A168273 -id:A168273 - cp's OEIS frontend

A114375 a(n) = (a(n-1) XOR a(n-2)) + 1, a(0) = a(1) = 0.

0, 0, 1, 2, 4, 7, 4, 4, 1, 6, 8, 15, 8, 8, 1, 10, 12, 7, 12, 12, 1, 14, 16, 31, 16, 16, 1, 18, 20, 7, 20, 20, 1, 22, 24, 15, 24, 24, 1, 26, 28, 7, 28, 28, 1, 30, 32, 63, 32, 32, 1, 34, 36, 7, 36, 36, 1, 38, 40, 15, 40, 40, 1, 42, 44, 7, 44, 44, 1, 46, 48, 31, 48, 48, 1, 50, 52, 7, 52, 52, 1

Offset: 0

Franklin T. Adams-Watters, Feb 09 2006

nonn

The function moving to the next overlapping pair in the sequence is f:(i,j) = (j, (i XOR j) + 1) is one-to one. This means that the only possible trajectories for the sequence are loops, "lines", and "rays". The inverse is f^{-1}: (i,j) = (i XOR (j-1), i) is defined except when j = 0. Thus the only infinite non-repeating trajectories are those starting with (i,0) for some i. If we define the size of a pair (i,j) to be the largest power of two <= max(i,j). It is trivial to see that the size of f(i,j) is always >= the size of (i,j). Coupled with the fact there are only finitely many pairs with a given size, this means that "line" trajectories are not possible. Any trajectory that goes to a larger size must be part of a ray, so that tracing back will eventually reach zero. - Franklin T. Adams-Watters, Mar 03 2014

			G.f. = x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 4*x^6 + 4*x^7 + x^8 + 6*x^9 + 8*x^10 + ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A003987, A001511, A074723, A168273.

a[ n_] := If[ n < 0, BitXor[ a[n + 1], a[n + 2] - 1], If[n < 2, 0, 1 + BitXor[ a[n - 1], a[n - 2]]]]; (* Michael Somos, Mar 03 2014 *)
a[ n_] := If[ Mod[n, 3] == 0, 2 n/3, If[ Mod[n, 3] == 1, 4 Quotient[n + 3, 6], If[ n == -1, -1, 2^IntegerExponent[ Fibonacci[n + 1], 2] - 1]]]; (* Michael Somos, Mar 03 2014 *)
nxt[{a_,b_}]:={b,BitXor[a,b]+1}; NestList[nxt,{0,0},80][[All,1]] (* Harvey P. Dale, Feb 26 2020 *)

a(3n)=2n. a(3n+1)=4*floor((n+1)/2). a(6n+2)=1. a(6n+5)=2^(A001511(n+1)+2)-1.
a(3*n + 1) = A168273(n+1). a(3*n - 1) = A074723(n) - 1.- Michael Somos, Mar 03 2014
a(-n) = -a(n) if n == 0 (mod 3), a(-1-n) = -a(n) if n == 1 (mod 3), a(-2-n) = a(n) if n == 2 (mod 3). - Michael Somos, Mar 03 2014

A333119 Triangle T read by rows: T(n, k) = (n - k)(1 - (-1)^k + 2k)/4, with 0 <= k < n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 2, 2, 0, 4, 3, 4, 2, 0, 5, 4, 6, 4, 3, 0, 6, 5, 8, 6, 6, 3, 0, 7, 6, 10, 8, 9, 6, 4, 0, 8, 7, 12, 10, 12, 9, 8, 4, 0, 9, 8, 14, 12, 15, 12, 12, 8, 5, 0, 10, 9, 16, 14, 18, 15, 16, 12, 10, 5, 0, 11, 10, 18, 16, 21, 18, 20, 16, 15, 10, 6

Offset: 1

Stefano Spezia, Mar 08 2020

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A332566(n).
The h-th subdiagonal of the triangle T gives 0 followed by the multiples of h+1 repeated.
For k > 0, the (2*k-1)-th and (2*k)-th columns of the triangle T give the multiples of k.

			n\k| 0 1 2 3 4 5
---+------------
1  | 0
2  | 0 1
3  | 0 2 1
4  | 0 3 2 2
5  | 0 4 3 4 2
6  | 0 5 4 6 4 3
...
For n = 4 the matrix M(4) is
      0 1 1 2
      1 0 1 1
      1 1 0 1
      2 1 1 0
and therefore T(4, 0) = 0, T(4, 1) = 3, T(4, 2) = 2 and T(4, 3) = 2.

