A001950 -id:A001950 - cp's OEIS frontend

A214742 Least m>0 such that L(n)-m divides U(n)+m, where L = A000201 and U = A001950 (lower and upper Wythoff sequences).

1, 3, 2, 1, 1, 10, 4, 13, 2, 2, 9, 10, 20, 3, 3, 26, 10, 29, 4, 4, 12, 36, 13, 5, 20, 42, 43, 35, 6, 49, 49, 52, 7, 7, 20, 28, 21, 8, 8, 65, 23, 68, 9, 9, 72, 75, 75, 10, 10, 39, 29, 84, 11, 11, 31, 91, 32, 12, 95, 97, 98, 48, 13, 50, 104, 107, 14, 14, 39, 91, 40, 15

Offset: 2

Clark Kimberling, Jul 28 2012

			Write x#y if x|y is false; then 11#21, 10#22, 9#23, 8|24, so a(8) = 4.

Clark Kimberling, Table of n, a(n) for n = 2..1000

Cf. A214741.

r=GoldenRatio;
Table[m = 1; While[! Divisible[Floor[n*r^2]+m, Floor[n*r] - m], m++]; m, {n, 2, 100}]

Typo in name corrected by Clark Kimberling, Jul 22 2015

A260317 Numbers not of the form v(m) + v(n), where v = A001950 (upper Wythoff numbers) and 1 <= m <= n - 1, for n >= 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 14, 16, 19, 21, 24, 26, 29, 32, 34, 37, 40, 42, 45, 50, 53, 55, 58, 63, 66, 68, 71, 76, 79, 84, 87, 89, 92, 97, 100, 105, 108, 110, 113, 118, 121, 126, 131, 134, 139, 142, 144, 147, 152, 155, 160, 165, 168, 173, 176, 178, 181

Offset: 1

Clark Kimberling, Jul 22 2015

It appears that the difference sequence consists entirely of Fibonacci numbers (A000045); see A260311.

In fact, the difference sequence consists only of the numbers 1,2,3,5. Proved with the Walnut theorem-prover. - Jeffrey Shallit, Oct 12 2022

Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025. See pp. 2-3.

Cf. A000045, A259556, A259600, A259601, A260311.

r = GoldenRatio; z = 1060;
u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
s[m_, n_] := v[m] + v[n];
t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}]; (* A259601 *)
w = Flatten[Table[Count[Flatten[t], n], {n, 1, z}]];
p0 = Flatten[Position[w, 0]]  (* A260317 *)
d = Differences[p0] (* A260311 *)

n <= a(n) < 5n, see Shallit comment. - Charles R Greathouse IV, Nov 22 2024

A289974 p-INVERT of the upper Wythoff sequence (A001950), where p(S) = 1 - S.

Original entry on oeis.org

2, 9, 35, 139, 549, 2169, 8571, 33866, 133817, 528755, 2089288, 8255476, 32620147, 128893113, 509299806, 2012413902, 7951720511, 31419907712, 124150565816, 490560415825, 1938368302977, 7659141579267, 30263830481105, 119582517950630, 472510530626342

Offset: 0

Clark Kimberling, Aug 15 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A289780 for a guide to related sequences.

Cf. A001950, A289973.

z = 60; r = 1 + GoldenRatio; s = Sum[Floor[k*r] x^k, {k, 1, z}]; p = 1 - s;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A001950 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289974 *)

A295540 Number of ways of writing n as the sum of a lower Wythoff number (A000201) and an upper Wythoff number (A001950), when zero is included in both sequences.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 5, 1, 5, 5, 2, 7, 3, 5, 8, 1, 9, 5, 5, 10, 2, 9, 9, 3, 12, 5, 8, 13, 1, 13, 10, 5, 15, 5, 11, 15, 2, 17, 9, 9, 18, 3, 16, 15, 5, 20, 8, 13, 21, 1, 22, 13, 10, 23, 5, 19, 20, 5, 25, 11, 15, 26, 2, 25, 19, 9, 28, 9, 20, 27, 3, 30, 16, 15, 31, 5, 27, 25, 8, 33, 13, 21, 34, 1, 34, 23, 13, 36, 10, 27, 33, 5, 38, 19, 20, 39, 5, 35, 30, 11, 41, 15, 27, 41, 2, 43, 25, 19, 44, 9, 36, 37, 9, 46, 20, 27

Offset: 0

Paul D. Hanna, Nov 30 2017

nonn

Note that floor(n*phi) and floor(n*phi^2), for n>=1, form Beatty sequences.

