A010716 -id:A010716 - cp's OEIS frontend

A382106 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.

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

Offset: 0

A.H.M. Smeets, Mar 27 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382106, this sequence
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.236926885056189087514264040719917362643260002212...

A.H.M. Smeets, 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 25.4, n=5.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372270.

Equals (322-13*sqrt(70))/900.

A382107 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Mar 27 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | this sequence, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.4679139345726910473898703439895509948116556057692...

A.H.M. Smeets, 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 25.4, n=6.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372271.

A382686 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, this sequence, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.36076157304813860756983351383771611166152189274674...

A.H.M. Smeets, 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 25.4, n=6.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372272.

A382687 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, this sequence
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.17132449237917034504029614217273289352682250148404...

A.H.M. Smeets, 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 25.4, n=6.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372273.

A382688 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | this sequence, A382689, A382690

			0.3818300505051189449503697754889751338783650835338627...

A.H.M. Smeets, 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 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372274.

A382689 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, this sequence, A382690

			0.279705391489276667901467771423779582486925065226598764...

A.H.M. Smeets, 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 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372275.

A382690 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, this sequence

			0.12948496616886969327061143267908201832858740225994666...

A.H.M. Smeets, 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 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372276.

A382105 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Mar 27 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | this sequence, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.47862867049936646804129151483563819291229555334...

A.H.M. Smeets, 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) Table 25.4, n=5.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372269.

Equals (322+13*sqrt(70))/900.

A010703 Period 2: repeat (3,5).

Original entry on oeis.org

3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, Dec 10 2009: (Start)
Interleaving of A010701 and A010716.
Also continued fraction expansion of (15+sqrt(285))/10.
Also decimal expansion of 35/99.
Binomial transform of 3 followed by A084633 without initial terms 1,0.
Inverse binomial transform of A171497. (End)

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010701 (all 3's sequence), A010716 (all 5's sequence), A084633 (inverse binomial transform of repeated odd numbers), A171497.

&cat[ [3, 5]: n in [1..53] ]; // Klaus Brockhaus, Dec 10 2009

Table[If[OddQ[n], 3, 5], {n, 1, 50}] (* Stefan Steinerberger, Apr 10 2006 *)
PadRight[{},120,{3,5}] (* Harvey P. Dale, Sep 03 2012 *)

a(n)=3+n%2*2 \\ Charles R Greathouse IV, Nov 20 2011

From Klaus Brockhaus, Dec 10 2009: (Start)
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 5.
G.f.: (3+5*x)/((1-x)*(1+x)). (End)
a(n) = 4 - (-1)^n. - Aaron J Grech, Aug 02 2024
E.g.f.: 3*cosh(x) + 5*sinh(x). - Stefano Spezia, Aug 04 2024

A180138 Table, t, read by antidiagonals: t(b,e) is the smallest k such that k*b^e is a sum of two successive primes.

Original entry on oeis.org

5, 5, 4, 5, 4, 2, 5, 2, 2, 1, 5, 1, 7, 6, 7, 5, 2, 4, 2, 2, 4, 5, 6, 1, 10, 9, 10, 2, 5, 1, 18, 1, 2, 8, 20, 1, 5, 2, 2, 10, 4, 8, 2, 26, 9, 5, 3, 2, 15, 30, 12, 12, 25, 22, 15, 5, 18, 1, 20, 2, 18, 2, 12, 11, 10, 8, 5, 1, 6, 6, 22, 19, 4, 1, 36, 6, 16, 4, 5, 4, 1, 24, 6, 16, 6, 28, 4, 12, 10, 8, 2

