A017113 -id:A017113 - cp's OEIS frontend

A166464 a(n) = (3 + 2n + 6n^2 + 4*n^3)/3.

1, 5, 21, 57, 121, 221, 365, 561, 817, 1141, 1541, 2025, 2601, 3277, 4061, 4961, 5985, 7141, 8437, 9881, 11481, 13245, 15181, 17297, 19601, 22101, 24805, 27721, 30857, 34221, 37821, 41665, 45761, 50117, 54741, 59641, 64825, 70301, 76077, 82161, 88561, 95285, 102341, 109737, 117481, 125581

Offset: 0

Paul Curtz, Oct 14 2009

Atomic number of first transition metal of period 2n (n>3) or of the element after n-th alkaline earth metal. This can be calculated by finding the sum of the first n even squares plus 1. - Natan Arie Consigli, Jul 03 2016

JANET,Charles, La structure du Noyau de l'atome,consideree dans la Classification periodique,des elements chimiques,1927 (Novembre),N. 2,BEAUVAIS,67 pages,3 leaflets.

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

Cf. A002492, A010731, A016742, A017113, A018227.

[(3+2*n+6*n^2+4*n^3)/3: n in [0..60]]; // G. C. Greubel, Jul 27 2024

Table[(3+2*n+6*n^2+4*n^3)/3, {n,0,60}] (* G. C. Greubel, May 15 2016 *)

a(n)=(3+2*n+6*n^2+4*n^3)/3 \\ Charles R Greathouse IV, Oct 07 2015

[(3+2*n+6*n^2+4*n^3)//3 for n in range(61)] # G. C. Greubel, Jul 27 2024

a(n) - a(n-1) = 4*(n+1)^2 = A016742(n+1).
a(n) - 2*a(n-1) + a(n-2) = -4 + 8*n = A017113(n+1).
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 8 = A010731(n).
a(n) - 4*a(n-1) + 6*a(n-2) - 4*a(n-3) + a(n-4) = 0.
Binomial transform of quasi-finite sequence 1,4,12,8,0,(0 continued).
G.f.: (1+x+7*x^2-x^3)/(1-x)^4. - R. J. Mathar, Feb 15 2010
From Natan Arie Consigli, Jul 03 2016: (Start)
a(n) = A018227(2*n) + 3.
a(n) = A002492(n) + 1. (End)
E.g.f.: (1/3)*(3 + 12*x + 18*x^2 + 4*x^3)*exp(x). - G. C. Greubel, Jul 27 2024

Edited by N. J. A. Sloane, Oct 17 2009

More terms a(11)-a(35) from Vincenzo Librandi, Oct 17 2009

A047463 Numbers that are congruent to {2, 4} mod 8.

Original entry on oeis.org

2, 4, 10, 12, 18, 20, 26, 28, 34, 36, 42, 44, 50, 52, 58, 60, 66, 68, 74, 76, 82, 84, 90, 92, 98, 100, 106, 108, 114, 116, 122, 124, 130, 132, 138, 140, 146, 148, 154, 156, 162, 164, 170, 172, 178, 180, 186, 188, 194, 196, 202, 204, 210, 212, 218, 220, 226, 228, 234

Offset: 1

N. J. A. Sloane

First differences in A010696.

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

Union of A017089 and A017113.

Cf. A014848.

[ n: n in [2..234 by 2] | n mod 8 in [2,4] ];  // Bruno Berselli, May 11 2011

Select[Range[250], MemberQ[{2, 4}, Mod[#, 8]] &] (* Amiram Eldar, Dec 18 2021 *)

a(n) = 8*n - a(n-1) - 10, with a(1)=2. - Vincenzo Librandi, Aug 06 2010
From Bruno Berselli, May 11 2011: (Start)
G.f.: 2*x*(1+x+2*x^2)/((1+x)*(1-x)^2).
a(n) = 4*n-(-1)^n-3.
Sum_{i=1..n} a(i) = 2*A014848(n).
a(n) = 2*A042963(n-1). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 + log(2)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A047395 Numbers that are congruent to {0, 2, 6} mod 8.

Original entry on oeis.org

0, 2, 6, 8, 10, 14, 16, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 48, 50, 54, 56, 58, 62, 64, 66, 70, 72, 74, 78, 80, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 112, 114, 118, 120, 122, 126, 128, 130, 134, 136, 138, 142, 144, 146, 150, 152, 154, 158

Offset: 1

N. J. A. Sloane

The members of this sequence together with the members of A017113 give the even numbers. - Wesley Ivan Hurt, Apr 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A017113, A042965, A047407, A047410, A047464.

