A000034 -id:A000034 - cp's OEIS frontend

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

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

A265892 Array read by ascending antidiagonals: A(n,k) = A265893(A265609(n,k)), with n as row >= 0, k as column >= 0; the number of significant digits counted without trailing zeros in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 3, 2, 3, 2, 2, 1, 1, 0, 1, 2, 3, 2, 2, 3, 1, 1, 1, 0, 1, 3, 1, 2, 3, 1, 2, 2, 1, 1, 0, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 3, 3, 4, 2, 2, 2, 3, 3, 2, 1, 1, 0, 1, 1, 3, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 0, 1, 3, 3, 4, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 0

Offset: 0

Antti Karttunen, Dec 20 2015

Square array A(row,col) is read by ascending antidiagonals as: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...

			The top left corner of the array A265609 with its terms shown in factorial base (A007623) looks like this:
1,   0,    0,     0,       0,        0,         0,          0,           0
1,   1,   10,   100,    1000,    10000,    100000,    1000000,    10000000
1,  10,  100,  1000,   10000,   100000,   1000000,   10000000,   100000000
1,  11,  200,  2200,   30000,   330000,   4000000,   44000000,   500000000
1,  20,  310, 10000,  110000,  1220000,  14000000,  160000000,  1830000000
1,  21, 1100, 13300,  220000,  3000000,  36000000,  452000000,  5500000000
1, 100, 1300, 24000,  411000,  6000000,  82000000, 1100000000, 13300000000
1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000
-
Counting such digits for each term, but without the trailing zeros gives us the top left corner of this array:
-
The top left corner of the array:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1
1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2
1, 2, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 3, 4, 3
1, 1, 2, 2, 3, 1, 2, 2, 3, 4, 3, 1, 2, 3, 4, 2, 3, 2, 3, 4, 1, 2, 3, 3, 4
1, 3, 3, 2, 1, 2, 3, 4, 4, 4, 3, 4, 2, 3, 3, 4, 3, 4, 3, 3, 4, 2, 4, 5, 4
1, 2, 1, 1, 2, 3, 4, 3, 3, 2, 3, 2, 4, 5, 4, 3, 4, 3, 3, 4, 5, 3, 4, 3, 4
1, 3, 2, 4, 3, 4, 3, 4, 2, 3, 4, 5, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 3
1, 2, 3, 2, 3, 4, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 5, 4, 5, 4, 5, 3, 4
1, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 3, 3, 5, 6, 6, 5, 6, 5, 6, 5, 6, 4, 5, 6
1, 1, 3, 3, 3, 2, 3, 3, 4, 4, 5, 3, 4, 5, 5, 4, 5, 4, 5, 4, 5, 6, 4, 5, 4
1, 3, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 4, 5, 6, 6, 5, 6, 5, 7, 6, 5, 5, 5, 5
1, 2, 3, 2, 4, 3, 4, 4, 4, 4, 5, 5, 6, 5, 5, 4, 6, 5, 6, 5, 4, 4, 4, 5, 6
1, 3, 1, 2, 3, 4, 5, 4, 3, 4, 4, 5, 5, 7, 6, 7, 6, 7, 5, 6, 7, 5, 4, 5, 6
1, 2, 4, 3, 5, 4, 3, 5, 6, 6, 5, 6, 6, 5, 6, 5, 6, 4, 5, 6, 4, 4, 6, 7, 8
1, 3, 3, 5, 4, 5, 5, 6, 5, 6, 5, 7, 6, 7, 6, 7, 4, 5, 6, 8, 5, 6, 7, 8, 6
1, 1, 3, 3, 4, 3, 5, 4, 5, 4, 6, 5, 6, 5, 6, 6, 7, 6, 7, 4, 5, 6, 7, 5, 6
...

Cf. A007623, A265609.
Row 0: A000007, rows 1-2: A000012, row 3: A000034 (see comment in A001710).
Column 0: A000012, column 1: A265893.
Cf. also array A265890.

(define (A265892 n) (A265892bi (A025581 n) (A002262 n)))
(define (A265892bi row col) (A265893 (A265609bi row col)))

A(n,k) = A265893(A265609(n,k)).

A292680 Rule 6: 000, ..., 111 -> 0, 1, 1, 0, 0, 0, 0, 0.

