A.H.M. Smeets - cp's OEIS frontend

A385533 Third prepended column of the tribonacci array of the second kind, A385436.

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

Offset: 1

A.H.M. Smeets, Jul 02 2025

sign

The numbers that occur thrice in this sequence are given by A353084(n)-1, i.e., the first prepended column of the tribonacci array of the first kind minus one.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A353084, A385436.

A385455 First prepended column of the tribonacci array of the second kind, A385436.

Original entry on oeis.org

-1, 0, 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81

Offset: 1

A.H.M. Smeets, Jun 29 2025

sign

The numbers that do not occur in this sequence are given by A278041.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A278041, A385436.

A385532 Second prepended column of the tribonacci array of the second kind, A385436.

Original entry on oeis.org

-1, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44

Offset: 1

A.H.M. Smeets, Jul 02 2025

sign

The numbers that occur twice in this sequence are the numbers in A278039.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A278039, A385436.

A385436 Tribonacci array of the second kind, read by upward antidiagonals.

Original entry on oeis.org

0, 2, 1, 4, 5, 3, 6, 8, 10, 7, 9, 12, 16, 20, 14, 11, 18, 23, 31, 38, 27, 13, 21, 34, 44, 58, 71, 51, 15, 25, 40, 64, 82, 108, 132, 95, 17, 29, 47, 75, 119, 152, 200, 244, 176, 19, 32, 54, 88, 139, 220, 281, 369, 450, 325, 22, 36, 60, 101, 163, 257, 406, 518, 680

Offset: 1

A.H.M. Smeets, Jun 28 2025

The array is, as a sequence, a permutation of the nonnegative integers; however it does not satisfy the conditions for interspersion and dispersion as given by Eric Weisstein's World of Mathematics. However, when all terms are increased by 1, it does satisfy the conditions for interspersion and dispersion!
Rows satisfy the recurrence: T(m,k) = 2*T(m,k-1) - T(m,k-4) for all k>4.
This array belongs to a family of Wythoff like arrays, based on binary number representations like the greedy and lazy Fibonacci number representations (see A035513 and A372501 for arrays), greedy and lazy Narayana number representations (A136189 for the array related to greedy representation).
The array is related to the lazy tribonacci number representation A352103. The first column lists the even numbers, i.e., for wich 0 suffix A352103(T(m,1)). The odd numbers are represented in the columns k > 1: A352103(T(m,k)) = A352103(T(m,1)) + 1^(k-1). Here + stands for concatenation and ^ stands for repeated concatenation.

			Array including some prepended columns (p = 1..4):
  p=4 p=3 p=2 p=1 | k=1 k=2 k=3  k=4  k=5  k=6  k=7   k=8   k=9  k=10
   -2  -1  -1  -1 |   0   1   3    7   14   27   51    95   176   325
   -2  -1   0   0 |   2   5  10   20   38   71  132   244   450   829
   -2   0   0   1 |   4   8  16   31   58  108  200   369   680
   -2   0   1   2 |   6  12  23   44   82  152  281   518
   -1   0   2   4 |   9  18  34   64  119  220  406   748
   -1   1   2   5 |  11  21  40   75  139  257  474   873
   -1   1   3   6 |  13  25  47   88  163  301  555  1022
   -1   1   4   7 |  15  29  54  101  187  345  636  1171
   -1   2   4   8 |  17  32  60  112  207  382  704  1296
   -1   2   5   9 |  19  36  67  125
    0   2   6  11 |  22  42  78  145
Each row of the array satisfies the recurrence relation T(m,k) = 2*T(m,k-1) - T(m,k-4); from this, the prepended columns are obtained by rowwise backward recursion.

A.H.M. Smeets, Table of n, a(n) for n = 1..20100 (first 200 antidiagonals).
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Eric Weisstein's World of Mathematics, Interspersion.
Eric Weisstein's World of Mathematics, Sequence Dispersion.

Cf. A035513, A136189, A372501, A027084 (m=1), A351631 (k=1), A352103.