[n : n in [0..150] | n mod 8 in [0, 2, 6]]; // Wesley Ivan Hurt, Jun 13 2016

A047395:=n->2*floor((4*n-3)/3); seq(A047395(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Table[2 Floor[(4 n - 3)/3], {n, 100}] (* Wesley Ivan Hurt, Apr 01 2014 *)
Flatten[Table[8n + {0, 2, 6}, {n, 0, 19}]] (* Alonso del Arte, Apr 11 2014 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 2, 6, 8}, 100] (* Vincenzo Librandi, Jun 14 2016 *)

From R. J. Mathar, Dec 05 2011: (Start)
G.f.: 2*x^2*(1+x)^2 / ((1+x+x^2)*(x-1)^2).
a(n) = 2 * A042965(n). (End)
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*(12*n-12+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-6, a(3k-2) = 8k-8. (End)
a(n) = 2*(n - 1 + floor(n/3)). - Wolfdieter Lang, Sep 11 2021
Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*log(sqrt(2)+2)/4 - (sqrt(2)-1)*log(2)/8. - Amiram Eldar, Dec 19 2021

A047406 Numbers that are congruent to {4, 6} mod 8.

Original entry on oeis.org

4, 6, 12, 14, 20, 22, 28, 30, 36, 38, 44, 46, 52, 54, 60, 62, 68, 70, 76, 78, 84, 86, 92, 94, 100, 102, 108, 110, 116, 118, 124, 126, 132, 134, 140, 142, 148, 150, 156, 158, 164, 166, 172, 174, 180, 182, 188, 190, 196, 198, 204, 206, 212, 214, 220, 222, 228

Offset: 1

N. J. A. Sloane

In groups of four, add the odd and even numbers (4=1+3, 6=2+4; 12=5+7, 14=6+8; etc.). - George E. Antoniou, Dec 12 2001

The first 250 terms (4 through 998) are the 250 non-occurring Fibonacci number residues modulo 1000; i.e., if leading zeros are supplied as necessary for the terms having fewer than three digits, these are the 250 sets of three digits that never appear as the last three digits of a Fibonacci number. - Jon E. Schoenfield, Jul 05 2010

			a(2) = 8*2 - 4 - 6 = 6;
a(3) = 8*3 - 6 - 6 = 12;
a(4) = 8*4 - 12 - 6 = 14.

David Lovler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A017113 and A017137.

Cf. A042964.

Select[Range[230], MemberQ[{4, 6}, Mod[#, 8]] &] (* Amiram Eldar, Dec 19 2021 *)

a(n)=4*n-1-(-1)^n \\ Charles R Greathouse IV, May 20 2013

a(n) = A042964(n)*2.
a(n) = (4*n - 1 - (-1)^n). - Jon E. Schoenfield, Jul 05 2010
a(n) = 8*n - a(n-1) - 6 (with a(1)=4). - Vincenzo Librandi, Aug 05 2010
G.f.: 2*x*(2+x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 18 2013: (Start)
a(n) = (8 * ceiling(n/2) - 4) * (n mod 2) + (8 * ceiling(n/2) - 2) * (n+1 mod 2).
a(n) = 8 * ceiling(n/2) - 3 + (-1)^n. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - log(2)/8. - Amiram Eldar, Dec 19 2021
E.g.f.: 2*(1 + 2*x*exp(x) - cosh(x)). - David Lovler, Sep 02 2022

A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

Original entry on oeis.org

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 54, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 135, 140, 144, 148, 156, 164, 172, 176, 180, 188, 189, 192, 196, 204, 208, 212, 216, 220, 228, 236, 240, 244, 252, 256, 260, 268, 270, 272, 276, 284, 292, 297, 300

Offset: 1

Amiram Eldar, Feb 27 2021

nonn

Numbers with a unitary divisor of the form p^(m*p) where p is a prime and m > 0.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 19, 188, 1883, 18825, 188244, 1882429, 18824297, 188242957, 1882429628, ...
The asymptotic density of this sequence is 1 - Product_{p prime} 1 - (p - 1)/(p*(p^p - 1)) = 0.18824296270011399086...

			4 = 2^2 is a term since 2 divides 2.
8 = 2^3 is not a term since 2 does not divide 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A013929.
Subsequences: A000302, A000312, A009971, A017113, A051674.
Cf. A072873, A369070 (characteristic function).

q[n_] := AnyTrue[FactorInteger[n], Divisible[Last[#], First[#]] &]; Select[Range[2, 300], q]

Wrong term 1 removed by Amiram Eldar, Jan 16 2024

A098502 a(n) = 16*n - 4.

