A082108 -id:A082108 - cp's OEIS frontend

A082109 Third row of number array A082105.

1, 13, 33, 61, 97, 141, 193, 253, 321, 397, 481, 573, 673, 781, 897, 1021, 1153, 1293, 1441, 1597, 1761, 1933, 2113, 2301, 2497, 2701, 2913, 3133, 3361, 3597, 3841, 4093, 4353, 4621, 4897, 5181, 5473, 5773, 6081, 6397, 6721, 7053, 7393, 7741, 8097, 8461

Offset: 0

Paul Barry, Apr 03 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Russ Cox et al., incorrect A082109 comment about A000217, SeqFan, 2024.
Takao Komatsu, Ritika Goel, and Neha Gupta, The Frobenius number for the triple of the 2-step star numbers, arXiv:2409.14788 [math.CO], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A082105, A082108, A000217.

Column 2 of array A188646.

[4*n^2+8*n+1: n in [0..60]]; // G. C. Greubel, Dec 22 2022

LinearRecurrence[{3,-3,1}, {1,13,33}, 51] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)

a(n)=4*n^2+8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

[4*n^2+8*n+1 for n in range(61)] # G. C. Greubel, Dec 22 2022

a(n) = 4*n^2 + 8*n + 1.
a(n) = a(n-1) + 8*n + 4, with a(0)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: (1 + 10*x - 3*x^2)/(1-x)^3. - Bruno Berselli, Apr 18 2011
E.g.f.: (1 + 12*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022
From Amiram Eldar, Jan 18 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/6 - cot(sqrt(3)*Pi/2)*sqrt(3)*Pi/12.
Sum_{n>=0} (-1)^n/a(n) = cosec(sqrt(3)*Pi/2)*sqrt(3)*Pi/12 - 1/6. (End)

A082046 Square array, A(n, k) = (kn)^2 + 3k*n + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 29, 19, 1, 1, 29, 55, 55, 29, 1, 1, 41, 89, 109, 89, 41, 1, 1, 55, 131, 181, 181, 131, 55, 1, 1, 71, 181, 271, 305, 271, 181, 71, 1, 1, 89, 239, 379, 461, 461, 379, 239, 89, 1, 1, 109, 305, 505, 649, 701, 649, 505, 305, 109, 1

Offset: 0

Paul Barry, Apr 03 2003

			Array, A(n, k), begins as:
  1,  1,   1,   1,   1,    1,    1,    1, ... A000012;
  1,  5,  11,  19,  29,   41,   55,   71, ... A028387;
  1, 11,  29,  55,  89,  131,  181,  239, ... A082108;
  1, 19,  55, 109, 181,  271,  379,  505, ... A069131;
  1, 29,  89, 181, 305,  461,  649,  869, ... ;
  1, 41, 131, 271, 461,  701,  991, 1331, ... ;
  1, 55, 181, 379, 649,  991, 1405, 1891, ... ;
  1, 71, 239, 505, 869, 1331, 1891, 2549, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,  1;
  1,  5,   1;
  1, 11,  11,   1;
  1, 19,  29,  19,   1;
  1, 29,  55,  55,  29,   1;
  1, 41,  89, 109,  89,  41,   1;
  1, 55, 131, 181, 181, 131,  55,  1;
  1, 71, 181, 271, 305, 271, 181, 71,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A028387, A057721, A069131, A072025, A082039, A082043, A082047, A082105, A082108.

[(k*(n-k))^2 + 3*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022

T[n_, k_]:= (k*(n-k))^2 + 3*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)

def A082046(n,k): return (k*(n-k))^2 + 3*(k*(n-k)) + 1
flatten([[A082046(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022

A(n, k) = (k*n)^2 + 3*k*n + 1 (square array).
A(k, n) = A(n, k).
A(n, n) = T(2*n, n) = A057721(n).
A(n, n+1) = A072025(n).
T(n, k) = (k*(n-k))^2 + 3*k*(n-k) + 1 (antidiagonals).
Sum_{k=0..n} T(n, k) = A082047(n) (antidiagonal sums).
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*(1 - 2*n).
T(2*n+1, n-1) = T(2*n-1, n-1) = A072025(n-1). (End)

A350470 Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 1, 7, 9, 11, 1, 1, 1, 9, 13, 29, 21, 1, 1, 1, 11, 17, 55, 65, 43, 1, 1, 1, 13, 21, 89, 133, 181, 85, 1, 1, 1, 15, 25, 131, 225, 463, 441, 171, 1, 1, 1, 17, 29, 181, 341, 937, 1261, 1165, 341, 1

