A163300 -id:A163300 - cp's OEIS frontend

A033996 8 times triangular numbers: a(n) = 4n(n+1).

0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 2400, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280

Offset: 0

N. J. A. Sloane, Dec 11 1999

Write 0, 1, 2, ... in a clockwise spiral; sequence gives numbers on one of 4 diagonals.
Also, least m > n such that T(m)*T(n) is a square and more precisely that of A055112(n). {T(n) = A000217(n)}. - Lekraj Beedassy, May 14 2004
Also sequence found by reading the line from 0, in the direction 0, 8, ... and the same line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Axis perpendicular to A195146 in the same spiral. - Omar E. Pol, Sep 18 2011
Number of diagonals with length sqrt(5) in an (n+1) X (n+1) square grid. Every 1 X 2 rectangle has two such diagonals. - Wesley Ivan Hurt, Mar 25 2015
Imagine a board made of squares (like a chessboard), one of whose squares is completely surrounded by square-shaped layers made of adjacent squares. a(n) is the total number of squares in the first to n-th layer. a(1) = 8 because there are 8 neighbors to the unit square; adding them gives a 3 X 3 square. a(2) = 24 = 8 + 16 because we need 16 more squares in the next layer to get a 5 X 5 square: a(n) = (2*n+1)^2 - 1 counting the (2n+1) X (2n+1) square minus the central square. - R. J. Cano, Sep 26 2015
The three platonic solids (the simplex, hypercube, and cross-polytope) with unit side length in n dimensions all have rational volume if and only if n appears in this sequence, after 0. - Brian T Kuhns, Feb 26 2016
The number of active (ON, black) cells in the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. - Robert Price, May 19 2016
The square root of a(n), n>0, has continued fraction [2n; {1,4n}] with whole number part 2n and periodic part {1,4n}. - Ron Knott, May 11 2017
Numbers k such that k+1 is a square and k is a multiple of 4. - Bruno Berselli, Sep 28 2017
a(n) is the number of vertices of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch, May 13 2018
a(n) is the number of vertices in conjoined n X n octagons which are arranged into a square array, a.k.a. truncated square tiling. - Donghwi Park, Dec 20 2020
a(n-2) is the number of ways to place 3 adjacent marks in a diagonal, horizontal, or vertical row on an n X n tic-tac-toe grid. - Matej Veselovac, May 28 2021

			Spiral with 0, 8, 24, 48, ... along lower right diagonal:
.
  36--37--38--39--40--41--42
   |                       |
  35  16--17--18--19--20  43
   |   |               |   |
  34  15   4---5---6  21  44
   |   |   |       |   |   |
  33  14   3   0   7  22  45
   |   |   |   | \ |   |   |
  32  13   2---1   8  23  46
   |   |           | \ |   |
  31  12--11--10---9  24  47
   |                   | \ |
  30--29--28--27--26--25  48
                            \
[Reformatted by _Jon E. Schoenfield_, Dec 25 2016]

Stuart M. Ellerstein, J. Recreational Math. 29 (3) 188, 1998.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10, 2016, 884-892.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Leo Tavares, Illustration: Centroid Diamonds.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Knight Graph.
Stephen Wolfram, A New Kind of Science
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata.
Index to Elementary Cellular Automata.

Cf. A000217, A016754, A002378, A024966, A027468, A028895, A028896, A045943, A046092, A049598, A088538, A124080, A008590 (first differences), A130809 (partial sums).
Sequences from spirals: A001107, A002939, A002943, A007742, A033951, A033952, A033953, A033954, A033988, A033989, A033990, A033991, A033996. - Omar E. Pol, Dec 12 2008
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

[ 4*n*(n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

seq(8*binomial(n+1, 2), n=0..46); # Zerinvary Lajos, Nov 24 2006
[seq((2*n+1)^2-1, n=0..46)];