Original entry on oeis.org

12, 28, 44, 60, 76, 92, 108, 124, 140, 156, 172, 188, 204, 220, 236, 252, 268, 284, 300, 316, 332, 348, 364, 380, 396, 412, 428, 444, 460, 476, 492, 508, 524, 540, 556, 572, 588, 604, 620, 636, 652, 668, 684, 700, 716, 732, 748, 764, 780, 796, 812, 828, 844

Offset: 1

Ralf Stephan, Sep 15 2004

For n > 3, the number of squares on the infinite 4-column chessboard at <= n knight moves from any fixed start point.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences which agree for a long time but are different.

Cf. A004767, A008590, A017113, A017137, A017629, A098498, A158953.

[16*n - 4: n in [1..60]]; // Vincenzo Librandi, Jul 24 2011

Range[12, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

a(n)=16*n-4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: 4*x*(3+x)/(1-x)^2. - Colin Barker, Jan 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3 - 2*sqrt(2)))/(16*sqrt(2)). - Amiram Eldar, Sep 01 2024
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 4*(exp(x)*(4*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = 4*A004767(n-1) = 2*A017137(n-1) = A017113(2*n-1). (End)

A288571 a(n) = Sum_{d|n} (-1)^(n/d+1)*tau(d), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 1, 3, 0, 3, 3, 3, -2, 6, 3, 3, 0, 3, 3, 9, -5, 3, 6, 3, 0, 9, 3, 3, -6, 6, 3, 10, 0, 3, 9, 3, -9, 9, 3, 9, 0, 3, 3, 9, -6, 3, 9, 3, 0, 18, 3, 3, -15, 6, 6, 9, 0, 3, 10, 9, -6, 9, 3, 3, 0, 3, 3, 18, -14, 9, 9, 3, 0, 9, 9, 3, -12, 3, 3, 18, 0, 9, 9, 3, -15, 15, 3, 3, 0, 9

Offset: 1

Ilya Gutkovskiy, Aug 23 2018

Dirichlet convolution of A048272 and A000012. - Vaclav Kotesovec, Jan 13 2024

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A001620, A007425, A017113 (positions of 0's), A048272, A288417, A317531.

with(numtheory): seq(add((-1)^(n/a+1)*tau(a),a=divisors(n)),n=1..85); # Paolo P. Lava, Aug 24 2018

Table[Sum[(-1)^(n/d + 1) DivisorSigma[0, d], {d, Divisors[n]}], {n, 85}]
nmax = 85; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := If[p == 2, (e + 1)*(2 - e)/2, (e + 1)*(e + 2)/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = sumdiv(n, d, (-1)^(n/d+1)*numdiv(d)); \\ Michel Marcus, Aug 24 2018

G.f.: Sum_{k>=1} tau(k)*x^k/(1 + x^k).
L.g.f.: log(Product_{k>=1} (1 + x^k)^(tau(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
Multiplicative with a(2^e) = (e+1)*(2-e)/2, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Sep 14 2023: (Start)
Dirichlet g.f.: (1- 1/2^(s-1)) * zeta(s)^3.
Sum_{k=1..n} a(k) ~ log(2) * n * (log(n) + 3*gamma - 1 - log(2)/2), where gamma is Euler's constant (A001620). (End)

A086570 Expansion of (1 + 3x + 5x^2 + 7x^3 + ...) / (1 - 2x + 3x^2 - 4x^3 + ...).

Original entry on oeis.org

1, 5, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420, 428

Offset: 0

Gary W. Adamson, Jul 22 2003

Row sums of number triangle A113128. - Paul Barry, Oct 14 2005

The Engel expansion of 1 + exp(1/8)*sqrt(2*Pi)*erf(1/(2*sqrt(2)))/5 = 1.2175306077808... - Benedict W. J. Irwin, Dec 16 2016

			a(6) = 44 = 8 + a(5) = 8 + 36.

Leo Tavares, Square illustration
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A078370, A016754.

CoefficientList[Series[(z^3 + 3*z^2 + 3*z + 1)/(z - 1)^2, {z, 0, 100}], z] (* and *) Join[{1, 5}, Table[4*(2*(n + 1) + 1), {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)

a(n)=if(n>1, 8*n-4, 4*n+1) \\ Charles R Greathouse IV, Dec 16 2016

a(0) = 1, a(1) = 5, a(2) = 12; then a(n+1) = a(n) + 8, n > 2.
From Paul Barry, Oct 14 2005: (Start)
G.f.: (1+x)^3/(1-x)^2;
a(n) = 8n - 4 + 4*C(0, n) + C(1, n);
a(n) = C(n+1, n) + 3*C(n, n-1) + 3*C(n-1, n-2) + C(n-2, n-3). (End)
a(n) = A017113(n-1), n > 1. - R. J. Mathar, Sep 12 2008

A188134 a(4n) = n, a(1+2n) = 4+8n, a(2+4n) = 2+4*n.