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 3*x^10 + 5*x^11 + x^12 + 5*x^13 + 5*x^14 + 2*x^15 + 7*x^16 + 3*x^17 + 5*x^18 + 8*x^19 + x^20 + 9*x^21 + 5*x^22 + 5*x^23 + 10*x^24 + 2*x^25 + 9*x^26 + 9*x^27 + 3*x^28 + 12*x^29 + 5*x^30 + 8*x^31 + 13*x^32 + x^33 + 13*x^34 + 10*x^35 + 5*x^36 + 15*x^37 + 5*x^38 + 11*x^39 + 15*x^40 + 2*x^41 + 17*x^42 + 9*x^43 + 9*x^44 + 18*x^45 + 3*x^46 + 16*x^47 + 15*x^48 + 5*x^49 + 20*x^50 +...+ a(n)*x^n +...
such that A(x) = WL(x) * WU(x) where
WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 + x^12 + x^14 + x^16 + x^17 + x^19 + x^21 + x^22 + x^24 + x^25 + x^27 + x^29 + x^30 +...+ x^A000201(n) +...
WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 + x^18 + x^20 + x^23 + x^26 + x^28 + x^31 + x^34 + x^36 + x^39 + x^41 + x^44 + x^47 + x^49 +...+ x^A001950(n) +...
Terms equal 1 only at positions:
[0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, ..., Fibonacci(n+1)-1, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..10000

Cf. A259598, A000201, A001950.

{a(n) = my(phi = (1 + sqrt(5))/2, WL=1, WU=1);
WL = sum(m=0,floor(n/phi)+1, x^floor(m*phi) +x*O(x^n));
WU = sum(m=0,floor(n/phi^2)+1, x^floor(m*phi^2) +x*O(x^n));
polcoeff(WL*WU,n)}
for(n=0,120, print1(a(n),", "))

G.f.: [ Sum_{n>=0} x^floor(n*phi) ] * [ Sum_{n>=0} x^floor(n*phi^2) ], where phi = (1+sqrt(5))/2.
G.f.: [1 + Sum_{n>=1} x^A000201(n) ] * [1 + Sum_{n>=1} x^A001950(n) ], where A000201 and A001950 are the lower and upper Wythoff sequences, respectively.
a(Fibonacci(n+1)-1) = 1 for n>=1.
a(Fibonacci(n+2)-2) = Fibonacci(n) for n>=1.

A307295 If n is even, a(n) = A001950(n/2+1), otherwise a(n) = A001950((n-1)/2+1) + 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 89, 90, 91, 92, 94, 95, 96, 97, 99, 100, 102, 103, 104, 105

Offset: 0

N. J. A. Sloane, Apr 12 2019

nonn

It follows from the definition that a(2i+1) = a(2i)+1 for all i.
From Jeffrey Shallit, Jun 06 2021: (Start)
This sequence consists of the nonzero distances between occurrences of 1 in the Fibonacci word A003849 (easily provable with the Walnut theorem-prover).
Alternatively, these are the positive n such that A003849(n-1) = 1 or A003849(n-2) = 1 (again, easily provable with the Walnut theorem-prover). (End)

Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date? [See Omega, a few lines below Table 2.]

Cf. A000201, A001950, A307294.

from math import isqrt
def A307295(n): return ((m:=(n>>1)+1)+isqrt(5*m**2)>>1)+m+(n&1) # Chai Wah Wu, Aug 10 2022

A003250 The number m such that A001950(m) = A003231(A003234(n)).

Original entry on oeis.org

4, 11, 15, 22, 26, 29, 33, 40, 44, 51, 58, 62, 69, 73, 76, 80, 87, 91, 98, 102, 105, 109, 116, 120, 127, 134, 138, 145, 149, 152, 156, 163, 167, 174, 178, 181, 185, 192, 196, 199, 203, 210, 214, 221, 225, 228, 232, 239, 243, 250, 257, 261, 268, 272, 275, 279

Offset: 1

N. J. A. Sloane

nonn

This is the function named z in [Carlitz]. - Eric M. Schmidt, Aug 14 2014

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

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

From Eric M. Schmidt, Aug 14 2014: (Start)
a(n) = ceiling((1/phi^2) * A003231(A003234(n))), where phi is the golden ratio.
a(n) = 5*k - 1 - A003231(k), where k = A003234(n). [Cf. Theorems 4.1(ii) and 5.9(vii) in Carlitz.]
Conjecture: a(n) = floor((3-phi)*A003234(n)).
(End)

More terms and a definition from Eric M. Schmidt, Aug 14 2014

A023654 Convolution of (F(2), F(3), F(4), ...) and A001950.

Original entry on oeis.org

2, 9, 23, 49, 95, 172, 300, 510, 853, 1412, 2319, 3790, 6174, 10034, 16283, 26397, 42765, 69253, 112114, 181468, 293688, 475267, 769072, 1244461, 2013660, 3258254, 5272052, 8530449

Offset: 1

Clark Kimberling

nonn

A023668 Convolution of A001950 and A023533.

