A001950 -id:A001950 - cp's OEIS frontend

A025123 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A001950 (upper Wythoff sequence), t = A023533.

0, 0, 2, 5, 7, 0, 0, 0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 43, 49, 54, 59, 13, 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, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62

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 >;
A025123:= func< n | (&+[Floor(k*(3+Sqrt(5))/2)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
[A025123(n): n in [1..100]]; // G. C. Greubel, Sep 14 2022

b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];
A025123[n_]:= A025123[n]= Sum[Floor[(n-j+2)*GoldenRatio^2]*b[j], {j, Floor[(n+4)/2], n+1}];
Table[A025123[n], {n,100}] (* G. C. Greubel, Sep 14 2022 *)

@CachedFunction
def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..13))
@CachedFunction
def A025123(n): return sum(floor((n-j+2)*golden_ratio^2)*b(j) for j in (((n+4)//2)..n+1))
[A025123(n) for n in (1..100)] # G. C. Greubel, Sep 14 2022

A184819 E.g.f.: A(x) = Sum_{n>=0} (-log(1-x))^[nphi^2] / [nphi^2]!, where [n*phi^2] = A001950(n), the upper Wythoff sequence, and phi = (1+sqrt(5))/2.

Original entry on oeis.org

1, 0, 1, 3, 11, 51, 289, 1940, 15056, 132579, 1305352, 14203398, 169179053, 2188695718, 30552880513, 457633893249, 7319707872140, 124497880667346, 2243512187621332, 42695546402663276, 855593102807351931

Offset: 0

Paul D. Hanna, Jan 22 2011

nonn

			E.g.f.: A(x) = 1 + x^2/2! + 3*x^3/3! + 11*x^4/4! + 51*x^5/5! +...
The series expansion begins:
A(x) = 1 + log(1-x)^2/2! - log(1-x)^5/5! - log(1-x)^7/7! + log(1-x)^10/10! - log(1-x)^13/13! +...+ (-log(1-x))^A001950(n)/A001950(n)! +...
The complementary series begins:
A(x) = 1/(1-x) + log(1-x) + log(1-x)^3/3! - log(1-x)^4/4! - log(1-x)^6/6! - log(1-x)^8/8! + log(1-x)^9/9! +...+ -(-log(1-x))^A000201(n)/A000201(n)! +...

Cf. A184818, A000201, A001950.

{a(n)=local(phi=(sqrt(5)+1)/2,A=1+x+x*O(x^n)); for(i=1, n,A=1+sum(k=1, n,(-log(1-x+x*O(x^n)))^floor(k*phi^2)/floor(k*phi^2)!+x*O(x^n))); n!*polcoeff(A, n)}

E.g.f.: A(x) = 1/(1-x) - Sum_{n>=1} (-log(1-x))^[n*phi] / [n*phi]!, where [n*phi] = A000201(n), the lower Wythoff sequence.

a(n) = n! - A184818(n) for n>0.

A247431 The largest integer m such that A001950(m) < A003231(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 90, 92

Offset: 1

Eric M. Schmidt, Sep 17 2014

nonn

This is the function named K in [Carlitz].

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

Cf. A001950, A003231.

a31(n) = (5*n+sqrtint(5*n^2))\2; \\ A003231
a50(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950
a(n) = my(m=1, N=a31(n)); while(a50(m) < N, m++); m-1; \\ Michel Marcus, Nov 14 2023

A252055 Number of products A000201(i)*A001950(j) = n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 23 2014

A000201 and A001950 are the lower and upper Wythoff sequences, which partition the nonnegative integers.

Does this sequence include every nonnegative integer? What is the maximal number of consecutive 0's? What is the maximal number of consecutive 1's?

			a(312) counts these 7 products:  3*104, 4*78, 6*52, 8*39, 12*26, 24*13, 156*2

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000201, A001950.

N:= 1000: # to get a(1) to a(N)
A201:= [seq(floor(n*phi),n=1..N)]:
A1950:= [seq(floor(n*phi^2),n=1..N)]:
A:= Vector(N):
for i from 1 to N do
  for j from 1 do
    m:= A201[i]*A1950[j];
    if m > N then break fi;
    A[m]:= A[m]+1;
   od
od:
convert(A,list); # Robert Israel, Dec 23 2014

r = (1 + Sqrt[5])/2; s = r/(r - 1); t = Flatten[Table[Floor[r*j]*Floor[s*k], {j, 1, 300}, {k, 1, 300}]]; a[n_] := Count[t, n]; u = Table[a[n], {n, 1, 300}]

A342716 Frobenius number of the upper Wythoff sequence (A001950), starting with the n-th term.

Original entry on oeis.org

3, 16, 19, 42, 42, 42, 55, 58, 76, 79, 79, 110, 110, 110, 118, 121, 144, 144, 144, 155, 160, 173, 181, 181, 207, 207, 207, 220, 223, 254, 254, 254, 275, 275, 275, 283, 283, 309, 309, 309, 320, 325, 343, 346, 346, 377, 377, 377, 385, 388, 406, 409, 409, 422

Offset: 1

Jeffrey Shallit, Mar 19 2021

nonn

The Frobenius number of a set S is the largest positive integer t such that t cannot be written as a nonnegative integer linear combination of the elements of S.

