A082110 -id:A082110 - cp's OEIS frontend

A082114 Diagonal sums of number array A082110.

1, 2, 9, 32, 89, 210, 441, 848, 1521, 2578, 4169, 6480, 9737, 14210, 20217, 28128, 38369, 51426, 67849, 88256, 113337, 143858, 180665, 224688, 276945, 338546, 410697, 494704, 591977, 704034, 832505, 979136, 1145793, 1334466, 1547273

Offset: 0

Paul Barry, Apr 04 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A082045, A082107, A082110.

[(n+1)*(n^4-n^3+26*n^2-26*n+30)/30: n in [0..50]]; // G. C. Greubel, Dec 22 2022

LinearRecurrence[{6,-15,20,-15,6,-1}, {1,2,9,32,89,210}, 51] (* G. C. Greubel, Dec 22 2022 *)

[(n+1)*(n^4-n^3+26*n^2-26*n+30)/30 for n in range(51)] # G. C. Greubel, Dec 22 2022

a(n) = (n+1)*(n^4 - n^3 + 26*n^2 - 26*n + 30)/30.
From G. C. Greubel, Dec 22 2022: (Start)
G.f.: (1 - 4*x + 12*x^2 - 12*x^3 + 7*x^4)/(1-x)^6.
E.g.f.: (1/30)*(30 + 30*x + 90*x^2 + 50*x^3 + 10*x^4 + x^5)*exp(x). (End)

A082111 a(n) = n^2 + 5*n + 1.

Original entry on oeis.org

1, 7, 15, 25, 37, 51, 67, 85, 105, 127, 151, 177, 205, 235, 267, 301, 337, 375, 415, 457, 501, 547, 595, 645, 697, 751, 807, 865, 925, 987, 1051, 1117, 1185, 1255, 1327, 1401, 1477, 1555, 1635, 1717, 1801, 1887, 1975, 2065, 2157, 2251, 2347, 2445, 2545, 2647

Offset: 0

Paul Barry, Apr 04 2003

From Gary W. Adamson, Jul 29 2009: (Start)
Let (a,b) = roots to x^2 - 5*x + 1 = 0 = 4.79128... and 0.208712...
Then a(n) = (n + a) * (n + b). Example: a(5) = 51 = (5 + 4.79128...) * (5 + 0.208712...) (End)
For n > 0: a(n) = A176271(n+2,n). - Reinhard Zumkeller, Apr 13 2010
a(n-2) = n*(n+1) - 5, n >= 0, with a(-2) = -5 and a(-1) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 21 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
Numbers m > 0 such that 4m+21 is a square. - Bruce J. Nicholson, Jul 19 2017
Numbers represented as 151 in number base B. If 'digits' from B upwards are allowed then 151(2)=15, 151(3)=25, 151(4)=37, 151(5)=51 also. - Ron Knott, Nov 14 2017
If A and B are sequences satisfying the recurrence t(n) = 5*t(n-1) - t(n-2) with initial values A(0) = 1, A(1) = n+5 and B(0) = -1, B(1) = n, then a(n) = A(i)^2 - A(i-1)*A(i+1) = B(j)^2 - B(j-1)*B(j+1) for i, j > 0. - Klaus Purath, Oct 18 2020
The prime terms in this sequence are listed in A089376. The prime factors are given in A038893. With the exception of 3 and 7, each prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -5 (mod p). - Klaus Purath, Nov 24 2020

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First row of A082110.

Cf. A002522, A014209, A028387, A028872, A338041.

Table[n^2 + 5*n + 1,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
LinearRecurrence[{3,-3,1},{1,7,15},80] (* Harvey P. Dale, Apr 22 2012 *)

a(n)=n^2+5*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n + a(n-1) + 4 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=7, a(2)=15. - Harvey P. Dale, Apr 22 2012
Sum_{n>=0} 1/a(n) = 8/15 + Pi*tan(sqrt(21)*Pi/2)/sqrt(21) = 1.424563592286456286... . - Vaclav Kotesovec, Apr 10 2016
From G. C. Greubel, Jul 19 2017: (Start)
G.f.: (1 + 4*x - 3*x^2)/(1 - x)^3.
E.g.f.: (x^2 + 6*x + 1)*exp(x). (End)
a(n) = A014209(n+1) - 2 = A338041(2*n+1). - Hugo Pfoertner, Oct 08 2020
a(n) = A249547(n+1) - A024206(n-4), n >= 5. - Klaus Purath, Nov 24 2020

New title (using given formula) from Hugo Pfoertner, Oct 08 2020

A082107 Diagonal sums of number array A082105.

