A010863 -id:A010863 - cp's OEIS frontend

A176910 Decimal expansion of sqrt(145).

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

Offset: 2

Klaus Brockhaus, Apr 28 2010

Continued fraction expansion of sqrt(145) is 12 followed by A010863.

			sqrt(145) = 12.04159457879229548012...

Index entries for algebraic numbers, degree 2.

Cf. A002163 (decimal expansion of sqrt(5)), Cf. A010484 (decimal expansion of sqrt(29)), A176907 (decimal expansion of (9+sqrt(145))/16), A176908 (decimal expansion of (7+sqrt(145))/16), A010863 (all 24's sequence).

RealDigits[Sqrt[145],10,120][[1]] (* Harvey P. Dale, Oct 15 2023 *)

A157371 a(n) = (n+1)(9-9n+5*n^2-n^3).

Original entry on oeis.org

9, 8, 9, 0, -55, -216, -567, -1216, -2295, -3960, -6391, -9792, -14391, -20440, -28215, -38016, -50167, -65016, -82935, -104320, -129591, -159192, -193591, -233280, -278775, -330616, -389367, -455616, -529975, -613080, -705591, -808192, -921591, -1046520, -1183735, -1334016, -1498167

Offset: 0

Paul Curtz, Feb 28 2009

This is the fourth in a family of sequences that appear in columns on pages 36 and 56 of the reference: (i) sequence n+1, A000029, (ii) sequence (n+1)*(1-n), A147998 and (iii) (n+1)*(5-5*n+2*n^2), A152064.

First differences along columns shown on page 56 of the reference are columns of what is shown on page 36 of the reference. Example: the third column of page 56, A152064, has first differences which constitute the third column p page 36, A140811.

Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[(n+1)*(9-9*n+5*n^2-n^3): n in [0..40] ]; // Vincenzo Librandi, Jul 14 2011

LinearRecurrence[{5,-10,10,-5,1},{9,8,9,0,-55},40] (* or *) Table[(n+1)(9-9n+5n^2-n^3),{n,0,40}] (* or *) CoefficientList[ Series[ (55x^3- 59x^2+ 37x-9)/ (x-1)^5,{x,0,40}],x] (* Harvey P. Dale, Jul 13 2011 *)

a(n)=(n+1)*(9-9*n+5*n^2-n^3) \\ Charles R Greathouse IV, Oct 16 2015

First differences: a(n+1)-a(n) = -A141530(n).
Fourth differences: a(n+4)-4*a(n+3)+6*a(n+2)-4*a(n+1)+a(n) = -24 = -A010863(n).
From Harvey P. Dale, Jul 13 2011: (Start)
a(0)=9, a(1)=8, a(2)=9, a(3)=0, a(4)=-55, a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
G.f.: (9-37*x+59*x^2-55*x^3)/(1-x)^5. (End)
E.g.f.: (9 - x + x^2 - 2*x^3 - x^4)*exp(x). - G. C. Greubel, Feb 02 2018

Edited, extended by R. J. Mathar, Sep 25 2009

A165563 a(n) = 1 + 2n + n^2 + 2n^3 + n^4.

Original entry on oeis.org

1, 7, 41, 151, 409, 911, 1777, 3151, 5201, 8119, 12121, 17447, 24361, 33151, 44129, 57631, 74017, 93671, 117001, 144439, 176441, 213487, 256081, 304751, 360049, 422551, 492857, 571591, 659401, 756959, 864961, 984127, 1115201, 1258951, 1416169, 1587671

Offset: 0

Paul Curtz, Sep 22 2009

Also binomial transform of the quasi-finite sequence 1,6,28,48,24,0 (0 continued).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

[1 +2*n +n^2 +2*n^3 +n^4: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011

Table[1+2n+n^2+2n^3+n^4,{n,0,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,7,41,151,409},50] (* Harvey P. Dale, Nov 13 2021 *)

a(n)=1+2*n+n^2+2*n^3+n^4 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24 -> 4th differences are 24 = A010863(n).
G.f.: (-1 - 2*x - 16*x^2 - 6*x^3 + x^4)/(x-1)^5.

Edited and extended by R. J. Mathar, Sep 25 2009

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 1, 2, 6, 24, 1, 1, 1, 2, 6, 24, 120, 1, 1, 1, 2, 6, 24, 120, 720, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880

Offset: 1

Johannes W. Meijer, Oct 05 2009

			Triangle starts:
1,
1, 1,
1, 1, 1,
1, 1, 1, 2,
1, 1, 1, 2, 6,
1, 1, 1, 2, 6, 24,
1, 1, 1, 2, 6, 24, 120,
1, 1, 1, 2, 6, 24, 120, 720,
1, 1, 1, 2, 6, 24, 120, 720, 5040,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,
...

Cf. A138771, A165675 and A000142.
A000012 (3x), A007395, A010722, A010863 equal the first six left hand columns.
A159333 equals, for n=>-1, all right hand columns.
A067078 equals the row sums.

nmax:=11: for n from 1 to nmax do a(n,1):=1 od: for n from 2 to nmax do for m from 2 to n do a(n,m):=(m-2)! od: od: for n from 1 to nmax do seq(a(n,m),m=1..n) od;

a(n) = A138771(n)/A165675(n-1).

cp's OEIS Frontend

A176910 Decimal expansion of sqrt(145).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A157371 a(n) = (n+1)*(9-9*n+5*n^2-n^3).

Views

Author

Keywords

Comments

References

Links

Programs

Magma

Mathematica

PARI

Formula

Extensions

A165563 a(n) = 1 + 2*n + n^2 + 2*n^3 + n^4.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

Extensions

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A157371 a(n) = (n+1)(9-9n+5*n^2-n^3).

A165563 a(n) = 1 + 2n + n^2 + 2n^3 + n^4.