A014601 -id:A014601 - cp's OEIS frontend

A212439 a(n) = 2*n + A181935(n) mod 2.

1, 3, 5, 6, 8, 11, 13, 15, 17, 19, 20, 22, 24, 27, 29, 30, 32, 35, 37, 38, 40, 42, 45, 47, 49, 51, 52, 54, 56, 59, 61, 63, 65, 67, 69, 70, 72, 74, 77, 79, 81, 83, 85, 86, 88, 90, 93, 94, 96, 99, 101, 102, 104, 106, 108, 111, 113, 115, 116, 118, 120, 123, 125

Offset: 0

Reinhard Zumkeller, May 17 2012

A212444 gives iterations starting from 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
Benjamin Chaffin, John P. Linderman, N. J. A. Sloane, and Allan R. Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
Index entries for sequences related to binary expansion of n.

Cf. A014601, A007088, A181935, A212412, A212444.

Haskell
```
a212439 n = 2 * n + a212412 n
```

f[n_, e_] := Module[{d = IntegerDigits[n, 2^e]}, Length[Split[d][[-1]]] - If[SameQ @@ d && Mod[n, 2^e] < 2^(e - 1), 1, 0]]; a[n_] := 2*n + Mod[Max[Table[f[n, e], {e, Range[Max[1, Floor[Log2[n]]]]}]], 2]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Apr 08 2025 *)

a(n) = 2*n + A212412(n): concatenation of binary representation of n and the parity of its curling number.

A222390 Nonnegative integers m such that 10m(m+1)+1 is a square.

Original entry on oeis.org

0, 3, 15, 132, 588, 5031, 22347, 191064, 848616, 7255419, 32225079, 275514876, 1223704404, 10462309887, 46468542291, 397292260848, 1764580902672, 15086643602355, 67007605759263, 572895164628660, 2544524437949340, 21754929612286743, 96624921036315675

Offset: 1

Bruno Berselli, Feb 18 2013

a(n+1)/a(n) tends alternately to (7+2*sqrt(10))/3 and (13+4*sqrt(10))/3; a(n+2)/a(n) tends to A176398^2.

Subsequence of A014601.

Bruno Berselli, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (1,38,-38,-1,1).

Cf. nonnegative integers m such that k*m*(m+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), this sequence (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), A222393 (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].

Cf. A221875.

m:=22; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(3*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2))));

I:=[0,3,15,132,588]; [n le 5 select I[n] else Self(n-1) +38*Self(n-2)-38*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013

LinearRecurrence[{1, 38, -38, -1, 1}, {0, 3, 15, 132, 588}, 23]
CoefficientList[Series[3 x (1 + 4 x + x^2)/((1 - x) (1 - 6 x - x^2) (1 + 6 x - x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 18 2013 *)

makelist(expand(-1/2+((5+(-1)^n*sqrt(10))*(3-sqrt(10))^(2*floor(n/2))+(5-(-1)^n*sqrt(10))*(3+sqrt(10))^(2*floor(n/2)))/20), n, 1, 23);

x='x+O('x^30); concat([0], Vec(3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)))) \\ G. C. Greubel, Jul 15 2018

G.f.: 3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)).
a(n) = a(-n+1) = a(n-1)+38*a(n-2)-38*a(n-3)-a(n-4)+a(n-5).
a(n) = -1/2+((5+t*(-1)^n)*(3-t)^(2*floor(n/2))+(5-t*(-1)^n)*(3+t)^(2*floor(n/2)))/20, where t=sqrt(10).
2*a(n)+1 = A221875(n).

A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

Original entry on oeis.org

1, 4, 2, 3, 5, 8, 6, 7, 9, 12, 10, 11, 13, 16, 14, 15, 17, 20, 18, 19, 21, 24, 22, 23, 25, 28, 26, 27, 29, 32, 30, 31, 33, 36, 34, 35, 37, 40, 38, 39, 41, 44, 42, 43, 45, 48, 46, 47, 49, 52, 50, 51, 53, 56, 54, 55, 57, 60, 58, 59, 61, 64, 62, 63, 65, 68, 66, 67

