A032793 -id:A032793 - cp's OEIS frontend

A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).

0, 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, 9, 18, 20, 11, 22, 24, 13, 26, 28, 15, 30, 32, 17, 34, 36, 19, 38, 40, 21, 42, 44, 23, 46, 48, 25, 50, 52, 27, 54, 56, 29, 58, 60, 31, 62, 64, 33, 66, 68, 35, 70, 72, 37, 74, 76, 39, 78, 80, 41, 82, 84, 43, 86, 88, 45

Offset: 0

Bruno Berselli, Dec 12 2015 - based on an idea by Paul Curtz

The inverse permutation is given by P(n) = A006368(n-1) + 1, for n >= 1, and P(0) = 0. - Wolfdieter Lang, Sep 21 2021

This permutation is given by A006369(n-1) + 1, with A006369(-1) = -1. Observed by Kevin Ryde. - Wolfdieter Lang, Sep 22 2021

			-------------------------------------------------------------------------
0, 1, 2, 3,  4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
+  +  +  +   +  +  +   +   +   +   +   +   +   +   +   +   +   +   +
0, 0, 0, 1, -1, 1, 2, -2,  2,  3, -3,  3,  4, -4,  4,  5, -5,  5,  6, ...
-------------------------------------------------------------------------
0, 1, 2, 4,  3, 6, 8,  5, 10, 12,  7, 14, 16,  9, 18, 20, 11, 22, 24, ...
-------------------------------------------------------------------------

Bruno Berselli, Table of n, a(n) for n = 0..1000
Peter Lynch and Michael Mackey, Parity and Partition of the Rational Numbers, arXiv:2205.00565 [math.NT], 2022. See set F p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
Index entries for permutations of the positive (or nonnegative) integers.

Cf. A001477, A006368, A006369, A008738, A032793.
Cf. A064455: n+floor(n/2)*(-1)^(n mod 2).
Cf. A265888: n+floor(n/4)*(-1)^(n mod 4).
Cf. A265734: n+floor(n/5)*(-1)^(n mod 5).

[n+Floor(n/3)*(-1)^(n mod 3): n in [0..70]];

Table[n + Floor[n/3] (-1)^Mod[n, 3], {n, 0, 70}]

[n+floor(n/3)*(-1)^mod(n,3) for n in (0..70)]

G.f.: x*(1 + 2*x + 4*x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-3) - a(n-6).
a(3*k)   = 4*k;
a(3*k+1) = 2*k+1, hence a(3*k+1) = a(3*k)/2 + 1;
a(3*k+2) = 4*k+2, hence a(3*k+2) = 2*a(3*k+1) = a(3*k) + 2.
Sum_{i=0..n} a(i) = A008738(A032793(n+1)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Mar 30 2023

A202803 a(n) = n(5n+1).

Original entry on oeis.org

0, 6, 22, 48, 84, 130, 186, 252, 328, 414, 510, 616, 732, 858, 994, 1140, 1296, 1462, 1638, 1824, 2020, 2226, 2442, 2668, 2904, 3150, 3406, 3672, 3948, 4234, 4530, 4836, 5152, 5478, 5814, 6160, 6516, 6882, 7258, 7644, 8040, 8446, 8862, 9288, 9724, 10170

Offset: 0

Jeremy Gardiner, Dec 24 2011

First bisection of A219190. - Bruno Berselli, Nov 15 2012

a(n)*Pi is the total length of 5 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A017341. The spiral length ratio rounded down [floor(L(n)/L(1))] is A032793. See illustration in links. - Kival Ngaokrajang, Dec 27 2013

			G.f. = 6*x + 22*x^2 + 48*x^3 + 84*x^4 + 130*x^5 +186*x^6 + 252*x^7 + 328*x^8 + ...

Jeremy Gardiner, Table of n, a(n) for n = 0..3000
Kival Ngaokrajang, Illustration of 5 points circle center spiral.
Leo Tavares, Illustration: Twin Trapezoids.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033429, A168668, A126264, A135705, A124080.
Cf. sequences listed in A254963.
Cf. A001622, A005475.