Original entry on oeis.org

0, 4, 2, 12, 1, 20, 6, 28, 2, 36, 10, 44, 3, 52, 14, 60, 4, 68, 18, 76, 5, 84, 22, 92, 6, 100, 26, 108, 7, 116, 30, 124, 8, 132, 34, 140, 9, 148, 38, 156, 10, 164, 42, 172, 11, 180, 46, 188, 12, 196, 50, 204, 13, 212, 54, 220, 14, 228, 58, 236, 15, 244, 62

Offset: 0

Paul Curtz, Mar 21 2011

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A016825, A017113, A060819, A061037, A064680, A146160, A176895, A177499.

[(64-3*(1+(-1)^n)*(9+(-1)^(n div 2)))*n/16 : n in [0..80]]; // Wesley Ivan Hurt, Jul 06 2016

A188134:=n->8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)): seq(A188134(n), n=0..100); # Wesley Ivan Hurt, Jul 06 2016

Table[8 n/(11 + 9 Cos[Pi*n] + 12 Cos[n*Pi/2]), {n, 0, 80}] (* Wesley Ivan Hurt, Jul 06 2016 *)
CoefficientList[Series[x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2, {x, 0, 50}], x] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,2,12,1,20,6,28},70] (* Harvey P. Dale, Aug 14 2019 *)

x='x+O('x^50); concat([0], Vec(x*(4+2*x+12*x^2+x^3+12*x^4+ 2*x^5 +4*x^6)/(1-x^4)^2)) \\ G. C. Greubel, Sep 20 2018

a(n) = 2*a(n-4) - a(n-8) for n>7.
a(n) = A176895(n) * A060819(n).
a(n) = (4*A061037(n+2))/(n+4).
a(n) = 4*n / A146160(n).
a(2*n) = A064680(n).
a(1+2*n) = A017113(n).
a(4*n) = a(-4+4*n) +  1.
a(1+4*n) = a(-3+4*n) + 16.
a(2+4*n) = a(-2+4*n) +  4.
a(3+4*n) = a(-1+4*n) + 16.  See A177499.
From Bruno Berselli, Mar 22 2011: (Start)
G.f.: x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2.
a(n) = (64-3*(1+(-1)^n)*(9+i^n))*n/16 with i=sqrt(-1).
a(n)/a(n-4) = n/(n-4) for n>4. (End)
a(n) = 8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)). - Wesley Ivan Hurt, Jul 06 2016
a(n) = lcm(4,n)/gcd(4,n). - R. J. Mathar, Feb 12 2019
Sum_{k=1..n} a(k) ~ (37/32)*n^2. - Amiram Eldar, Oct 07 2023

A209675 Radon function at even positions: a(n) = A003484(2*n).

Original entry on oeis.org

2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 12, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, 2, 4, 2, 8, 2

Offset: 1

Reinhard Zumkeller, Mar 11 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003484, A003485, A005408, A016825, A017113, A051062, A053381.

Haskell
```
a209675 = a003484 . (* 2)
```

a[n_] := 8*Floor[(e = IntegerExponent[n, 2] + 1)/4] + 2^Mod[e, 4]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

a(n) = A053381(n-1) + 1.
a(n) > 1.
a(A005408(n)) = 2; a(A016825(n)) = 4; a(A017113(n)) = 8; a(A051062(n)) = 9.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/3. - Amiram Eldar, Nov 29 2022

cp's OEIS Frontend

A166464 a(n) = (3 + 2*n + 6*n^2 + 4*n^3)/3.

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A047463 Numbers that are congruent to {2, 4} mod 8.

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A047395 Numbers that are congruent to {0, 2, 6} mod 8.

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A047406 Numbers that are congruent to {4, 6} mod 8.

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A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

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A098502 a(n) = 16*n - 4.

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A288571 a(n) = Sum_{d|n} (-1)^(n/d+1)*tau(d), where tau = number of divisors (A000005).

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A166464 a(n) = (3 + 2n + 6n^2 + 4*n^3)/3.

A188134 a(4n) = n, a(1+2n) = 4+8n, a(2+4n) = 2+4*n.