Offset: 1

Guenther Schrack, Mar 24 2017

Partition the natural number sequence into quadruples starting with (1, 2, 3, 4); swap the third and fourth element, then swap the second and third element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699.
Subsequences:
  elements with odd index: A042963(n), n > 0
  elements with even index: A014601(A103889(n)), n > 0
  odd elements: A005408(n-1), n > 0
  indices of odd elements: A042948(n), n > 0
  even elements: 2*A103889(n), n > 0
  indices of even elements: A042964(n), n > 0
Sequence of fixed points: A016813(n-1), n > 0
Every fourth element starting at:
  n=1: a(4n-3) = 4n-3 = A016813(n-1), n > 0
  n=2: a(4n-2) = 4n = A008586(n), n > 0
  n=3: a(4n-1) = 4n-2 = A016825(n-1), n > 0
  n=4: a(4n) = 4n-1 = A004767(n-1), n > 0
Difference between pairs of elements:
  a(2n+1)-a(2n-1) = A010684(n-1), n > 0
Compositions:
  a(n) = A133256(A116966(n-1)), n > 0
  a(A042948(n)) = A005408(n-1), n > 0
  A067060(a(n)) = A092486(n), n > 0

a = [1 4 2 3];
max = (specify);
for n = 5:max
   a(n) = a(n-4) + 4;
end;

Table[n + ((-1)^n - (-1)^(n (n - 1)/2) (1 + 2 (-1)^n))/2, {n, 68}] (* Michael De Vlieger, Mar 28 2017 *)
LinearRecurrence[{1,0,0,1,-1},{1,4,2,3,5},70] (* or *) {#[[1]],#[[4]], #[[2]],#[[3]]}&/@Partition[Range[70],4]//Flatten(* Harvey P. Dale, Sep 27 2017 *)

for(n=1, 68, print1(n + ((-1)^n - (-1)^(n*(n - 1)/2)*(1 + 2*(-1)^n))/2,", ")) \\ Indranil Ghosh, Mar 29 2017

a(1)=1, a(2)=4, a(3)=2, a(4)=3, a(n) = a(n-4) + 4, n > 4.
O.g.f.: (x^4 + x^3 - 2*x^2 + 3x - 1)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^n - (-1)^(n*(n-1)/2)*(1 + 2*(-1)^n))/2.
a(n) = n + (-1)^n*(1 - (-1)^(n*(n-1)/2) - (i^n - (-i)^n))/2.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5), n > 5.
First differences, periodic: (3, -2, 1, 2), repeat.
a(n) = (2*n - 3*cos(n*Pi/2) + cos(n*Pi) + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Apr 01 2017

A301505 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-1)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 3, 2, 0, 2, 5, 2, 0, 4, 7, 3, 1, 7, 10, 4, 2, 11, 14, 5, 4, 17, 19, 6, 8, 25, 25, 9, 13, 36, 33, 12, 21, 50, 43, 16, 33, 69, 55, 23, 49, 93, 70, 32, 71, 124, 89, 45, 102, 163, 112, 64, 142, 212, 141, 89, 195, 273, 177, 123, 265, 349

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 3 mod 4.

			a(11) = 3 because we have [11], [8, 3] and [7, 4].

Index entries for sequences related to partitions

Cf. A014601, A035364, A035457, A131795, A147599, A301504, A301507, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k)) (1 + x^(4 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[x QPochhammer[-1, x^4] QPochhammer[-x^(-1), x^4]/(2 (1 + x)), {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 3}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A014601(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A305474 Coefficients of Hilbert class polynomial H_D(x) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Original entry on oeis.org

0, 1, -1728, 1, 3375, 1, -8000, 1, 32768, 1, -54000, 1, -121287375, 191025, 1, -287496, 1, 884736, 1, -681472000, -1264000, 1, 12771880859375, -5151296875, 3491750, 1, 14670139392, -4834944, 1, 12288000, 1, -16581375, 1, 1566028350940383, -58682638134

Offset: 1

Seiichi Manyama, Jun 02 2018

			D   |                0             1         2  3
----+---------------------------------------------
-3  |                0,            1;
-4  |            -1728,            1;
-7  |             3375,            1;
-8  |            -8000,            1;
-11 |            32768,            1;
-12 |           -54000,            1;
-15 |       -121287375,       191025,        1;
-16 |          -287496,            1;
-19 |           884736,            1;
-20 |       -681472000,     -1264000,        1;
-23 |   12771880859375,  -5151296875,  3491750, 1;
-24 |      14670139392,     -4834944,        1;
-27 |         12288000,            1;
-28 |        -16581375,            1;
-31 | 1566028350940383, -58682638134, 39491307, 1;
-32 |      12167000000,    -52250000,        1;
-35 |    -134217728000,    117964800,        1;
-36 |   -1790957481984,   -153542016,        1;