Offset: 0

Peter Luschny, Mar 19 2022

			Array starts:
n\k 0, 1,  2,  3,   4,    5,    6,     7,      8,      9, ...
---------------------------------------------------------------------
[0] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ... A000012
[1] 1, 1,  3,  5,  11,   21,   43,    85,    171,    341, ... A001045
[2] 1, 1,  5,  9,  29,   65,  181,   441,   1165,   2929, ... A006131
[3] 1, 1,  7, 13,  55,  133,  463,  1261,   4039,  11605, ... A015441
[4] 1, 1,  9, 17,  89,  225,  937,  2737,  10233,  32129, ... A015443
[5] 1, 1, 11, 21, 131,  341, 1651,  5061,  21571,  72181, ... A015446
[6] 1, 1, 13, 25, 181,  481, 2653,  8425,  40261, 141361, ... A053404
[7] 1, 1, 15, 29, 239,  645, 3991, 13021,  68895, 251189, ... A350468
[8] 1, 1, 17, 33, 305,  833, 5713, 19041, 110449, 415105, ... A168579
[9] 1, 1, 19, 37, 379, 1045, 7867, 26677, 168283, 648469, ... A350469
      A005408 | A082108 |
           A016813   A014641

Rows: A000012, A001045, A006131, A015441, A015443, A015446, A053404, A350468, A168579, A350469.
Columns: A000012, A005408, A016813, A082108, A014641.
Cf. A350467 (main diagonal), A352361 (Fibonacci polynomials), A352362 (Lucas polynomials).

J := (n, x) -> add(2^k*binomial(n - k, k)*x^k, k = 0..n):
seq(seq(J(k, n-k), k = 0..n), n = 0..10);

T[n_, k_] := Hypergeometric2F1[(1 - k)/2, -k/2, -k, -8 n];
Table[T[n, k], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[n_, k_] := With[{s = Sqrt[8*n+1]}, ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s)];
Table[Simplify[T[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm

T(n, k) = ([1, 2; k, 0]^n)[1, 1] ;
export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))

T(n, k) = Sum_{j=0..k} binomial(k - j, j)*(2*n)^j.
T(n, k) = ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s) where s = sqrt(8*n + 1).
T(n, k) = [x^k] (1 / (1 - x - 2*n*x^2)).
T(n, k) = hypergeom([1/2 - k/2, -k/2], [-k], -8*n).

A206531 a(n) = (2(n+1)(2n+1)-1)a(n-1) + 2n(2n-1)a(n-2), a(0)=0, a(1)=2.

Original entry on oeis.org

0, 2, 58, 3250, 292498, 38609738, 7026972314, 1686473355362, 516060846740770, 196103121761492602, 90599642253809582122, 50011002524102889331346, 32507151640666878065374898, 24575406640344159817423422890

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

The numerators of the fractions limiting to the value of A206533.

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

G. C. Greubel, Table of n, a(n) for n = 0..220

Cf. A002939, A082108, A206532, A206533.

[n le 2 select 2*(n-1) else (2*n*(2*n-1)-1)*Self(n-1) + 2*(n-1)*(2*n-3)*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 21 2022

RecurrenceTable[{a[n]==(2(n+1)(2n+1)-1)a[n-1]+2n(2n-1)a[n-2],a[0]==0,a[1]==2},a,{n,15}]

@CachedFunction
def a(n): # a = A206531
    if (n<2): return 2*n
    else: return (2*(n+1)*(2*n+1)-1)*a(n-1) + 2*n*(2*n-1)*a(n-2)
[a(n) for n in range(31)] # G. C. Greubel, Dec 21 2022

a(n) = A082108(n)*a(n-1) + A002939(n)*a(n-2), a(0) = 0, a(1) = 2.

A206532 a(n) = (2(n+1)(2n+1)-1) * a(n-1) + 2n(2n-1) * a(n-2), a(0) = 1, a(1) = 11.

Original entry on oeis.org

1, 11, 331, 18535, 1668151, 220195931, 40075659443, 9618158266319, 2943156429493615, 1118399443207573699, 516700542761899048939, 285218699604568275014327, 185392154742969378759312551, 140156468985684850342040288555

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

The denominators of the fractions limiting to the value of A206533.