Table[(2n - 1)^2 - 1, {n, 50}] (* Alonso del Arte, Mar 31 2013 *)

nsqm1(n) = { forstep(x=1,n,2, y = x*x-1; print1(y, ", ") ) }

a(n) = 4*n^2 + 4*n = (2*n+1)^2 - 1.
G.f.: 8*x/(1-x)^3.
a(n) = A016754(n) - 1 = 2*A046092(n) = 4*A002378(n). - Lekraj Beedassy, May 25 2004
a(n) = A049598(n) - A046092(n); a(n) = A124080(n) - A002378(n). - Zerinvary Lajos, Mar 06 2007
a(n) = 8*A000217(n). - Omar E. Pol, Dec 12 2008
a(n) = A005843(n) * A163300(n). - Juri-Stepan Gerasimov, Jul 26 2009
a(n) = a(n-1) + 8*n (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
For n > 0, a(n) = A058031(n+1) - A062938(n-1). - Charlie Marion, Apr 11 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Mar 25 2015
a(n) = A000578(n+1) - A152618(n). - Bui Quang Tuan, Apr 01 2015
a(n) - a(n-1) = A008590(n), n > 0. - Altug Alkan, Sep 26 2015
From Ilya Gutkovskiy, May 19 2016: (Start)
E.g.f.: 4*x*(2 + x)*exp(x).
Sum_{n>=1} 1/a(n) = 1/4. (End)
Product_{n>=1} a(n)/A016754(n) = Pi/4. - Daniel Suteu, Dec 25 2016
a(n) = A056220(n) + A056220(n+1). - Bruce J. Nicholson, May 29 2017
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^2. - Seiichi Manyama, Dec 23 2018
a(n)*a(n+k) + 4*k^2 = m^2 where m = (a(n) + a(n+k))/2 - 2*k^2; for k=1, m = 4*n^2 + 8*n + 2 = A060626(n). - Ezhilarasu Velayutham, May 22 2019
Sum_{n>=1} (-1)^n/a(n) = 1/4 - log(2)/2. - Vaclav Kotesovec, Dec 21 2020
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(4/Pi)*cos(Pi/sqrt(2)).
Product_{n>=1} (1 + 1/a(n)) = 4/Pi (A088538). (End)

A299174 The positive even integers.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144

Offset: 1

Joss Langford, Feb 04 2018

Possible periods of Post's {00, 1101} tag system. - Charles R Greathouse IV, Dec 13 2021

Numbers m such that 2^m - m is divisible by 2. - Bernard Schott, Dec 15 2021

Joss Langford, Simple Integer Sequences
James B. Shearer, Periods of strings, American Scientist Vol. 84, No. 3 (May-June 1996), p. 207. (See final page of pdf, and see also pp. 3-4 for an introduction.)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Equals A005843 without the leading zero.
Bisection of A000027. Complement of A004273. - Omar E. Pol, Feb 25 2018
First row of A083140.
Cf. A005408.
Cf. A000325, A047257.
Essentially the same as A163300, A103517, A051755, A005843 and A004277.

Maple
```
A299174 := n->2*n;
```
Mathematica
```
Range[2, 180, 2]
```

a(n)=2*n \\ Charles R Greathouse IV, Dec 13 2021

R
```
seq(2, 180, 2)
```

a(n) = 2*n, n >= 1.
G.f.: 2*x/(1 - x)^2; corrected by Ilya Gutkovskiy, Mar 29 2018
a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Jul 17 2025

A059270 a(n) is both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.