Cf. A332566.

Cf. A000004: 1st column; A000027: 2nd and 3rd column; A004526: diagonal; A005843: 4th and 5th column; A052928: 1st subdiagonal; A168237: 2nd subdiagonal; A168273: 3rd subdiagonal; A173196: row sums.

T[n_,k_]:=(n-k)(1-(-1)^k+2k)/4; Flatten[Table[T[n,k],{n,1,12},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2 *(1 - y)^3 (1 + y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 12]]

O.g.f.: y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2*(1 - y)^3*(1 + y)^2).
T(n, k) = k*(n - k)/2 for k even.
T(n, k) = (1 + k)*(n - k)/2 for k odd.

A248800 a(n) = (2*n^2 + 3 + (-1)^n)/2.

Original entry on oeis.org

2, 2, 6, 10, 18, 26, 38, 50, 66, 82, 102, 122, 146, 170, 198, 226, 258, 290, 326, 362, 402, 442, 486, 530, 578, 626, 678, 730, 786, 842, 902, 962, 1026, 1090, 1158, 1226, 1298, 1370, 1446, 1522, 1602, 1682, 1766, 1850, 1938, 2026, 2118

Offset: 0

Paul Curtz, Oct 14 2014

Numbers belonging to A016825: a(n) = A016825( A002620(n) ). - Bruno Berselli, Oct 15 2014

For n>1, a(n) is the number of row vectors of length 2n with entries in [1,n], first entry 1, with maximum inner distance. That is, vectors where the modular distance between adjacent entries is at least (n-2)/2. Modular distance is the minimum of remainders of (x - y) and (y - x) when dividing by n. Geometrically, this metric counts how far the integers mod n are from each other if 1 and n are adjacent as on a circle. - Omar Aceval Garcia, Jan 30 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Omar Aceval Garcia, On the Number of Maximum Inner Distance Latin Squares, arXiv:2112.13912 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000034, A000290, A002620, A016825, A042964, A080827, A168273, A005899, A069894.

[n^2+3/2+(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Oct 15 2014

Table[n^2 + 3/2 + (-1)^n/2, {n, 0, 50}] (* Bruno Berselli, Oct 15 2014 *)
CoefficientList[Series[2(x^3+x^2-x+1)/((1-x)^3 (x+1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2014 *)
LinearRecurrence[{2,0,-2,1},{2,2,6,10},60] (* Harvey P. Dale, Apr 08 2019 *)

Vec(-2*(x^3+x^2-x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Oct 15 2014

[(2*n^2 +3 +(-1)^n)/2 for n in (0..50)] # G. C. Greubel, Dec 14 2021

a(n) = A000290(n) + A000034(n+1) = 4*A002620(n) + 2.
a(n+1) = 2*A080827(n+1) = (n+2)^2 - A042964(n+1) = a(n) + 2*n + 1 -(-1)^n.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Colin Barker, Oct 15 2014
G.f.: 2*(1-x+x^2+x^3) / ((1-x)^3*(x+1)). - Colin Barker, Oct 15 2014
E.g.f.: cosh(x) + (1 + x + x^2)*exp(x). - G. C. Greubel, Dec 14 2021
a(2n) = A005899(n) for n > 0, a(2n+1) = A069894(n). - Omar Aceval Garcia, Dec 30 2021

Typo in data fixed by Colin Barker, Oct 15 2014

A159469 Maximum remainder when (k + 1)^n + (k - 1)^n is divided by k^2 for variable n and k > 2.

Original entry on oeis.org

6, 8, 20, 24, 42, 48, 72, 80, 110, 120, 156, 168, 210, 224, 272, 288, 342, 360, 420, 440, 506, 528, 600, 624, 702, 728, 812, 840, 930, 960, 1056, 1088, 1190, 1224, 1332, 1368, 1482, 1520, 1640, 1680, 1806, 1848, 1980, 2024, 2162, 2208, 2352, 2400, 2550, 2600

Offset: 3

Gaurav Kumar, Apr 13 2009

			For n = 3, maxr => 3*3 - 3 = 6 since 3 is odd.
For n = 4, maxr => 4*4 - 2*4 = 8 since 4 is even.