Prepended columns: A385455 (p=1), A385532 (p=2), A385533 (p=3).

def ToDual_111_Zeck(n):
    if n == 0:
        return "0"
    f0, f1, f2, sf = 1, 0, 0, 0
    while n > sf:
        f0, f1, f2 = f0+f1+f2, f0, f1
        sf += f0
    r, s = sf-n, "1"
    while f0 > 1:
        f0, f1, f2 = f1, f2, f0-f1-f2
        r, s = r%f0, s+str(1-r//f0)
    return s
def From_111_Zeck(s):
    f0, f1, f2, i, n = 1, 1, 0, len(s), 0
    while i > 0:
        i -= 1
        f0, f1, f2, n = f0+f1+f2, f0, f1, n+int(s[i])*f0
    return n
d, a, n, c1 = 0, 0, 0, []
while d < 11:
    s = ToDual_111_Zeck(a)
    if s[len(s)-1] == "0": # == even
        n, d = n+1, d+1
        print(a, end = ", ")
        i, c1, p1 = d-1, c1+[s], ""
        while i > 0:
            n, i, p1 = n+1, i-1, p1+"1"
            print(From_111_Zeck(c1[i]+p1), end = ", ")
    a += 1

A384590 a(n) = floor(X(n,n)), where X(n,n) is the largest zero of the Laguerre polynomial of degree n.

Original entry on oeis.org

1, 3, 6, 9, 12, 15, 19, 22, 26, 29, 33, 37, 40, 44, 48, 51, 55, 59, 62, 66, 70, 73, 77, 81, 85, 89, 92, 96, 100, 104, 107, 111, 115, 119, 123, 126, 130, 134, 138, 142, 146, 149, 153, 157, 161, 165, 169, 172, 176, 180, 184, 188, 192, 196, 199, 203, 207, 211

Offset: 1

A.H.M. Smeets, Jun 14 2025

nonn

For X(k,n), the k-th smallest zero of the Laguerre polynomial of degree n, see formula section of A091476, for large n and relative small k, k << n.
Some terms for large n:
a(1000) = floor(3943.2473948452...), a(2000) = floor(7927.9014222639...), a(4000) = floor(15908.5812117320...), a(8000) = floor(31884.2511300626...), a(16000) = floor(63853.6067816122...), a(32000) = floor(127815.0051094389...), a(64000) = floor(255766.3763209512...), a(128000) = floor(511705.1129209706...), a(256000) = floor(1023627.9299056501...), a(512000) = floor(2047530.6886230061...).

A.H.M. Smeets, Table of n, a(n) for n = 1..12245
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], Table 25.9.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.

Cf. A091476.

Cf. 1+A014176 (n=2), A384279 (n=3), A384587 (n=4).

A384590[n_] := Floor[Root[LaguerreL[n, #] &, n]];
Array[A384590, 70] (* Paolo Xausa, Jun 26 2025 *)

Limit_{n -> oo} X(n,n)/n = 4.

a(n) ~ floor(4*n + 2 - 5.8917*n^(1/3)).

A384589 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384587.

Original entry on oeis.org

0, 0, 0, 5, 3, 9, 2, 9, 4, 7, 0, 5, 5, 6, 1, 3, 2, 7, 4, 5, 0, 1, 0, 3, 7, 9, 0, 5, 6, 7, 6, 2, 0, 5, 9, 3, 2, 1, 2, 2, 7, 7, 2, 5, 6, 9, 6, 6, 4, 3, 3, 2, 4, 4, 0, 8, 5, 4, 6, 6, 4, 9, 9, 4, 7, 7, 9, 0, 1, 0, 9, 1, 7, 5, 6, 9, 3, 7, 2, 3, 0, 2, 7, 8, 5, 7, 9, 1, 1, 6

Offset: 0

A.H.M. Smeets, Jun 14 2025

			0.00053929470556132745010379056762059321227725696643324...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

Cf. A384590.
There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, this sequence

First[RealDigits[Root[1990656*#^4 - 1990656*#^3 + 504576*#^2 - 16960*# + 9 &, 1], 10, 100, -1]] (* Paolo Xausa, Jun 26 2025 *)

solve(x = 0.0, 0.01, 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9)

Smallest root of 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9 = 0.

A384588 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384586.

Original entry on oeis.org

0, 3, 8, 8, 8, 7, 9, 0, 8, 5, 1, 5, 0, 0, 5, 3, 8, 4, 2, 7, 2, 4, 3, 8, 1, 6, 8, 1, 5, 6, 2, 0, 9, 9, 1, 3, 7, 2, 2, 3, 0, 7, 1, 9, 1, 3, 4, 8, 2, 7, 6, 9, 0, 2, 1, 8, 1, 6, 3, 5, 2, 9, 2, 4, 0, 4, 5, 2, 5, 7, 6, 2, 9, 1, 0, 1, 7, 6, 9, 8, 0, 9, 9, 9, 8, 4, 3, 3

Offset: 0

A.H.M. Smeets, Jun 07 2025

			0.038887908515005384272438168156209913722307191348276...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, this sequence, A384589

First[RealDigits[Root[1990656*#^4 - 1990656*#^3 + 504576*#^2 - 16960*# + 9 &, 2], 10, 100, -1]] (* Paolo Xausa, Jun 26 2025 *)

solve(x = 0.1, 0.04, 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9)

Second smallest root of 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9 = 0.