Original entry on oeis.org

0, 3, 15, 42, 90, 165, 273, 420, 612, 855, 1155, 1518, 1950, 2457, 3045, 3720, 4488, 5355, 6327, 7410, 8610, 9933, 11385, 12972, 14700, 16575, 18603, 20790, 23142, 25665, 28365, 31248, 34320, 37587, 41055, 44730, 48618, 52725, 57057, 61620

Offset: 0

Henry Bottomley, Jan 24 2001

Group the non-multiples of n as follows, e.g., for n = 4: (1,2,3), (5,6,7), (9,10,11), (13,14,15), ... Then a(n) is the sum of the members of the n-th group. Or, the sum of (n-1)successive numbers preceding n^2. - Amarnath Murthy, Jan 19 2004
Convolution of odds (A005408) and multiples of three (A008585). G.f. is the product of the g.f. of A005408 by the g.f. of A008585. - Graeme McRae, Jun 06 2006
Sums of rows of the triangle in A126890. - Reinhard Zumkeller, Dec 30 2006
Corresponds to the Wiener indices of C_{2n+1} i.e., the cycle on 2n+1 vertices (n > 0). - K.V.Iyer, Mar 16 2009
Also the product of the three numbers from A005843(n) up to A163300(n), divided by 8. - Juri-Stepan Gerasimov, Jul 26 2009
Partial sums of A033428. - Charlie Marion, Dec 08 2013
For n > 0, sum of multiples of n and (n+1) from 1 to n*(n+1). - Zak Seidov, Aug 07 2016
A generalization of Ianakiev's formula, a(n) = A005408(n)*A000217(n), follows. A005408(n+k)*A000217(n) is the sum of n+1 consecutive integers and, after skipping k integers, the sum of the n immediately higher consecutive integers. For example, for n = 3 and k = 2, 9*6 = 54 = 12+13+14+15 = 17+18+19. - Charlie Marion, Jan 25 2022

			a(5) = 25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35 = 165.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Roger B. Nelsen, Proof Without Words: Consecutive Sums of Consecutive Integers, Math. Mag., 63 (1990), 25.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A059255 for analog for sum of squares.
Cf. A222716 for the analogous sum of triangular numbers.
Cf. A234319 for nonexistence of analogs for sums of n-th powers, n > 2. - Jonathan Sondow, Apr 23 2014
Cf. A098737 (first subdiagonal).
Bisection of A109900.

I:=[0, 3, 15, 42]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 23 2012

A059270 := proc(n) n*(n+1)*(2*n+1)/2 ; end proc: # R. J. Mathar, Jul 10 2011

# (#+1)(2#+1)/2 &/@ Range[0,39] (* Ant King, Jan 03 2011 *)
CoefficientList[Series[3 x (1 + x)/(x - 1)^4, {x, 0, 39}], x]
LinearRecurrence[{4,-6,4,-1},{0,3,15,42},50] (* Vincenzo Librandi, Jun 23 2012 *)

a(n) = n*(n+1)*(2*n+1)/2 \\ Charles R Greathouse IV, Mar 08 2013

[bernoulli_polynomial(n+1,3) for n in range(0, 41)] # Zerinvary Lajos, May 17 2009

a(n) = n*(n+1)*(2*n+1)/2.
a(n) = A000330(n)*3 = A006331(n)*3/2 = A055112(n)/2 = A000217(A002378(n)) - A000217(A005563(n-1)) = A000217(A005563(n)) - A000217(A002378(n)).
a(n) = A110449(n+1, n-1) for n > 1.
a(n) = Sum_{k=A000290(n) .. A002378(n)} k = Sum_{k=n^2..n^2+n} k.
a(n) = Sum_{k=n^2+n+1 .. n^2+2*n} k = Sum_{k=A002061(n+1) .. A005563(n)} k.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6 = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Ant King, Jan 03 2011
G.f.: 3*x*(1+x)/(1-x)^4. - Ant King, Jan 03 2011
a(n) = A000578(n+1) - A000326(n+1). - Ivan N. Ianakiev, Nov 29 2012
a(n) = A005408(n)*A000217(n) = a(n-1) + 3*A000290(n). -Ivan N. Ianakiev, Mar 08 2013
a(n) = n^3 + n^2 + A000217(n). - Charlie Marion, Dec 04 2013
From Ilya Gutkovskiy, Aug 08 2016: (Start)
E.g.f.: x*(6 + 9*x + 2*x^2)*exp(x)/2.
Sum_{n>=1} 1/a(n) = 2*(3 - 4*log(2)) = 0.4548225555204375246621... (End)
a(n) = Sum_{k=0..2*n} A001318(k). - Jacob Szlachetka, Dec 20 2021
a(n) = Sum_{k=0..n} A000326(k) + A005449(k). - Jacob Szlachetka, Dec 21 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi-3). - Amiram Eldar, Sep 17 2022