List([0..50], n-> n*(5*n+1)); # G. C. Greubel, Jul 04 2019

[n*(5*n+1):n in [0..50]]; // Vincenzo Librandi, Aug 11 2014

CoefficientList[Series[2x(3+2x)/(1-x)^3, {x, 0, 50}] ,x] (* Vincenzo Librandi, Aug 11 2014 *)
Table[5*n^2+n, {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)

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

[n*(5*n+1) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = 5*n^2 + n.
a(n) = A033429(n) + n. - Omar E. Pol, Dec 24 2011
G.f.: 2*x*(3+2*x)/(1-x)^3. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 6, a(2) = 22. - Philippe Deléham, Mar 27 2013
a(n) = A131242(10n+5). - Philippe Deléham, Mar 27 2013
a(n) = 2*A005475(n). - Philippe Deléham, Mar 27 2013
a(n) = A168668(n) - n. - Philippe Deléham, Mar 27 2013
a(n) = (n+1)^3 - (1 + n + n*(n-1) + n*(n-1)*(n-2)). - Michael Somos, Aug 10 2014
E.g.f.: x*(6+5*x)*exp(x). - G. C. Greubel, Aug 22 2017
Sum_{n>=1} 1/a(n) = 5*(1-log(5)/4) - sqrt(1+2/sqrt(5))*Pi/2 -sqrt(5)*log(phi)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Jul 19 2022

A032794 Positive integers of the form n(n+1)(n+2)(n+3)(n+4)/(n+(n+1)+(n+2)+(n+3)+(n+4)) that are a multiple of n.

Original entry on oeis.org

8, 36, 224, 756, 1232, 2808, 5544, 7488, 12852, 20672, 25704, 38456, 55440, 65780, 90720, 122148, 140616, 183744, 236096, 266112, 334628, 415584, 461168, 563472, 681912, 747684, 893376, 1059380, 1150560, 1350440, 1575288, 1697696, 1963764, 2259936, 2419992

Offset: 1

Patrick De Geest, May 15 1998

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4,0,-6,6,0,4,-4,0,-1,1).

Cf. A032793, A032795.

Vec(4*x*(2 + 7*x + 47*x^2 + 125*x^3 + 91*x^4 + 206*x^5 + 164*x^6 + 52*x^7 + 47*x^8 + 9*x^9) / ((1 - x)^5*(1 + x + x^2)^4) + O(x^40)) \\ Colin Barker, May 30 2019

From Colin Barker, May 30 2019: (Start)
G.f.: 4*x*(2 + 7*x + 47*x^2 + 125*x^3 + 91*x^4 + 206*x^5 + 164*x^6 + 52*x^7 + 47*x^8 + 9*x^9) / ((1 - x)^5*(1 + x + x^2)^4).
a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4) - 6*a(n-6) + 6*a(n-7) + 4*a(n-9) - 4*a(n-10) - a(n-12) + a(n-13) for n>13.
(End)

Edited and offset changed by Alois P. Heinz, May 29 2019

A023054 Simon Plouffe's conjectured extension of sequence A008368.

Original entry on oeis.org

1, 1, 3, 4, 7, 8, 13, 14, 20, 22, 29, 31, 40, 42, 52, 55, 66, 69, 82, 85, 99, 103, 118, 122, 139, 143, 161, 166, 185, 190, 211, 216, 238, 244, 267, 273, 298, 304, 330, 337, 364, 371, 400, 407, 437, 445, 476, 484, 517, 525, 559, 568, 603, 612, 649, 658, 696, 706, 745, 755

Offset: 0

N. J. A. Sloane and J. H. Conway

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 8*x^5 + 13*x^6 + 14*x^7 + 20*x^8 + ...

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).

Cf. A032793, A042706.

CoefficientList[Series[(1-x^5)/((1-x)*(1-x^2)^2*(1-x^3)), {x, 0, 59}], x] (* Georg Fischer, Oct 13 2020 *)

{a(n) = if( n%2, (n + 1) * (5*n + 7) + 8 * (n%6 == 3), (n + 2) * (5*n + 8) + 8 * (n%6 == 0) ) / 24}; /* Michael Somos, May 22 2014 */

{a(n) = if( n<0, n = -3 - n); polcoeff( (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3)) + x * O(x^n), n)}; /* Michael Somos, May 22 2014 */

G.f.: (1-x^5)/((1-x)*(1-x^2)^2*(1-x^3)).
Euler transform of length 5 sequence [ 1, 2, 1, 0, -1]. - Michael Somos, May 22 2014
a(-3 - n) = a(n). - Michael Somos, May 22 2014
a(2*n + 2) - a(2*n) = A032793(n + 2). a(2*n + 3) - a(2*n + 1) = A042706(n + 2). - Michael Somos, May 22 2014

A032795 Positive numbers k such that (k+1)(k+2)(k+3)*(k+4)/(k+(k+1)+(k+2)+(k+3)+(k+4)) is an integer.