Original entry on oeis.org

0, 6, 12, 8, 24, 26, 16, 16, 48, 54, 52, 48, 32, 34, 32, 32, 96, 102, 108, 104, 104, 106, 96, 96, 64, 70, 68, 64, 64, 66, 64, 64, 192, 198, 204, 200, 216, 218, 208, 208, 208, 214, 212, 208, 192, 194, 192, 192, 128, 134, 140, 136, 136, 138, 128, 128, 128, 134, 132, 128

Offset: 0

M. F. Hasler, Oct 09 2017

The orbit of 1 under this rule is A266180.
Rule 6 is the smallest rule which is even (otherwise infinitely many bits would be switched on at step 1, for any finite starting value) and nontrivial (i.e., does not lead to extinction nor simple reproduction, possibly shifted left or right, of a single-bit initial state).
As is customary in the context of elementary cellular automata, the result is the bitmap obtained from the argument extended by one bit to the right (as to consider the cell which has bit 0 of the input as left neighbor), cf. example. Since the rule has a value < 16, no cell having its left neighbor 'on' will be on. Therefore all values a(n) are even. See A292681 for the variant without this extension beyond bit 0, i.e., a(n)/2.

			     n        |          a(n)
   0 =   0[2] |       0[2] =  0
   1 =   1[2] |     110[2] =  6 (bits below 001 and 010 are on, below 100 is off)
   2 =  10[2] |    1100[2] = 12 (as above, plus an additional bit 0 below 000)
   3 =  11[2] |    1000[2] =  8 (1 below 001, 0 below 011, 110 and 100.)
   4 = 100[2] |   11000[2] = 24 (as n = 1 and n = 2, shifted right once more)
   5 = 101[2] |   11010[2] = 26 (1 below 001 and 010 (twice), 0 below 101 and 100)
   6 = 110[2] |   10000[2] = 16 (as n = 3, shifted right once)
   7 = 111[2] |   10000[2] = 16 (1 below 001, 0 below 011, 111, 110 and 100).

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A266178, A266179, A266180, A019590, A003953, A003945, A000034, A032766, A042948, A035608.

apply( A292680(n,r=6)=sum(i=0,logint(!n+n<<=2,2)+1,bittest(r,(n>>i)%8)<

A306277 Numbers congruent to 1 or 8 mod 10.

Original entry on oeis.org

1, 8, 11, 18, 21, 28, 31, 38, 41, 48, 51, 58, 61, 68, 71, 78, 81, 88, 91, 98, 101, 108, 111, 118, 121, 128, 131, 138, 141, 148, 151, 158, 161, 168, 171, 178, 181, 188, 191, 198, 201, 208, 211, 218, 221, 228, 231, 238, 241, 248, 251, 258, 261, 268, 271, 278, 281, 288, 291, 298, 301, 308, 311, 318, 321

Offset: 1

Davis Smith, Feb 02 2019

A007310(a(n)+1) is always a multiple of 5.
a(1) = 1, a(n+1) = a(n)+7 when n is odd, a(n+1) = a(n)+3 when n is even.
a(n) mod 6 follows the following pattern: 1,2,5,0,3,4,1,2,5,0,3,4, and so on.
A020639(A007310(a(n)+1)) = 5.

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

Cf. A001622, A020639, A007310, A000034.
Cf. A017281, A017365 (bisections).
One less than A273669.

Maple
```
seq(seq(10*i+j, j=[1, 8]), i=0..350);
```

Select[Range[350], MemberQ[{1, 8}, Mod[#, 10]] &]

for(n=1, 350, if((n%10==1) || (n%10==8), print1(n, ", ")))

Vec(x*(1 + 7*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 09 2019

a(n) = 5*n - 2*A000034(n+1).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = A273669(n) - 1. - Antti Karttunen, Feb 07 2019
G.f.: x*(1 + 7*x + 2*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Feb 09 2019
E.g.f.: 2 + (5*x - 3)*exp(x) + exp(-x). - David Lovler, Sep 07 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (5+sqrt(5))^(3/2)*phi*Pi/(100*sqrt(2)) + log(phi)/(2*sqrt(5)) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A321119 a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118.

Original entry on oeis.org

4, 2, 8, 10, 28, 38, 104, 142, 388, 530, 1448, 1978, 5404, 7382, 20168, 27550, 75268, 102818, 280904, 383722, 1048348, 1432070, 3912488, 5344558, 14601604, 19946162, 54493928, 74440090, 203374108, 277814198, 759002504, 1036816702, 2832635908, 3869452610

Offset: 0

Franck Maminirina Ramaharo, Nov 01 2018

			a(0) = ((1 - sqrt(3))^0 + (1 + sqrt(3))^0)/2^floor((0 - 1)/2) = 2*(1 + 1) = 4.

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

Encyclopedia of Mathematics, Quadrature formula
John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.
Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A002176 (common denominators of Cotesian numbers).
Cf. A321118, A321120.
Cf. A001834, A002530, A002531, A003500, A005246, A048788, A083336, A125905.