A087156 Nonnegative numbers excluding 1.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

N. J. A. Sloane, Oct 11 2008

The old entry with this sequence number was a duplicate of A026835.
A063524(a(n)) = 0. - Reinhard Zumkeller, Oct 11 2008
Inverse binomial transform of A006589. - Philippe Deléham, Nov 25 2008
a(n) = maximum value of j, where 1 <= j <= n-1, such that floor(j^2 / n) > 0 for each n.

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

Cf. A000027, A166373.

[n mod n^2: n in [1..100]]; // Vincenzo Librandi, Aug 22 2015

A087156[n_] := Mod[n, DivisorSigma[1, n]] (* Enrique Pérez Herrero, Nov 11 2009 *)
Drop[Range[0,80],{2}] (* Harvey P. Dale, Dec 13 2011 *)

a(n)=n-(n==1) \\ Charles R Greathouse IV, Aug 26 2011

def A087156(n): return n if n>1 else 0 # Chai Wah Wu, Jul 31 2024

G.f.: x^2*(2-x)/(1-x)^2 . E.g.f.: x*(exp(x)-1). - Philippe Deléham, Nov 25 2008
a(n) = A163300(n)/2. - Juri-Stepan Gerasimov, Aug 14 2009
a(n) = n mod sigma_k(n), where sigma_k is the k divisor sigma function. -Enrique Pérez Herrero, Nov 11 2009
a(n+1) = floor((n+sqrt(n^2+8n))/2). - Philippe Deléham, Oct 03 2011
a(n) = n mod n^2. - Andrew Secunda, Aug 21 2015

Comment and cross-reference added by Christopher Hunt Gribble, Oct 14 2009, Oct 17 2009

A163395 a(n) = (n-th even nonprime)^(n-th even number).

Original entry on oeis.org

1, 16, 1296, 262144, 100000000, 61917364224, 56693912375296, 72057594037927936, 121439531096594251776, 262144000000000000000000, 705429498686404044207947776, 2315513501476187716057433112576

Offset: 1

Juri-Stepan Gerasimov, Jul 26 2009

nonn

Here n-th even nonprime = A163300(n), n-th even number = A005843(n), (A014076 U A163300 = A141468).

			a(1) = 0^0 = 1, a(2) = 4^2 = 16, a(3) = 6^4 = 1296.

G. C. Greubel, Table of n, a(n) for n = 1..100

Cf. A005843, A014076, A141468, A163300.

A163300 := proc(n) if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a; end if; end do; end if; end proc: A005843 := proc(n) 2*n ; end: A163395 := proc(n) A163300(n)^A005843(n-1) ; end: seq(A163395(n),n=1..13) ; # R. J. Mathar, Oct 14 2009

Join[{1}, Table[(2*(n - 1 + Mod[(n + 1), n]))^(2 n - 2), {n, 2, 10}]] (* G. C. Greubel, Dec 21 2016 *)
With[{nn=30},Range[2,nn,2]^Range[0,nn-2,2]] (* Harvey P. Dale, Nov 09 2017 *)

a(n) = A163300(n)^A005843(n).