Offset: 1

Zak Seidov & Robert G. Wilson v, Aug 15 2010

1st row: A180130, 2nd row: A180131, 3rd row: bisection of A180130, 4th row: A180132, 5th row: A180133, 6th row: A180134, 7th row: trisection of A180130, 8th row: bisection of A180131, 9th row: A179975, 10th row: A180135, 11th row: A180136 and 12th row: A180137; 1st column: A010716.
The k-th term == 1 10, 12, 24, 30, 32, 36, 58, 68, 74, 81, 105, 155, 278, 303, 315, 331, 419, 437, 439, 632, 638, 752, 857, 863, 906, 924, 950, ..., .
Increasing terms: {5, 6, 10, 20, 26, 72, 104, 118, 306, 320, 348, 572, 824, 828, 972, 1054, 1110, 1540, 5, 7, 10, 18, 20, 26, 30, 36, 52, 66, 72, 120, 132, 168, 266, 574, 640, 776, 1600, 1938, 2616, 3124, 3306, 4440, ...,
which occurs at the k-th term: 5, 6, 10, 20, 26, 72, 104, 118, 306, 320, 348, 572, 824, 828, 972, 1054, 1110, 1540, 5, 7, 10, 18, 20, 26, 30, 36, 52, 66, 72, 120, 132, 168, 266, 574, 640, 776, 1600, 1938, 2616, 3124, 3306, 4440, 1, 13, 25, 31, 35, 44, 50, 75, 114, 117, 119, 166, 187, 267, 289, 615, 1416, 1575, 2069, 3463, 4840, 5968, 7709, 9695, ..., .
Increasing terms by antidiagonals: t(2,0)=5, t(4,2)=t(2,4)=7, t(5,3)=t(3,5)=10, t(3,6)=20, t(3,7)=26, t(7,4)=30, t(5,8)=36, t(3,13)=72, t(7,12)=120, t(5,15)=132, t(11,13)=168, t(13,12)=266, t(17,19)=574, t(17,37)=640, t(23,34)=776, t(13,52)=1600, t(25,59)=1938, t(13,86)=2616. t(29,81)=3124, t(43,82)=3306, t(37,103)=4440..., .
Corresponding primes are twin primes for t(18,2), t(24,2), t(54,6), t(60,5), t(72,6), t(102,8), t(114,1), t=(126,1), ..., .

			.\e..0...1...2...3...4...5...6...7...8...9..10..11..12..13..14..15..16..17..18..19..20..21..22..23..24..25
.b\
.2...5...4...2...1...7...4...2...1...9..15...8...4...2...1..25..19..11..12...6...3..10...5..35..33..52..26
.3...5...4...2...6...2..10..20..26..22..10..16...8...8..72..24...8..18...6...2...6...2..10..20..20..22..20
.4...5...2...7...2...9...8...2..25..11...6..10..35..52..13..14..15..19..47..13..84..21..35...9..23..49..52
.5...5...1...4..10...2...8..12..12..36..12..28..66..30...6..18.132..36.108..34..14..48..60..12..22.150..30
.6...5...2...1...1...4..12...2...1...4...3...5...8...7..34...8..11..33..26..13...9..13..90..15..40..30...5
.7...5...6..18..10..30..18...4..28...4..30..30..60.120..38..12...6..52.120..70..10.102..60..70..10.186.174
.8...5...1...2..15...2..19...6...5..52..28..15..45..13..42..35..46..49..26..24...3..18..15..21..62..32...4
.9...5...2...2..20..22..16...8..24..18...2...2..20..22..52.104..84..38.102.100..30.192..46..22..84.176..30
10...5...3...1...6...6...6..14...6...9..19..21..21..42..93..21...6..11...2..12.111..37..39..63..38..42..24
11...5..18...6..24...6..32..40..26..20..94..50..26..10.168..30..18.196.126..70.166..30..54.130..26..50..10
12...5...1...1...2..18...8..13...6...2..11..11..39..20..12...1...8...9..31.182..24...2.126.128..66...9..86
13...5...4..24...4...8..22..40...4..14..16..28..10.266..40..20..46.112.156..12..20.228..26...2.220..60.140
...

Robert G. Wilson v, Table of n, a(n) for n = 1..10153

Cf. A180130, A180131, A180132, A180133, A180134, A179975, A180135, A180136, A180137.

t[b_, e_] := Block[{k = 1, hnp = b^e/2}, While[ h = k*hnp; PrimeQ@h || NextPrime[h, -1] + NextPrime@h != 2 h, k++ ]; k]; Table[ t[b - e, e], {b, 2, 14}, {e, 0, b - 2}] // Flatten
(* to find twins other than 2&3 *) gQ[b_, e_, k_] := Block[{h = k*b^e/2}, NextPrime@h - NextPrime[h, -1] < 3 ]; Do[ If[ gQ[b - e, e, k], Print[{b - e, e}]], {b, 2, 143}, {e, 0, b - 2}]

from sympy import isprime, nextprime, prevprime
def sum2succ(n):
  if n <= 5: return n == 5
  return not isprime(n//2) and n == prevprime(n//2) + nextprime(n//2)
def T(b, e):
  k, powb = 1, b**e
  while not sum2succ(k*powb): k += 1
  return k
def atodiag(maxd): # maxd antidiagonals
  return [T(b-e, e) for b in range(2, maxd+2) for e in range(b-1)]
print(atodiag(13)) # Michael S. Branicky, May 05 2021

cp's OEIS Frontend

A382106 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.

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A382107 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.

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A382686 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.

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A382687 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.

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A382688 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.

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A382689 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.

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A382690 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.

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A382105 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.

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A010703 Period 2: repeat (3,5).

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A180138 Table, t, read by antidiagonals: t(b,e) is the smallest k such that k*b^e is a sum of two successive primes.