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc.,1966.

Robert Israel, Table of n, a(n) for n = 0..223

Cf. A002939, A082108, A206531, A206533.

f:= gfun:-rectoproc({a(n) = (2*(n+1)*(2*n+1)-1) * a(n-1) + 2*n*(2*n-1) * a(n-2), a(0) = 1, a(1) = 11},a(n),remember):
map(f, [$0..40]); # Robert Israel, Sep 16 2018

RecurrenceTable[{a[n]==(2(n+1)(2n+1)-1)a[n-1]+2n(2n-1)a[n-2],a[0]==1,a[1]==11},a,{n,15}]

a(n) = A082108*a(n-1) + A002939*a(n-2), a(0) = 1, a(1) = 11.

a(n) = -4*n*(-1)^n*(n+1)*LommelS1(2*n+1/2, 3/2, 1)-2*(-1)^n*(n+1)*LommelS1(2*n+3/2, 1/2, 1)+(1-cos(1))*(2*n+2)!+(-1)^n. - Robert Israel, Sep 16 2018

A206533 Decimal expansion of 1/(1-cos(1)).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Feb 11 2012

The value of the fractional limit of the numerators(A206531+2*A206532) and the denominators(A206532).

Abs(A206532/(1-cos(1)) - (A206531+2*A206532)) -> 0.

			2.17534264967002141077678675965606997584844...

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Cf. A002939, A082108, A206531, A206532, A049470, A371936.

Mathematica
```
RealDigits[N[1/(1-Cos[1]), 150]][[1]]
```

1/(1 - cos(1)) \\ Stefano Spezia, Apr 21 2025

Equals 1/(1-A049470).
A206531/A206532+2 -> 1/(1-A049470).
Equals 1/A371936. - Hugo Pfoertner, Apr 21 2025

Incorrect a(86)=9 removed by Georg Fischer, Apr 04 2020

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

A102094 a(n) = (2n-1)(2*n+1)^2.

Original entry on oeis.org

9, 75, 245, 567, 1089, 1859, 2925, 4335, 6137, 8379, 11109, 14375, 18225, 22707, 27869, 33759, 40425, 47915, 56277, 65559, 75809, 87075, 99405, 112847, 127449, 143259, 160325, 178695, 198417, 219539, 242109, 266175, 291785, 318987, 347829, 378359, 410625

Offset: 1

Gerald McGarvey, Feb 13 2005

Numbers which are both the sum of 2n+1 consecutive odd integers and, after skipping one odd integer, the sum of the 2n-1 immediately higher consecutive odd integers. See A082108(n-1) for the smallest of the 2n+1 odd integers, and A054569(n+1) for the skipped number. Odd integer counterpart to A059270. - Charlie Marion, Apr 30 2020

G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, p. 123.
J. Ewell, An Eulerian Method for Representing Pi^2 by Series, The Rocky Mountain Journal of Mathematics 1992 v.22, pp. 165-168.

G. C. Greubel, Table of n, a(n) for n = 1..1000
J. Ewell, An Eulerian Method for Representing Pi^2 by Series, The Rocky Mountain Journal of Mathematics 1992 v.22, pp. 165-168.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002388.

List([1..40], n-> (2*n-1)*(2*n+1)^2); # G. C. Greubel, Oct 27 2019

[(2*n-1)*(2*n+1)^2: n in [1..40]]; // G. C. Greubel, Oct 27 2019

seq((2*n-1)*(2*n+1)^2, n=1..40); # G. C. Greubel, Oct 27 2019

Table[(2n-1)(2n+1)^2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{9,75,245,567},40] (* Harvey P. Dale, Jul 24 2012 *)

vector(40, n, (2*n-1)*(2*n+1)^2) \\ G. C. Greubel, Oct 27 2019

[(2*n-1)*(2*n+1)^2 for n in (1..40)] # G. C. Greubel, Oct 27 2019

Sum_{n>=1} 1/a(n) = (12 - Pi^2)/16.
Sum_{n>=1} n/a(n) = (Pi^2 - 4)/32. - Sign flipped by Bernard Schott, May 06 2020
From Harvey P. Dale, Jul 24 2012: (Start)
a(1)=9, a(2)=75, a(3)=245, a(4)=567, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (9 + 39*x - x^2 + x^3)/(1-x)^4. (End)
E.g.f.: 1 + (-1 + 10*x + 28*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Oct 27 2019

More terms from Harvey P. Dale, Jul 24 2012

A324017 Square array A(m,n) (m>=1, n>=1) read by antidiagonals: A(m,n) = (2n - 1)^^m mod (2n)^m (see Comments for definition of ^^).

