   3   5
  3   4   6   7
  4   5   7   8  10

Cf. A001950, A054770, A214971, A284624.

t[n_, k_] := Floor[n + k*GoldenRatio];
Grid[Table[t[n, k], {n, 0, 10}, {k, 0, 10}]] (* A332529 array *)
Table[t[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten  (* A332529 sequence *)

T(n,k) = floor(n + k*r), where r = (golden ratio)^2 = (3+sqrt(5))/2.

Definition corrected by Harvey P. Dale, Jun 14 2022

A104857 Positive integers that cannot be represented as the sum of distinct Lucas 3-step numbers (A001644).

Original entry on oeis.org

2, 5, 6, 9, 13, 16, 17, 20, 23, 26, 27, 30, 34, 37, 38, 41, 44, 45, 48, 52, 55, 56, 59, 62, 65, 66, 69, 73, 76, 77, 80, 84, 87, 88, 91, 94, 97, 98, 101, 105, 108, 109, 112, 115, 116, 119, 123, 126, 127, 130, 133, 136, 137, 140, 144, 147, 148, 151, 154

Offset: 1

Jonathan Vos Post, Apr 24 2005

Similar to A054770 "Numbers that are not the sum of distinct Lucas numbers (A000204)" but with Lucas 3-step numbers (A001644). Wanted: equivalent of David W. Wilson conjecture (A054770) as proved by Ian Agol. Note that all positive integers can be presented as the sum of distinct Fibonacci numbers in A000119 way. Catalani called Lucas 3-step numbers "generalized Lucas numbers" but that is quite ambiguous. These are also called tribonacci-Lucas numbers.

			In "base Lucas 3-step numbers" we can represent 1 as "1", but cannot represent 2 because there is no next Lucas 3-step number until 3 and we can't have two instances of 1 summed here. We can represent 3 as "10" (one 3 and no 1's), 4 as "11" (one 3 and one 1). Then we cannot represent 5 or 6 because there is no next Lucas 3-step number until 7 and we can't sum two 3s or six 1's. 7 becomes "100" (one 7, no 3s and no 1's), 8 becomes "101" and so forth.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.

Cf. A000119, A001644, A054770.

More terms from T. D. Noe, Apr 26 2005

A301653 Expansion of x(1 + 2x)/((1 - x)(1 + x)(1 - x - x^2)).

Original entry on oeis.org

0, 1, 3, 5, 10, 16, 28, 45, 75, 121, 198, 320, 520, 841, 1363, 2205, 3570, 5776, 9348, 15125, 24475, 39601, 64078, 103680, 167760, 271441, 439203, 710645, 1149850, 1860496, 3010348, 4870845, 7881195, 12752041, 20633238, 33385280, 54018520, 87403801, 141422323, 228826125, 370248450

Offset: 0

Ilya Gutkovskiy, Mar 25 2018

Apparently (for n > 0), numbers that have a unique partition into a sum of distinct Lucas numbers (A000204).

Eric Weisstein's World of Mathematics, Lucas Number
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A000071, A000204, A001622, A003263, A014217, A054770, A294203.

CoefficientList[Series[x (1 + 2 x)/((1 - x) (1 + x) (1 - x - x^2)) , {x, 0, 40}], x]
LinearRecurrence[{1, 2, -1, -1}, {0, 1, 3, 5}, 41]
Table[LucasL[n + 1] - (3 - (-1)^n)/2, {n, 0, 40}]
Table[Floor[GoldenRatio^(n + 1)] - 1, {n, 0, 40}]

a(n) = fibonacci(n) + fibonacci(n+2) + ((-1)^n - 3)/2; \\ Altug Alkan, Mar 25 2018

G.f.: x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
a(n) = Lucas(n+1) - (3 - (-1)^n)/2.
a(n) = floor(phi^(n+1)) - 1, where phi = (1 + sqrt(5))/2 is the golden ratio (A001622).
a(n) = Sum_{k>=0} A051601(n-k,k) (conjectured). - Greg Dresden, May 18 2023

A306683 Integers k for which the base-phi representation of k does not include 1 or phi.

Original entry on oeis.org

3, 5, 7, 10, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 34, 36, 39, 41, 43, 45, 47, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 79, 81, 83, 86, 88, 90, 92, 94, 97, 99, 101, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 126, 128, 130, 133, 135, 137, 139, 141, 144

Offset: 1

Michel Dekking, May 06 2019

nonn

Let b = A214970 be the sequence of the integers k for which the base phi representation includes 1, and let c be the sequence of integers k for which the base phi representation includes phi.
Note that a, b and c form a complementary triple (since consecutive digits 11 do not occur in a base phi representation).
Conjecture (Moses 2012/Baruchel 2018): b is the generalized Beatty sequence b(n) = floor(n*phi) + 2*n + 1.
Conjecture (Kimberling 2012): c = A054770 = A000201(n) + 2*n - 1 = floor(n*phi) + 2*n - 1.
One can prove that the Moses/Baruchel conjecture and the Kimberling conjecture are equivalent.
Conjecture: (a(n)) is a union of two generalized Beatty sequences v and w, given by v(n) = floor(n*phi) + 2*n = A003231(n), and w(n) = 3*floor(n*phi) + n + 1 = A190509(n) + 1.
This conjecture is compatible with the Moses/Baruchel/Kimberling conjecture.
These three conjectures are proved in my paper 'Base phi representations and golden mean beta-expansions'. - Michel Dekking, Jun 26 2019

			3 = phi^2 + phi^{-2}, 5 = phi^3 + phi^{-1} + phi^{-4}.

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
George Bergman, A Number System with an Irrational Base, Mathematics Magazine, Vol. 31, No. 2 (Nov. - Dec., 1957), pp. 98-110.
M. Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.

Cf. A214970, A054770, A000201.

cp's OEIS Frontend

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

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A332529 Rectangular array by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

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A104857 Positive integers that cannot be represented as the sum of distinct Lucas 3-step numbers (A001644).

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Extensions

A301653 Expansion of x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).

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A306683 Integers k for which the base-phi representation of k does not include 1 or phi.

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A301653 Expansion of x(1 + 2x)/((1 - x)(1 + x)(1 - x - x^2)).