The sequence a(n) is "Fibonacci-synchronized"; there is an automaton that recognizes the Fibonacci representation of the pairs (n, a(n)) in parallel. This means specific values of a(n) are easily computed.

Jeffrey Shallit, Frobenius numbers and automatic sequences, arXiv:2103.10904 [math.NT], 2021.

Cf. A001950, A342715 (analog for the lower Wythoff numbers).

A374863 Obverse convolution A000201**A001950; see Comments.

Original entry on oeis.org

0, 2, 45, 840, 23040, 823680, 29729700, 1319205888, 60949324800, 3307060721664, 207247208103936, 13289727219664896, 959973192342388224, 77222763982780416000, 6318019834620558704640, 568402264910884213555200, 52282854778694683852800000

Offset: 0

Clark Kimberling, Aug 18 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

A000201 and A001950 are assumed to start with a(0) = 0.

Cf. A000201, A001950, A001622, A374848.

r = GoldenRatio;
s[n_] := Floor[n*r]; t[n_] := Floor[n*r^2];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 20}]

A374864 Obverse convolution (1)**A001950; see Comments.

Original entry on oeis.org

1, 3, 18, 144, 1584, 22176, 354816, 6741504, 141571584, 3397718016, 91738386432, 2660413206528, 85133222608896, 2979662791311360, 110247523278520320, 4409900931140812800, 185215839107914137600, 8334712759856136192000, 400066212473094537216000

Offset: 0

Clark Kimberling, Aug 18 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences. a(n+1)/a(n) is an integer for n>=0, so (a(n)) is a divisibility sequence.

Cf. A000012, A000201, A001622, A026352, A374848.

r = GoldenRatio;
s[n_] := 1; t[n_] := Floor[n*r^2];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 20}]

a(n) = Product_{k=0..n} A026352(k).

A022879 The number of numbers [ [ ix ]jx ] that equal n, where i >= 1, j >= 1 and x=(1+sqrt(5))/2. a(n)=0 iff n is in Beatty sequence A001950.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 0, 1, 3, 0, 1, 3, 0, 3, 0, 1, 2, 0, 5, 0, 1, 2, 0, 2, 4, 0, 2, 0, 4, 2, 0, 2, 3, 0, 3, 0, 1, 7, 0, 2, 0, 1, 4, 0, 3, 2, 0, 4, 0, 1, 5, 0, 4, 0, 2, 2, 0, 6, 2, 0, 3, 0, 2, 4, 0, 1, 6, 0, 2, 0, 4, 4, 0, 2, 0, 1, 9, 0, 1, 3, 0, 4, 0, 2, 2, 0, 5, 3, 0, 5

Offset: 1

Clark Kimberling

nonn

Index entries for sequences related to Beatty sequences

A023542 Convolution of natural numbers with Beatty sequence for tau^2 A001950.

Original entry on oeis.org

2, 9, 23, 47, 84, 136, 206, 296, 409, 548, 715, 913, 1145, 1413, 1720, 2068, 2460, 2899, 3387, 3927, 4521, 5172, 5883, 6656, 7494, 8400, 9376, 10425, 11549, 12751, 14034, 15400, 16852, 18393, 20025, 21751, 23573, 25494, 27517, 29644

Offset: 1

Clark Kimberling

nonn

Index entries for sequences related to Beatty sequences

A023564 Convolution of A023531 and A001950.

Original entry on oeis.org

0, 2, 5, 7, 12, 18, 22, 28, 35, 43, 51, 58, 67, 77, 87, 97, 108, 119, 129, 141, 155, 167, 181, 194, 206, 221, 235, 251, 266, 282, 299, 314, 329, 345, 363, 382, 399, 418, 438, 454, 473, 491, 509, 531, 551, 571, 594, 614, 635, 655, 676, 699, 719

Offset: 1

Clark Kimberling

nonn

cp's OEIS Frontend

A025123 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A001950 (upper Wythoff sequence), t = A023533.

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A184819 E.g.f.: A(x) = Sum_{n>=0} (-log(1-x))^[n*phi^2] / [n*phi^2]!, where [n*phi^2] = A001950(n), the upper Wythoff sequence, and phi = (1+sqrt(5))/2.

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A247431 The largest integer m such that A001950(m) < A003231(n).

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PARI

A252055 Number of products A000201(i)*A001950(j) = n.

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Maple

Mathematica

A342716 Frobenius number of the upper Wythoff sequence (A001950), starting with the n-th term.

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A374863 Obverse convolution A000201**A001950; see Comments.

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Mathematica

A374864 Obverse convolution (1)**A001950; see Comments.

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Mathematica

Formula

A022879 The number of numbers [ [ ix ]jx ] that equal n, where i >= 1, j >= 1 and x=(1+sqrt(5))/2. a(n)=0 iff n is in Beatty sequence A001950.

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A023542 Convolution of natural numbers with Beatty sequence for tau^2 A001950.

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A023564 Convolution of A023531 and A001950.

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A025123 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A001950 (upper Wythoff sequence), t = A023533.

A184819 E.g.f.: A(x) = Sum_{n>=0} (-log(1-x))^[nphi^2] / [nphi^2]!, where [n*phi^2] = A001950(n), the upper Wythoff sequence, and phi = (1+sqrt(5))/2.