keyword:less - cp's OEIS frontend

A141270 (0, 1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, ...) transform in (01, 23, 22, 52, 37, 23, 32, 25, 112, 2*3, ...).

0, 6, 4, 10, 21, 6, 6, 10, 22, 6, 26, 21, 10, 68, 6, 38, 4, 15, 14, 253, 6, 15, 4, 39, 6, 14, 58, 15, 62, 15, 22, 85, 14, 6, 74, 38, 39, 6, 205, 6, 301, 4, 33, 10, 46, 94, 12, 14, 10, 6, 34, 26, 106, 9, 55, 6, 21, 38, 1711, 4, 15, 122, 93, 14, 12, 65, 6, 737, 4, 51, 46, 35, 142, 9

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2008

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A136734, A141269.

Map[Times @@ # &, Partition[Flatten[{0}~Join~Array[DeleteCases[Flatten@ FactorInteger[#], 1] &, 72] /. {} -> {1}], 2, 2]] (* Michael De Vlieger, Oct 20 2021 *)

A 65 replaced with 85, a 54 with 74, etc. by R. J. Mathar, Feb 21 2009

A153521 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1), read by rows.

Original entry on oeis.org

2, 11, 11, 2, 238, 2, 2, 1329, 1329, 2, 2, 1353, 5276, 1353, 2, 2, 1377, 21248, 21248, 1377, 2, 2, 1401, 37508, 100532, 37508, 1401, 2, 2, 1425, 54056, 371768, 371768, 54056, 1425, 2, 2, 1449, 70892, 838412, 1849388, 838412, 70892, 1449, 2, 2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 2

Offset: 1

Roger L. Bagula, Dec 28 2008

			Triangle begins as:
   2;
  11,   11;
   2,  238,     2;
   2, 1329,  1329,       2;
   2, 1353,  5276,    1353,       2;
   2, 1377, 21248,   21248,    1377,       2;
   2, 1401, 37508,  100532,   37508,    1401,       2;
   2, 1425, 54056,  371768,  371768,   54056,    1425,     2;
   2, 1449, 70892,  838412, 1849388,  838412,   70892,  1449,    2;
   2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 2;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), this sequence (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).

Cf. A151617 (row sums).

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,p,q,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j);
  end if; return T;
end function;
[T(n,k,0,1,5): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021

T[n_, k_, p_, q_, j_]:= T[n,k,p,q,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]];
Table[T[n,k,0,1,5], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,p,q,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
flatten([[T(n,k,0,1,5) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 2021

T(n, k)= T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1).
From G. C. Greubel, Mar 04 2021: (Start)
T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (0,1,5).
Sum_{k=1..n} T(n,k,p,q,j) = 2*(prime(j)-3)*[n=1] -2*prime(j)*(prime(j)-3)*[n=2] +2*prime(j)^2*(i*sqrt(prime(j)))^(n-3)*(ChebyshevU(n-3, -i/Sqrt(prime(j))) -((prime(j) -2)*i/sqrt(prime(j)))*ChebyshevU(n-4, -i/sqrt(prime(j)))) for (p,q,j)=(0,1,5) = A151617(n).
Row sums satisfy the recurrence relation S(n) = 2*S(n-1) + prime(j)*S(n-2), for n > 4, with S(1) = 2, S(2) = 2*prime(j), S(3) = 2*prime(j)^2, S(4) = 2*prime(j)^3 for j=5. (End)

Edited by G. C. Greubel, Mar 04 2021

A171386 The characteristic function of 2 and 3: 1 if n is prime such that either n-1 or n+1 is prime, else 0.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Dec 07 2009

A181354(n) + A181376(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + a(n) = A052268(n).

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

Cf. A000007, A010051, A063524.

Join[{0, 1, 1},LinearRecurrence[{1},{0},102]] (* Ray Chandler, Sep 23 2015 *)

a(n)=n==2||n==3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A130130(n) - A130130(n-1), for n>0.

Edited by Charles R Greathouse IV, Mar 23 2010

A172332 Floor(n*(sqrt(13)+sqrt(5))).

Original entry on oeis.org

0, 5, 11, 17, 23, 29, 35, 40, 46, 52, 58, 64, 70, 75, 81, 87, 93, 99, 105, 110, 116, 122, 128, 134, 140, 146, 151, 157, 163, 169, 175, 181, 186, 192, 198, 204, 210, 216, 221, 227, 233, 239, 245, 251, 257, 262, 268, 274, 280, 286, 292

Offset: 0

Vincenzo Librandi, Feb 01 2010

Also integer part of n*5.8416192529..., where the constant is the largest root of x^4 -36*x^2 +64.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A110117, A172323-A172332, A172334, A172336-A172338.

[Floor(n*(Sqrt(13)+Sqrt(5))): n in [0..60]];

With[{c = Sqrt[13] + Sqrt[5]}, Floor[c Range[0, 70]]] (* Vincenzo Librandi, Aug 01 2013 *)

A172334 Floor(n*(sqrt(13)+sqrt(3))).