Original entry on oeis.org

8, 18, 56, 126, 176, 312, 504, 624, 918, 1292, 1512, 2024, 2640, 2990, 3780, 4698, 5208, 6336, 7616, 8316, 9842, 11544, 12464, 14448, 16632, 17802, 20304, 23030, 24480, 27560, 30888, 32648, 36366, 40356, 42456, 46872, 51584, 54054, 59228

Offset: 1

Patrick De Geest, May 15 1998

Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A032793, A032794.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 2*x*(4+ 5*x +19*x^2+23*x^3+10*x^4+11*x^5+3*x^6)/((1-x)*(1-x^3)^3) )); // G. C. Greubel, May 29 2019

CoefficientList[Series[2*x*(4+5x+19x^2+23x^3+10x^4+11x^5+3x^6)/((1-x)^4*(1+x+x^2)^3), {x, 0, 39}], x] (* Georg Fischer, May 27 2019 *)

Vec(2*x*(4+5*x+19*x^2+23*x^3+10*x^4+11*x^5+3*x^6)/((1-x)^4*(1+x+x^2)^3) + O(x^20)) \\ Felix Fröhlich, May 27 2019

a=(2*x*(4+ 5*x +19*x^2+23*x^3+10*x^4+11*x^5+3*x^6)/((1-x)*(1-x^3)^3) ).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 29 2019

a(n) = A032794(n)/A032793(n).

O.g.f.: 2*x*(4+5*x+19*x^2+23*x^3+10*x^4+11*x^5+3*x^6)/((1-x)^4* (1+x+x^2)^3). [Corrected by Georg Fischer, May 27 2019]

Definition amended and offset changed by Georg Fischer, May 27 2019

A241013 Semiprimes congruent to {1, 2, 4} mod 5.

Original entry on oeis.org

4, 6, 9, 14, 21, 22, 26, 34, 39, 46, 49, 51, 57, 62, 69, 74, 77, 82, 86, 87, 91, 94, 106, 111, 119, 121, 122, 129, 134, 141, 142, 146, 159, 161, 166, 169, 177, 187, 194, 201, 202, 206, 209, 214, 217, 219, 221, 226, 237, 247, 249, 254, 259, 262, 267, 274, 287, 289

Offset: 1

K. D. Bajpai, Aug 07 2014

Semiprimes in A032793.

			21 = 3 * 7 which is semiprime and 21 = 1 mod 5.
39 = 3 * 13 which is semiprime and 39 = 4 mod 5.

K. D. Bajpai, Table of n, a(n) for n = 1..11280

Cf. A001358, A032793, A045371.

Select[Range[500], PrimeOmega[#] == 2 && MemberQ[{1, 2, 4}, Mod[#, 5]] &](* Bajpai *)
Select[Complement[Range[100], 5Range[20] - 2, 5Range[20]], PrimeOmega[#] == 2 &] (* Alonso del Arte, Aug 07 2014 *)

for(n=1,10^4,if(n!=Mod(0,5)&&n!=Mod(3,5),if(bigomega(n)==2,print1(n,", ")))) \\ Derek Orr, Aug 07 2014

A271779 a(n) = n^3 + 2n^2 + 5n + 11.

Original entry on oeis.org

11, 19, 37, 71, 127, 211, 329, 487, 691, 947, 1261, 1639, 2087, 2611, 3217, 3911, 4699, 5587, 6581, 7687, 8911, 10259, 11737, 13351, 15107, 17011, 19069, 21287, 23671, 26227, 28961, 31879, 34987, 38291, 41797, 45511, 49439, 53587, 57961, 62567, 67411, 72499

Offset: 0

Vincenzo Librandi, Apr 14 2016

Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3, page 24.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A001651, A032793.

Magma
```
[n^3+2*n^2+5*n+11: n in [0..50]];
```

Table[n^3 + 2 n^2 + 5 n + 11, {n, 0, 50}]

vector(50, n, n--; n^3+2*n^2+5*n+11) \\ Altug Alkan, Apr 14 2016

O.g.f.: (11 - 25*x + 27*x^2 - 7*x^3)/(1 - x)^4.
E.g.f.: (11 + 8*x + 5*x^2 + x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

cp's OEIS Frontend

A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).

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A202803 a(n) = n*(5*n+1).

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A032794 Positive integers of the form n(n+1)(n+2)(n+3)(n+4)/(n+(n+1)+(n+2)+(n+3)+(n+4)) that are a multiple of n.

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PARI

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A023054 Simon Plouffe's conjectured extension of sequence A008368.

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Mathematica

PARI

PARI

Formula

A032795 Positive numbers k such that (k+1)*(k+2)*(k+3)*(k+4)/(k+(k+1)+(k+2)+(k+3)+(k+4)) is an integer.

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Author

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Magma

Mathematica

PARI

Sage

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Extensions

A241013 Semiprimes congruent to {1, 2, 4} mod 5.

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Comments

Examples

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Programs

Mathematica

PARI

A271779 a(n) = n^3 + 2*n^2 + 5*n + 11.

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Author

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A202803 a(n) = n(5n+1).

A032795 Positive numbers k such that (k+1)(k+2)(k+3)*(k+4)/(k+(k+1)+(k+2)+(k+3)+(k+4)) is an integer.

A271779 a(n) = n^3 + 2n^2 + 5n + 11.