Original entry on oeis.org

2, 5, 7, 12, 18, 22, 28, 33, 38, 46, 53, 61, 70, 77, 85, 93, 100, 109, 116, 126, 137, 147, 158, 168, 178, 190, 199, 210, 221, 230, 242, 252, 262, 274, 285, 299, 312, 324, 339, 350, 364, 377, 390, 404, 416, 429, 444, 455, 469, 482, 494, 509, 521

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001622, A001950, A023533, A023665.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Floor(k*(3+Sqrt(5))/2)*A023533(n-k+1): k in [1..n]]): n in [1..80]]; // G. C. Greubel, Jul 18 2022

A023668[n_, k_]:= A023668[n, k]= Sum[Floor[(k+1 +Binomial[n+2,3] -Binomial[j+2, 3])*GoldenRatio^2], {j, n}];
Table[A023668[n, k], {n, 7}, {k,0,n*(n+3)/2}] (* G. C. Greubel, Jul 18 2022 *)

def A023668(n, k): return sum( floor((k+1 + binomial(n+2,3) - binomial(j+2,3))*golden_ratio^2) for j in (1..n) )
flatten([[A023668(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # G. C. Greubel, Jul 18 2022

a(n) = Sum_{j=1..n} A001950(j) * A023533(n-j+1).

T(n, k) = Sum_{j=1..n} A001950(k+1 +binomial(n+2,3) -binomial(j+2,3)), for 0 <= k <= n*(n+3)/2, n >= 1 (as an irregular triangle). - G. C. Greubel, Jul 18 2022

A023867 a(n) = 1t(n) + 2t(n-1) + ...+ k*t(n+1-k), where k=floor((n+1)/2) and t is A001950 (upper Wythoff sequence).

Original entry on oeis.org

2, 5, 17, 24, 54, 71, 127, 153, 242, 279, 409, 465, 645, 717, 954, 1052, 1354, 1473, 1848, 1989, 2444, 2620, 3164, 3367, 4007, 4239, 4983, 5260, 6116, 6426, 7402, 7764, 8868, 9269, 10509, 10950, 12333, 12835, 14370, 14917, 16611, 17226, 19087, 19752, 21788, 22504

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

f:= func< n | n + Floor(n*(1+Sqrt(5))/2) >;
[(&+[j*f(n+1-j): j in [1..Floor((n+1)/2)]]): n in [1..50]]; // G. C. Greubel, Jun 12 2019

f[n_]:= n +Floor[n*GoldenRatio]; Table[Sum[j*f[n+1-j], {j,1,Floor[(n + 1)/2]}], {n, 1, 50}] (* G. C. Greubel, Jun 12 2019 *)

f(n) = n + floor(n*(1+sqrt(5))/2);
a(n) = sum(j=1, floor((n+1)/2), j*f(n+1-j)); \\ G. C. Greubel, Jun 12 2019

def f(n): return n + floor(n*golden_ratio)
[sum(j*f(n+1-j) for j in (1..floor((n+1)/2))) for n in (1..50)] # G. C. Greubel, Jun 12 2019

Title simplified by Sean A. Irvine, Jun 12 2019

A024690 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A001950 (upper Wythoff sequence), t = A023533.

Original entry on oeis.org

2, 0, 0, 2, 5, 7, 10, 0, 0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 43, 49, 54, 59, 65, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 59, 65, 69, 75, 81, 85, 91, 95, 101, 107, 111, 117, 123, 127, 39, 41, 44, 47, 49, 52

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001950, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A001950:= func< n |  Floor(n*(3+Sqrt(5))/2) >;
A024690:= func< n | (&+[A001950(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]) >;
[A024690(n): n in [1..130]]; // G. C. Greubel, Sep 07 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A001950[n_]:= Floor[n*GoldenRatio^2];
A024690[n_]:= A024690[n]= Sum[A001950[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A024690[n], {n, 130}] (* G. C. Greubel, Sep 07 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
def A001950(n): return floor(n*golden_ratio^2)
def A024690(n): return sum(A001950(k)*A023533(n-k+1) for k in (1..((n+1)//2)))
[A024690(n) for n in (1..130)] # G. C. Greubel, Sep 07 2022

cp's OEIS Frontend

A214742 Least m>0 such that L(n)-m divides U(n)+m, where L = A000201 and U = A001950 (lower and upper Wythoff sequences).

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A260317 Numbers not of the form v(m) + v(n), where v = A001950 (upper Wythoff numbers) and 1 <= m <= n - 1, for n >= 2.

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A289974 p-INVERT of the upper Wythoff sequence (A001950), where p(S) = 1 - S.

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A295540 Number of ways of writing n as the sum of a lower Wythoff number (A000201) and an upper Wythoff number (A001950), when zero is included in both sequences.

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A307295 If n is even, a(n) = A001950(n/2+1), otherwise a(n) = A001950((n-1)/2+1) + 1.

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A003250 The number m such that A001950(m) = A003231(A003234(n)).

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A023654 Convolution of (F(2), F(3), F(4), ...) and A001950.

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A023668 Convolution of A001950 and A023533.

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A023867 a(n) = 1*t(n) + 2*t(n-1) + ...+ k*t(n+1-k), where k=floor((n+1)/2) and t is A001950 (upper Wythoff sequence).

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A023867 a(n) = 1t(n) + 2t(n-1) + ...+ k*t(n+1-k), where k=floor((n+1)/2) and t is A001950 (upper Wythoff sequence).

A024690 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A001950 (upper Wythoff sequence), t = A023533.