Iain Fox, Table of n, a(n) for n = 3..10000
Project Euler, Problem 120: Square remainders
Iain Fox, Proof for recursive definition and generating function
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A050187.

LinearRecurrence[{1,2,-2,-1,1},{6,8,20,24,42},50] (* Harvey P. Dale, Apr 18 2018 *)

a(n) = if (n % 2, n^2 - n, n^2 - 2*n); \\ Michel Marcus, Aug 26 2013

first(n) = Vec(x^3*(-6-2*x)/((x+1)^2*(x-1)^3) + O(x^(n+3))) \\ Iain Fox, Nov 26 2017

maxr(n) = n*n - 2*n if n is even, and n*n - n if n is odd.
G.f.: x^3*(-6-2*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 (proved by Iain Fox, Nov 26 2017)
a(n) = 2*A050187(n). - R. J. Mathar, Aug 08 2009 (proved by Iain Fox, Nov 27 2017)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 7. - Colin Barker, Oct 29 2017 (proved by Iain Fox, Nov 26 2017)
a(n) = n^2 - n*(3 + (-1)^n)/2. - Iain Fox, Nov 26 2017
From Iain Fox, Nov 27 2017: (Start)
a(n) = A000290(n) - A022998(n).
a(n) = 2*A093005(n-2) + A168273(n-1).
a(n) = (4*(A152749(n-2)) + A091574(n-1) - A010719(n-1))/3.
E.g.f.: x*(exp(x)*x - sinh(x)).
(End)

A242371 Modified eccentric connectivity index of the cycle graph with n vertices, C[n].

Original entry on oeis.org

12, 32, 40, 72, 84, 128, 144, 200, 220, 288, 312, 392, 420, 512, 544, 648, 684, 800, 840, 968, 1012, 1152, 1200, 1352, 1404, 1568, 1624, 1800, 1860, 2048, 2112, 2312, 2380, 2592, 2664, 2888, 2964, 3200, 3280, 3528, 3612, 3872, 3960, 4232, 4324, 4608, 4704

Offset: 3

Nilanjan De, Jun 08 2014

nonn

The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph. This is a generalization of eccentric connectivity index.

a(n) = 4*A093353(n-1) = n*A168273(n) for n>2. - Alois P. Heinz, Jun 26 2014

			a(3) = 3*4 = 12 because there are 3 vertices and each vertex has eccentricity 1 and the total degree of neighboring vertices is 4.

Nilanjan De, Table of n, a(n) for n = 3..100
N. De, S. M. A. Nayeem and A. Pal, Bounds for modified eccentric connectivity index, Advanced Modeling and Optimization, 16(1) (2014) 133-142.
N. De, S. M. A. Nayeem and A. Pal, Bounds for modified eccentric connectivity index, arXiv:1402.1870 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Eccentricity

Cf. A093353, A168273, A206490, A238410, A238411.

a:= n-> n*(2*n-1+(-1)^n):
seq(a(n), n=3..60);  # Alois P. Heinz, Jun 26 2014

a[n_] := 2n(n-Boole[OddQ[n]]);
Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Nov 28 2018 *)

a(n) = if (n % 2, 2*n*(n-1), 2*n^2); \\ Michel Marcus, Jun 20 2014

a(n) = 2*n*(n-1) if n is odd; and a(n) = 2*n^2 if n is even (n>2).

G.f.: -4*x^3*(3+5*x-4*x^2-2*x^3+2*x^4)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 26 2014

A248812 Repeated terms of (2n)! (A010050).

Original entry on oeis.org

1, 1, 2, 2, 24, 24, 720, 720, 40320, 40320, 3628800, 3628800, 479001600, 479001600, 87178291200, 87178291200, 20922789888000, 20922789888000, 6402373705728000, 6402373705728000, 2432902008176640000, 2432902008176640000, 1124000727777607680000

Offset: 0

Wesley Ivan Hurt, Oct 16 2014

For n>1, a(n) is the product of the smallest parts in the partitions of 4*floor(n/2) = A168273(n) into two parts.

Index entries for sequences related to partitions

Cf. A000142, A052928, A010050, A168273, A211374.