Original entry on oeis.org

0, 5, 10, 16, 21, 26, 32, 37, 42, 48, 53, 58, 64, 69, 74, 80, 85, 90, 96, 101, 106, 112, 117, 122, 128, 133, 138, 144, 149, 154, 160, 165, 170, 176, 181, 186, 192, 197, 202, 208, 213, 218, 224, 229, 234, 240, 245, 250, 256, 261, 266

Offset: 0

Vincenzo Librandi, Feb 01 2010

Also integer part of n*5.3376020830..., where the constant is the largest root of x^4 -32*x^2 +100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A110117, A172323-A172332, A172336-A172338.

[ Floor(n*(Sqrt(13)+Sqrt(3))): n in [0..60] ];

With[{c = Sqrt[13] + Sqrt[3]}, Table[Floor[c n], {n, 0, 50}]] (* Harvey P. Dale, Apr 25 2011 *)

A172336 a(n) = floor(n*(sqrt(13)+sqrt(2))).

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 256, 261, 266, 271, 276

Offset: 0

Vincenzo Librandi, Feb 01 2010

a(n) = integer part of n*(sqrt(13)+sqrt(2)), where the constant is the largest root of x^4 -30*x^2 +121.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A110117, A172323-A172332, A172334, A172337, A172338.

[Floor(n*(Sqrt(13)+Sqrt(2))): n in [0..60]];

With[{c = Sqrt[13] + Sqrt[2]}, Floor[c Range[0, 70]]] (* Vincenzo Librandi, Aug 01 2013 *)

for(n=0,50, print1(floor(n*(sqrt(13)+sqrt(2))), ", ")) \\ G. C. Greubel, Jul 05 2017

A172338 a(n) = floor(n*(sqrt(5)+sqrt(3))).

Original entry on oeis.org

0, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218

Offset: 0

Vincenzo Librandi, Feb 01 2010

sqrt(5)+sqrt(3) = 3.96811878506867... is the largest root of x^4 - 16*x^2 + 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Beatty sequences

Cf. A110117, A172323-A172332, A172334, A172336, A172337.

[Floor(n*Sqrt(5)+Sqrt(3)): n in [0..60] ];

With[{c = Sqrt[5] + Sqrt[3]}, Floor[c Range[0,60]]] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=sqrtint(sqrtint(60*n^4)+8*n^2) \\ Charles R Greathouse IV, Jan 24 2022

A131713 Period 3: repeat [1, -2, 1].

Original entry on oeis.org

1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1

Offset: 0

Paul Curtz, Sep 14 2007

Second differences of A131534. Binomial transform of 1, -3, 6, -9, 9, 0, ..., A057083 signed.

Nonsimple continued fraction expansion of sqrt(2)-1 = 0.414213562... - R. J. Mathar, Mar 08 2012

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

Cf. A057083, A061347, A131534.

&cat [[1, -2, 1]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([1, -2, 1]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

f[n_] := If[ Mod[n, 3] == 1, -2, 1]; Array[f, 105, 0]
CoefficientList[Series[(1 - x)/(1 + x + x^2), {x, 0, 104}], x]
PadRight[{}, 120, {1,-2,1}] (* Harvey P. Dale, Jan 25 2014 *)

a(n)=[1,-2,1][1+n%3] \\ Jaume Oliver Lafont, Mar 24 2009

a(n)=1-3*(n%3==1) \\ Jaume Oliver Lafont, Mar 24 2009

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^n) = 0^n, b(3^n) = 3 * 0^n - 2, b(p^n) = 1 if p > 3. - Michael Somos, Jan 02 2011
G.f.: (1-x)/(x^2+x+1). - R. J. Mathar, Nov 14 2007
a(n) = 2*cos((2n+1)*Pi/3). - Jaume Oliver Lafont, Nov 23 2008
a(n) = A117188(2*n). - R. J. Mathar, Jun 13 2011
a(n) + a(n-1) + a(n-2) = 0 for n>1, a(n) = a(n-3) for n>2. - Wesley Ivan Hurt, Jul 01 2016
a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*(-3)^k. - Peter Bala, Feb 06 2019
E.g.f.: 2*exp(-x/2)*cos(sqrt(3)*x/2 + Pi/3). - Fabian Pereyra, Oct 17 2024

Corrected and extended by Michael Somos, Jan 02 2011

A152811 a(n) = 2(n^2 + 2n - 2).