Original entry on oeis.org

1, 2, 8, 28, 79, 190, 406, 792, 1437, 2458, 4004, 6260, 9451, 13846, 19762, 27568, 37689, 50610, 66880, 87116, 112007, 142318, 178894, 222664, 274645, 335946, 407772, 491428, 588323, 699974, 828010, 974176, 1140337, 1328482, 1540728

Offset: 0

Paul Barry, Apr 03 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A082043, A082045, A082046, A082105, A082106, A082109, A082110.

[(n^5+20*n^3+9*n+30)/30: n in [0..50]]; // G. C. Greubel, Dec 22 2022

LinearRecurrence[{6,-15,20,-15,6,-1}, {1,2,8,28,79,190}, 51] (* G. C. Greubel, Dec 22 2022 *)

[(n^5+20*n^3+9*n+30)/30 for n in range(51)] # G. C. Greubel, Dec 22 2022

a(n) = (n^5 + 20*n^3 + 9*n + 30)/30.
G.f.: (1-4*x+11*x^2-10*x^3+6*x^4)/(1-x)^6 . - R. J. Mathar, Mar 27 2019
E.g.f.: (1/30)*(30 +30*x +75*x^2 +45*x^3 +10*x^4 +x^5)*exp(x). - G. C. Greubel, Dec 22 2022

A082112 a(n) = 4n^2 + 10n + 1.

Original entry on oeis.org

1, 15, 37, 67, 105, 151, 205, 267, 337, 415, 501, 595, 697, 807, 925, 1051, 1185, 1327, 1477, 1635, 1801, 1975, 2157, 2347, 2545, 2751, 2965, 3187, 3417, 3655, 3901, 4155, 4417, 4687, 4965, 5251, 5545, 5847, 6157, 6475, 6801, 7135, 7477, 7827, 8185, 8551

Offset: 0

Paul Barry, Apr 04 2003

A row of number array A082110.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A082108, A082109, A082110.

[4*n^2 + 10*n + 1 : n in [0..50]]; // Wesley Ivan Hurt, Dec 22 2021

Table[n +(n+1)^2 -4, {n,1,200, 2}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
LinearRecurrence[{3,-3,1},{1,15,37},50] (* Harvey P. Dale, Dec 18 2014 *)

a(n)=4*n^2+10*n+1 \\ Charles R Greathouse IV, Jun 17 2017

[4*n^2+10*n+1 for n in range(51)] # G. C. Greubel, Dec 22 2022

a(n) = a(n-1) + 8*n + 6 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: (1+12*x-5*x^2) / (1-x)^3. - R. J. Mathar, Dec 03 2014
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Wesley Ivan Hurt, Dec 22 2021
E.g.f.: (1 + 14*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022

A082113 a(n) = n^4 + 5*n^2 + 1.

Original entry on oeis.org

1, 7, 37, 127, 337, 751, 1477, 2647, 4417, 6967, 10501, 15247, 21457, 29407, 39397, 51751, 66817, 84967, 106597, 132127, 162001, 196687, 236677, 282487, 334657, 393751, 460357, 535087, 618577, 711487, 814501, 928327, 1053697, 1191367

Offset: 0

Paul Barry, Apr 04 2003

Main diagonal of number array A082110.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A082044, A082047, A082106.

[n^4+5*n^2+1: n in [0..40]]; // G. C. Greubel, Dec 22 2022

Table[n^4+5n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,7,37,127,337},40] (* Harvey P. Dale, May 16 2019 *)

[n^4+5*n^2+1 for n in range(41)] # G. C. Greubel, Dec 22 2022

a(n) = n^4 + 5*n^2 + 1.
G.f.: (1+2*x+12*x^2+2*x^3+7*x^4) / (1-x)^5. - R. J. Mathar, Dec 03 2014
E.g.f.: (1 + 6*x + 12*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 22 2022

cp's OEIS Frontend

A082114 Diagonal sums of number array A082110.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A082111 a(n) = n^2 + 5*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A082107 Diagonal sums of number array A082105.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A082112 a(n) = 4*n^2 + 10*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A082113 a(n) = n^4 + 5*n^2 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A082112 a(n) = 4n^2 + 10n + 1.