[Factorial(2*Floor(n/2)) : n in [0..20]];

A248812:=n->(2*floor(n/2))!: seq(A248812(n), n=0..20);

Mathematica
```
Table[(2*Floor[n/2])!, {n, 0, 20}]
```

a(n) = ( 2*floor(n/2) )! = A000142(A052928(n)).
a(2n) = a(2n+1) = A010050(n) = A211374(2n-1).
E.g.f.: log((1+x)/(1-x))/2+1/(1-x^2). - Robert Israel, Oct 19 2014

A267654 Irregular triangle of palindromic subsequences. Every row has 2n+1 terms. From the second row, there are only two alternated numbers: 2n+4 and 2*n+2.

Original entry on oeis.org

2, 4, 2, 4, 6, 4, 6, 4, 6, 8, 6, 8, 6, 8, 6, 8, 10, 8, 10, 8, 10, 8, 10, 8, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16

Offset: 0

Paul Curtz, Jan 19 2016

Row sums = 2, 10, 26, 50, ... = A069894(n).
Starting from A053186(n) =
0,                            for b(n)
0, 1, 2,                              for c(n)
0, 1, 2, 3, 4,                                for d(n)
0, 1, 2, 3, 4, 5, 6,
etc,
a(n) is used for
1) b(n+1) = b(n) + (a(0)=2) i.e. 0, 2, 4, 6, ... = A005843(n).
2) c(n+3) = c(n) + (period 3:repeat 4, 2, 4) i.e. 0, 1, 2, 4, 3, 6, 8, ... =  A265667(n).
3) d(n+5) = d(n) + (period 5:repeat 6, 4, 6, 4, 6) i.e. 0, 1, 2, 3, 4, 6, 5, 8, 7, 10, ... = A265734(n).
Etc.
a(n) has a companion with the same terms,differently distributed,yielding permutations of the nonnegative numbers. See A265672.
a(n) other writing (by pairs):
2,   4,  2,  4,
6,   4,  6,  4,
6,   8,  6,  8,  6,  8,  6,  8,
10   8, 10,  8, 10,  8, 10,  8,
10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12,
14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12,
etc.
First column: A168276(n+2). Second column: A168273(n+2).
Row sums: 12, 20, 56, 72, ... = 4*A074378(n+1).
The last term of the successive rows is the number of their terms.
Main diagonal: A005843(n+1).

			The triangle is
2,
4, 2, 4,
6, 4, 6, 4, 6,
8, 6, 8, 6, 8, 6, 8,
etc.

Cf. A007395, A005843, A008586, A016825, A053186, A069894, A074378, A086520, A168273, A168276, A265667, A265672, A265734.

Table[2 (n - 1) + 2 (Boole@ OddQ@ k + 1), {n, 0, 7}, {k, 2 n + 1}] // Flatten (* Michael De Vlieger, Jan 19 2016 *)

a(n) = 2 * A086520(n+2).
a(2n) = 4*n + 2 times 4*n + 2 = 2, 2, 6, 6, 6, 6, 6, 6, 10,....
a(2n+1) = 4*(n+1) times 4*(n+1) = 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 12, ....

cp's OEIS Frontend

A114375 a(n) = (a(n-1) XOR a(n-2)) + 1, a(0) = a(1) = 0.

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A333119 Triangle T read by rows: T(n, k) = (n - k)*(1 - (-1)^k + 2*k)/4, with 0 <= k < n.

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A248800 a(n) = (2*n^2 + 3 + (-1)^n)/2.

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Magma

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A159469 Maximum remainder when (k + 1)^n + (k - 1)^n is divided by k^2 for variable n and k > 2.

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Mathematica

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A242371 Modified eccentric connectivity index of the cycle graph with n vertices, C[n].

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Maple

Mathematica

PARI

Formula

A248812 Repeated terms of (2n)! (A010050).

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Magma

Maple

Mathematica

Formula

A267654 Irregular triangle of palindromic subsequences. Every row has 2*n+1 terms. From the second row, there are only two alternated numbers: 2*n+4 and 2*n+2.

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A333119 Triangle T read by rows: T(n, k) = (n - k)(1 - (-1)^k + 2k)/4, with 0 <= k < n.

A267654 Irregular triangle of palindromic subsequences. Every row has 2n+1 terms. From the second row, there are only two alternated numbers: 2n+4 and 2*n+2.