A384586 Decimal expansion of the second largest zero of the Laguerre polynomial of degree 4.

Original entry on oeis.org

4, 5, 3, 6, 6, 2, 0, 2, 9, 6, 9, 2, 1, 1, 2, 7, 9, 8, 3, 2, 7, 9, 2, 8, 5, 3, 8, 4, 9, 5, 7, 1, 3, 7, 8, 8, 0, 1, 2, 5, 7, 8, 4, 3, 5, 3, 3, 8, 6, 8, 0, 4, 6, 4, 9, 7, 4, 8, 0, 5, 7, 5, 8, 7, 5, 5, 5, 8, 2, 8, 4, 5, 0, 8, 7, 5, 1, 4, 3, 1, 5, 8, 9, 7, 6, 5, 3

Offset: 1

A.H.M. Smeets, Jun 04 2025

			4.53662029692112798327928538495713788012578435338680...

Paolo Xausa, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
V. I. Krylov, Approximate calculation of integrals (Dover publications) (1962) page 347 n=4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281, this sequence, A384587 | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[4, #] &, 3], 10, 100]] (* Paolo Xausa, Jun 18 2025 *)

solve(x = 2, 6, x^4 - 16*x^3 + 72*x^2 - 96*x + 24)

Second largest root of x^4 - 16 x^3 + 72 x^2 - 96 x + 24 = 0.

A384587 Decimal expansion of the largest zero of the Laguerre polynomial of degree 4.

Original entry on oeis.org

9, 3, 9, 5, 0, 7, 0, 9, 1, 2, 3, 0, 1, 1, 3, 3, 1, 2, 9, 2, 3, 3, 5, 3, 6, 4, 4, 3, 4, 2, 0, 5, 4, 7, 6, 1, 6, 4, 5, 6, 5, 8, 3, 9, 0, 6, 6, 0, 7, 8, 2, 7, 0, 8, 1, 2, 8, 0, 7, 0, 7, 8, 9, 7, 6, 3, 8, 7, 4, 6, 8, 1, 2, 9, 7, 4, 9, 5, 5, 6, 6, 7, 0, 1, 4, 7, 4

Offset: 1

A.H.M. Smeets, Jun 07 2025

			9.39507091230113312923353644342054761645658390660782...

Paolo Xausa, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281, A384586, this sequence | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[4, #] &, 4], 10, 100]] (* Paolo Xausa, Jun 18 2025 *)

solve(x = 6, 16, x^4 - 16*x^3 + 72*x^2 - 96*x + 24)

Second largest root of x^4 - 16 x^3 + 72 x^2 - 96 x + 24 = 0.

A384467 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384281.

Original entry on oeis.org

3, 5, 7, 4, 1, 8, 6, 9, 2, 4, 3, 7, 7, 9, 9, 6, 8, 6, 6, 4, 1, 4, 9, 2, 0, 1, 7, 4, 5, 8, 0, 9, 1, 2, 8, 1, 7, 6, 3, 5, 7, 8, 3, 6, 4, 9, 1, 9, 3, 4, 0, 9, 2, 1, 7, 4, 8, 2, 2, 5, 0, 4, 6, 6, 7, 5, 7, 6, 4, 1, 5, 9, 2, 0, 7, 0, 2, 7, 1, 1, 5, 1, 4, 3, 6, 2, 8

Offset: 0

A.H.M. Smeets, May 30 2025

			0.35741869243779968664149201745809128176357836491934...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281                         | A384466, this sequence

First[RealDigits[Root[1990656*#^4 - 1990656*#^3 + 504576*#^2 - 16960*# + 9 &, 3], 10, 100]] (* Paolo Xausa, Jun 26 2025 *)

solve(x = 0.3, 0.4, 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9)

Second largest root of 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9 = 0.

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User: A.H.M. Smeets

A385533 Third prepended column of the tribonacci array of the second kind, A385436.

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A385455 First prepended column of the tribonacci array of the second kind, A385436.

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A385532 Second prepended column of the tribonacci array of the second kind, A385436.

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A385436 Tribonacci array of the second kind, read by upward antidiagonals.

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A384590 a(n) = floor(X(n,n)), where X(n,n) is the largest zero of the Laguerre polynomial of degree n.

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A384589 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384587.

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A384588 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384586.

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A384586 Decimal expansion of the second largest zero of the Laguerre polynomial of degree 4.

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A384587 Decimal expansion of the largest zero of the Laguerre polynomial of degree 4.

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