Seiichi Manyama, Rows n = 1..250, flattened

Cf. A014600, A014601, A032354, A305475 (constant).

d(n) = 2*n+n%2;
T(n, k) = polcoef(polclass(-d(n)), k);
tabf(nn) = for(n=1, nn, for(k=0, poldegree(polclass(-d(n))), print1(T(n, k), ", ")); print)

A078636 a(n) = rad(n*(n+1)).

Original entry on oeis.org

2, 6, 6, 10, 30, 42, 14, 6, 30, 110, 66, 78, 182, 210, 30, 34, 102, 114, 190, 210, 462, 506, 138, 30, 130, 78, 42, 406, 870, 930, 62, 66, 1122, 1190, 210, 222, 1406, 1482, 390, 410, 1722, 1806, 946, 330, 690, 2162, 282, 42, 70, 510, 1326, 1378, 318, 330, 770, 798

Offset: 1

Jon Perry, Dec 12 2002

			a(3) = 6 as rad(3*4) = rad(12) = rad(2*2*3) = 2*3 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A002378, A007947, A008683, A014601, A139131, A307868.

A078636 := proc(n)
    A007947(n)*A007947(n+1) ;
end proc:
seq( A078636(n),n=1..10) ; # R. J. Mathar, Mar 15 2023

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := rad[n(n+1)];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2024 *)

rad(n)=local(p,i); p=factor(n)[,1]; prod(i=1,length(p),p[i])
for (k=1,100,print1(rad(k*(k+1))", "))

From Reinhard Zumkeller, Aug 05 2003: (Start)
a(n) = rad(n*(n+1)) = rad(n)*rad(n+1).
mu(a(n)) = mu(rad(n*(n+1))) = mu(rad(n))*mu(rad(n+1)), where rad=A007947 and mu=A008683. (End)
From Reinhard Zumkeller, Apr 10 2008: (Start)
a(A014601(n)) = A139131(A014601(n)).
a(n) = A139131(n) * A014695(n). (End)
From Amiram Eldar, Apr 04 2025: (Start)
a(n) = A007947(A002378(n)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 2/(p*(p+1))) = 0.4716806... (A307868). (End)

A129600 Array A(i,j) of binary run-length encoded product of i and j, read by ascending antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 3, 4, 7, 6, 7, 4, 5, 8, 8, 8, 8, 5, 6, 11, 12, 15, 12, 11, 6, 7, 12, 9, 16, 16, 9, 12, 7, 8, 15, 14, 23, 24, 23, 14, 15, 8, 9, 16, 16, 24, 19, 19, 24, 16, 16, 9, 10, 19, 24, 31, 28, 13, 28, 31, 24, 19, 10, 11, 20, 17, 32, 32, 17, 17, 32, 32, 17, 20, 11, 12

Offset: 0

Antti Karttunen, May 01 2007

Cf. A003991, A129601.

Cf. A129602 (center diagonal), A014601 (row 1 apart from the first term).

A(i,j) = A075158(((A075157(i)+1)*(A075157(j)+1)) - 1).

Name edited by Michel Marcus, Dec 01 2021

A279169 a(n) = floor( 4*n^2/5 ).

Original entry on oeis.org

0, 0, 3, 7, 12, 20, 28, 39, 51, 64, 80, 96, 115, 135, 156, 180, 204, 231, 259, 288, 320, 352, 387, 423, 460, 500, 540, 583, 627, 672, 720, 768, 819, 871, 924, 980, 1036, 1095, 1155, 1216, 1280, 1344, 1411, 1479, 1548, 1620, 1692, 1767, 1843, 1920, 2000, 2080, 2163, 2247

Offset: 0

Bruno Berselli, Dec 07 2016

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A090223: floor(4*n/5).
Subsequence of A008728, A014601, A118015, A131242.
Cf. similar sequences with closed form floor(k*n^2/5): A118015 (k=1), A033437 (k=2), A184535 (k=3).