Extended by R. J. Mathar, Oct 14 2009

A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

Original entry on oeis.org

1, 0, 2, 2, 3, 2, 5, 2, 5, 4, 7, 2, 11, 2, 9, 8, 9, 2, 17, 2, 17, 10, 13, 2, 23, 6, 15, 10, 23, 2, 29, 2, 17, 14, 19, 12, 35, 2, 21, 16, 37, 2, 41, 2, 35, 38, 25, 2, 47, 8, 47, 20, 41, 2, 53, 16, 51, 22, 31, 2, 59, 2, 33, 52, 33, 18, 65, 2, 53, 26, 67, 2, 71, 2, 39, 68, 59, 18, 77, 2, 77, 28, 43, 2, 83, 22, 45, 32, 79

Offset: 0

Alois P. Heinz, Dec 04 2018

Numbers where a(n) + A000010(n) != n + 1: A102467. - Robert G. Wilson v, Aug 23 2021

			a(6) = |{0,2,3,4,6}| = 5.
a(9) = |{0,3,6,9}| = 4.
a(10) = |{0,2,4,5,6,8,10}| = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Matt Baker, The Balanced Centrifuge Problem, Math Blog, 2018.
Holly Krieger and Brady Haran, The Centrifuge Problem, Numberphile video (2018)
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.
Robert G. Wilson v, Graph of the first 100001 terms.

Cf. A000040, A001248, A001748, A001750, A004280, A008588, A008864, A100484, A103306, A103314, A163300, A182986, A272470, A306275.

a:= proc(n) option remember; local f, b; f, b:=
       map(i-> i[1], ifactors(n)[2]),
       proc(m, i) option remember; m=0 or i>0 and
        (b(m, i-1) or f[i]<=m and b(m-f[i], i))
       end; forget(b); (t-> add(
      `if`(b(j, t) and b(n-j, t), 1, 0), j=0..n))(nops(f))
    end:
seq(a(n), n=0..100);

$RecursionLimit = 4096;
a[1] = 0;
a[n_] := a[n] = Module[{f, b}, f = FactorInteger[n][[All, 1]];
     b[m_, i_] := b[m, i] = m == 0 || i > 0 &&
     (b[m, i - 1] || f[[i]] <= m && b[m - f[[i]], i]);
     With[{t = Length[f]}, Sum[
     If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]]];
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz, corrected and updated Aug 07 2021 *)
f[n_] := Block[{c = 2, k = 2, p = First@# & /@ FactorInteger@ n}, While[k < n, If[ IntegerPartitions[k, All, p, 1] != {} && IntegerPartitions[n - k, All, p, 1] != {}, c++]; k++]; c]; f[0] = 1; f[1] = 0; Array[f, 75] (* Robert G. Wilson v, Aug 22 2021 *)

a(n) = |{ k : k and n-k can be written as a sum of prime factors of n }|.
a(n) = 2 <=> n is prime (A000040).
a(n) >= n-1 <=> n in {1,2,3,4} union { A008588 }.
a(n) = (n+4)/2 <=> n in { A100484 } minus { 4 }.
a(n) = (n+9)/3 <=> n in { A001748 } minus { 9 }.
a(n) = (n+25)/5 <=> n in { A001750 } minus { 25 }.
a(n) = (n+49)/7 <=> n in { A272470 } minus { 49 }.
a(n^2) = n+1 <=> n = 0 or n is prime <=> n in { A182986 }.
a(A001248(n)) = A008864(n).
a(n) is odd <=> n in { A163300 }.
a(n) is even <=> n in { A004280 }.

cp's OEIS Frontend

A174010 Primes p of the form p = A000040(k) - A163300(k) for some k (includes duplicates).

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A174189 Numbers k such that A000040(k) +/- A163300(k) are both prime.

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A033996 8 times triangular numbers: a(n) = 4*n*(n+1).

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A299174 The positive even integers.

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R

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A059270 a(n) is both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.

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A087156 Nonnegative numbers excluding 1.

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A163395 a(n) = (n-th even nonprime)^(n-th even number).

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A033996 8 times triangular numbers: a(n) = 4n(n+1).