Original entry on oeis.org

2, 12, 26, 44, 66, 92, 122, 156, 194, 236, 282, 332, 386, 444, 506, 572, 642, 716, 794, 876, 962, 1052, 1146, 1244, 1346, 1452, 1562, 1676, 1794, 1916, 2042, 2172, 2306, 2444, 2586, 2732, 2882, 3036, 3194, 3356, 3522, 3692, 3866, 4044, 4226, 4412, 4602, 4796, 4994

Offset: 1

Vincenzo Librandi, Dec 17 2008

Positive numbers k such that 2*k + 12 is a square. [Comment simplified by Zak Seidov, Jan 14 2009]
Sequence gives positive x values of solutions (x,y) to the Diophantine equation 2*x^3 + 12*x^2 = y^2. Corresponding y values are 8*A154560. There are three other solutions: (0,0), (-4,8) and (-6,0).
From a(2) onwards, third subdiagonal of triangle defined in A144562.
Also, nonnegative numbers of the form (m+sqrt(-3))^2 + (m-sqrt(-3))^2. - Bruno Berselli, Mar 13 2015
a(n-1) is the maximum Zagreb index over all maximal 2-degenerate graphs with n vertices.  The extremal graphs are 2-stars, so the bound also applies to 2-trees.  (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024

			a(4) = 2*(4^2 + 2*4 - 2) = 44 = 2*22 = 2*A028872(5); 2*44^3 + 12*44^2 = 193600 = 440^2 is a square.
The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs, Vol. 0(1) (2024), Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin., Vol. 89(1) (2024), pp. 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J. Comb. Optim., Vol. 27 (2014), pp. 271-291.
I. Gutman and K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., No. 50 (2004), pp. 83-92.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028872 (n^2-3), A154560 ((n+3)^2*n/2+1), A144562 (triangle T(m,n) = 2m*n+m+n-1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).

Magma
```
[ 2*(n^2+2*n-2) : n in [1..47] ];
```

Table[2*n*(n + 2) - 4, {n, 50}] (* Paolo Xausa, Aug 08 2024 *)

{m=4700; for(n=1, m, if(issquare(2*n^3+12*n^2), print1(n, ",")))}

G.f.: 2*x*(1 + 3*x - 2*x^2)/(1-x)^3. [corrected by Elmo R. Oliveira, Nov 17 2024]
a(n) = 2*A028872(n+1).
a(n) = a(n-1) + 4*n + 2 for n > 1, a(1)=2.
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/3 - cot(sqrt(3)*Pi)*Pi/(4*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = -(2 + sqrt(3)*Pi*cosec(sqrt(3)*Pi))/12. (End)
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: 2*(exp(x)*(x^2 + 3*x - 2) + 2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Edited and extended by Klaus Brockhaus, Jan 12 2009

A153516 Triangle T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (pj+q)prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j) and (p,q,j) = (0,1,2), read by rows.

Original entry on oeis.org

2, 3, 3, 2, 14, 2, 2, 25, 25, 2, 2, 33, 92, 33, 2, 2, 41, 200, 200, 41, 2, 2, 49, 340, 676, 340, 49, 2, 2, 57, 512, 1616, 1616, 512, 57, 2, 2, 65, 716, 3148, 5260, 3148, 716, 65, 2, 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2

Offset: 1

Roger L. Bagula, Dec 28 2008

			Triangle begins as:
  2;
  3,  3;
  2, 14,   2;
  2, 25,  25,    2;
  2, 33,  92,   33,     2;
  2, 41, 200,  200,    41,     2;
  2, 49, 340,  676,   340,    49,    2;
  2, 57, 512, 1616,  1616,   512,   57,   2;
  2, 65, 716, 3148,  5260,  3148,  716,  65,  2;
  2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Sequences with variable (p,q,j): this sequence (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).

Cf. A000244.

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,p,q,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j);
  end if; return T;
end function;
[T(n,k,0,1,2): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021

T[n_, k_, p_, q_, j_]:= T[n,k,p,q,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]];
Table[T[n,k,0,1,2], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,p,q,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
flatten([[T(n,k,0,1,2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021

T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p, q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 - 4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 - 2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (0,1,2).

Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for (p,q,j) = (0,1,2) = 2*A000244(n-1).

Edited by G. C. Greubel, Mar 03 2021

cp's OEIS Frontend

A141270 (0, 1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2*5, 11, 2^2*3, ...) transform in (0*1, 2*3, 2*2, 5*2, 3*7, 2*3, 3*2, 2*5, 11*2, 2*3, ...).

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A153521 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1), read by rows.

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A171386 The characteristic function of 2 and 3: 1 if n is prime such that either n-1 or n+1 is prime, else 0.

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A172332 Floor(n*(sqrt(13)+sqrt(5))).

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A172334 Floor(n*(sqrt(13)+sqrt(3))).

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A172336 a(n) = floor(n*(sqrt(13)+sqrt(2))).

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A172338 a(n) = floor(n*(sqrt(5)+sqrt(3))).

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Mathematica

PARI

A131713 Period 3: repeat [1, -2, 1].

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A141270 (0, 1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, ...) transform in (01, 23, 22, 52, 37, 23, 32, 25, 112, 2*3, ...).

A152811 a(n) = 2(n^2 + 2n - 2).

A153516 Triangle T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (pj+q)prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j) and (p,q,j) = (0,1,2), read by rows.