Original entry on oeis.org

1, 3, 1, 5, 11, 1, 7, 29, 59, 1, 9, 55, 29, 59, 1, 11, 89, 119, 1109, 827, 1, 13, 131, 289, 3703, 3701, 2875, 1, 15, 181, 563, 5289, 7799, 34805, 15163, 1, 17, 239, 965, 16115, 45289, 138871, 128117, 31547, 1, 19, 305, 1519, 25661, 57587, 745289, 1711735, 687989, 97083, 1

Offset: 1

Davis Smith, Mar 28 2019

Tetration (x^^n) is defined as x^^0 = 1 and x^^n = x^(x^^(n - 1)). Another way to put this is that x^^n = x^x^x^...x with n x's.
Conjecture: For any three integers (greater than 1), m, n, and k, such that (2*n - 1)^^m == k (mod (2*n)^m), (2*n - 1)^k == k (mod (2*n)^m). For example, 5^^2 == 29 (mod 6^2) and 5^29 == 29 (mod 6^2).
Conjecture: For n > 1 and m >= 2, floor(((2*n - 1)^^m)/(2*n)) ==  2*(n - 1) (mod 2*n). For example, floor((13^^3)/14) == 12 (mod 14) and floor((15^^4)/16) == 14 (mod 16).
Conjecture: For m > 1, where (2*n - 1)^^m == j (mod (2*n)^(m + 1)), A(m + 1,n) = j. For example, A(6,3) = 563 and A(6,4) = 16115; 11^^3 == 563 (mod 12^3) and 11^^3 == 16115 (mod 12^4).

			Square array A(m,n) begins:
  \n  1     2      3       4        5          6         7          8 ...
  m\
   1| 1     3      5       7        9         11        13         15 ...
   2| 1    11     29      55       89        131       181        239 ...
   3| 1    59     29     119      289        563       965       1519 ...
   4| 1    59   1109    3703     5289      16115     25661      13807 ...
   5| 1   827   3701    7799    45289      57587    332989     669167 ...
   6| 1  2875  34805  138871   745289    1799411   4635581     669167 ...
   7| 1 15163 128117 1711735  2745289   25687283  49812797   67778031 ...
   8| 1 31547 687989 8003191 92745289  419837171 155226301 3557438959 ...
.
Examples of columns in this array:
A(m,1) = A000012(m - 1).
A(m,5) = A306686(m) with a note about how this sequence repeats terms rather than skipping.
Examples of rows in this array:
A(1,n) = A005408(n - 1).
A(2,n) = A082108(n - 1).

Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.
Eric Weisstein's World of Mathematics,Power Tower.
Wikipedia, Tetration.

Cf. A000012, A005408, A082108, A306686.

tetrmod(b,n,m)=my(t=b);i=0;while(i<=n, i++&&if(i>1, t=lift(Mod(b,m)^t), t)); t
tetrmatrix(lim)= matrix(lim,lim,x,y,tetrmod((2*y)-1,x,(2*y)^x))

cp's OEIS Frontend

A082109 Third row of number array A082105.

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A082046 Square array, A(n, k) = (k*n)^2 + 3*k*n + 1, read by antidiagonals.

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A350470 Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials.

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A206531 a(n) = (2*(n+1)*(2*n+1)-1)*a(n-1) + 2*n*(2*n-1)*a(n-2), a(0)=0, a(1)=2.

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A206532 a(n) = (2(n+1)(2n+1)-1) * a(n-1) + 2n(2n-1) * a(n-2), a(0) = 1, a(1) = 11.

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A206533 Decimal expansion of 1/(1-cos(1)).

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

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A082046 Square array, A(n, k) = (kn)^2 + 3k*n + 1, read by antidiagonals.

A206531 a(n) = (2(n+1)(2n+1)-1)a(n-1) + 2n(2n-1)a(n-2), a(0)=0, a(1)=2.

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

A102094 a(n) = (2n-1)(2*n+1)^2.

A324017 Square array A(m,n) (m>=1, n>=1) read by antidiagonals: A(m,n) = (2n - 1)^^m mod (2n)^m (see Comments for definition of ^^).