Magma
```
[4*n^2 div 5: n in [0..60]];
```

Table[Floor[4 n^2/5], {n, 0, 60}]
LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,3,7,12,20,28},60] (* Harvey P. Dale, Nov 07 2020 *)

PARI
```
vector(60, n, n--; floor(4*n^2/5))
```
Python
```
[int(4*n**2/5) for n in range(60)]
```
Sage
```
[floor(4*n^2/5) for n in range(60)]
```

O.g.f.: x^2*(3 + x + x^2 + 3*x^3)/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
a(5*m+r) = 4*m*(5*m + 2*r) + a(r), where m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*4*(5*4 + 2*3) + a(3) = 416 + 7 = 423.
a(n) = A118015(2*n) = A008728(4*n+2) = A131242(4*n+4) = A014601(floor(2*n^2/5)).
Sum_{n>=2} 1/a(n) = Pi^2/120 + sqrt(29 - 62/sqrt(5))*Pi/8 + 5/16. - Amiram Eldar, Sep 26 2022

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Original entry on oeis.org

3, 1, 2, 4, 7, 5, 6, 8, 11, 9, 10, 12, 15, 13, 14, 16, 19, 17, 18, 20, 23, 21, 22, 24, 27, 25, 26, 28, 31, 29, 30, 32, 35, 33, 34, 36, 39, 37, 38, 40, 43, 41, 42, 44, 47, 45, 46, 48, 51, 49, 50, 52, 55, 53, 54, 56, 59, 57, 58, 60, 63, 61, 62

Offset: 1

Guenther Schrack, Sep 19 2017

nonn

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the first and the second element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699(n+1) - 1 for n > 0.
Sequence of fixed points: A008586(n) for n > 0.
Subsequences:
   elements with odd index: A042964(A103889(n)) for n > 0.
   elements with even index: A042948(n) for n > 0.
   odd elements: A166519(n) for n>0.
   indices of odd elements: A042963(n) for n > 0.
   even elements: A005843(n) for n>0.
   indices of even elements: A014601(n) for n > 0.
Sum of pairs of elements:
   a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.
Difference between pairs of elements:
   a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.
Compound relations:
   a(n) = A284307(n+1) - 1 for n > 0.
   a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.
Compositions:
   a(n) = A116966(A080412(n)) for n > 0.
   a(A284307(n)) = A256008(n) for n > 0.
   a(A042963(n)) = A166519(n-1) for n > 0.
   A256008(a(n)) = A056699(n) for n > 0.

a = [3 1 2 4]; % Generate b-file
max = 10000;
for n := 5:max
   a(n) = a(n-4) + 4;
end;

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

a(1)=3, a(2)=1, a(3)=2, a(4)=4, a(n) = a(n-4) + 4 for n > 4.
O.g.f.: (2*x^3 + x^2 - 2*x + 3)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^(n*(n-1)/2)*(2-(-1)^n) - (-1)^n)/2.
a(n) = n + (cos(n*Pi/2) - cos(n*Pi) + 3*sin(n*Pi/2))/2.
a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.

A305475 Constant of Hilbert class polynomial H_D(x) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Original entry on oeis.org

0, -1728, 3375, -8000, 32768, -54000, -121287375, -287496, 884736, -681472000, 12771880859375, 14670139392, 12288000, -16581375, 1566028350940383, 12167000000, -134217728000, -1790957481984, 20919104368024767633, 9103145472000, 884736000

Offset: 1

Seiichi Manyama, Jun 02 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A014601, A032354, A305474.

{a(n) = polcoeff(polclass(-2*n-n%2), 0)}

cp's OEIS Frontend

A212439 a(n) = 2*n + A181935(n) mod 2.

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A222390 Nonnegative integers m such that 10*m*(m+1)+1 is a square.

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A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

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A301505 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-1)).

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A305474 Coefficients of Hilbert class polynomial H_D(x) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

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A078636 a(n) = rad(n*(n+1)).

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A129600 Array A(i,j) of binary run-length encoded product of i and j, read by ascending antidiagonals.

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A279169 a(n) = floor( 4*n^2/5 ).

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A222390 Nonnegative integers m such that 10m(m+1)+1 is a square.

A301505 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-1)).