LinearRecurrence[{0, 4, 0, -1}, {4, 2, 8, 10}, 50]

a(n) := ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2)$
makelist(ratsimp(a(n)), n, 0, 50);

a(n) = (((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)*((2 - sqrt(2))*(-1)^n + 2 + sqrt(2))/2.
a(-n) = (-1)^n*a(n).
a(n) = 2*A000034(n+1)*A002531(n).
a(2*n) = 2*A001834(n).
a(2*n+1) = 2*A003500(n).
a(n) = 4*a(n-2) - a(n-4) with a(0) = 4, a(1) = 2, a(2) = 8, a(3) = 10.
a(2*n+3) = a(2*n+1) + a(2*n+2).
a(2*n+2) = a(2*n) + 2*a(2*n+1).
G.f.: 2*(1 - x)*(2 + 3*x - x^2)/(1 - 4*x^2 + x^4).
E.g.f.: (1 + exp(-sqrt(6)*x))*((2 - sqrt(2))*exp(sqrt(2 - sqrt(3))*x) + (2 + sqrt(2))*exp(sqrt(2 + sqrt(3))*x))/2.
Lim_{n->infinity} a(2*n+1)/a(2*n) = (1 + sqrt(3))/2.

A327767 Period 2: repeat [1, -2].

Original entry on oeis.org

1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2

Offset: 1

Michael Somos, Sep 24 2019

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

Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000034, A040001, A134451, A168361, A280193.

&cat [[1, -2]^^50]; // Vincenzo Librandi, Feb 29 2020

a[ n_] := If[ n < 1, 0, -2 + 3 Mod[n, 2]];
a[ n_] := Which[ n < 1, 0, OddQ[n], 1, True, -2];
a[ n_] := SeriesCoefficient[ (x - 2*x^2) / (1 - x^2), {x, 0, n}];
PadRight[{}, 100, {1, -2}] (* Vincenzo Librandi, Feb 29 2020 *)

{a(n) = if( n<1, 0, -(1 + 3*(-1)^n)/2)};

PARI
```
{a(n) = if( n<1, 0, -2 + 3*(n%2))};
```

{a(n) = if( n<1, 0, [-2, 1][n%2 + 1])};

{a(n) = if( n<0, 0, polcoeff( (x - 2*x^2) / (1 - x^2) + x * O(x^n), n))};

G.f.: x * (1 - 2*x) / (1 - x^2) = x / (1 + 2*x / (1 - 3*x / (2 - x))).
E.g.f.: (exp(x) - 1)*(3/exp(x) - 1)/2.
a(n) is multiplicative with a(2^e) = -2 if e>0, a(p^e) = 1 otherwise.
Moebius transform is length 2 sequence [1, -3].
a(n) = -(1 + 3*(-1)^n)/2 if n>=1.
a(2*n) = -2, a(2*n + 1) = 1, a(0) = 0.
a(n) = -(-1)^n * A134451(n) for all n in Z.
a(n) = a(n+2) = -(-1)^n * A000034(n-1) = -A168361(n+1) for n>=1.
Dirichlet g.f.: zeta(s)*(1-3/2^s). - Amiram Eldar, Jan 03 2023

A346741 Irregular triangle read by rows which is constructed in row n replacing the first A000070(n-1) terms of A336811 with their divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 2, 3, 6, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Jul 31 2021

The terms in row n are also all parts of all partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246.
The terms of row n in nondecreasing order give the n-th row of A302247.
For further information about the correspondence divisor/part see A336811 and A338156.

			Triangle begins:
[1];
[1],[1, 2];
[1],[1, 2],[1, 3],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1],[1, 5],[1, 3],[1, 2],[1],[1];
...
Below the table shows the correspondence divisor/part.
|---|-----------------|-----|-------|---------|-----------|-------------|
| n |                 |  1  |   2   |    3    |     4     |      5      |
|---|-----------------|-----|-------|---------|-----------|-------------|
| P |                 |     |       |         |           |             |
| A |                 |     |       |         |           |             |
| R |                 |     |       |         |           |             |
| T |                 |     |       |         |           |  5          |
| I |                 |     |       |         |           |  3 2        |
| T |                 |     |       |         |  4        |  4 1        |
| I |                 |     |       |         |  2 2      |  2 2 1      |
| O |                 |     |       |  3      |  3 1      |  3 1 1      |
| N |                 |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |                 |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
----|-----------------|-----|-------|---------|-----------|-------------|
.
|---|-----------------|-----|-------|---------|-----------|-------------|
|   |         A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |                 |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I |         A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |                 |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K |         A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |                 |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   |         A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|-----------------|-----|-------|---------|-----------|-------------|
.
.   |-------|
.   |Section|
|---|-------|---------|-----|-------|---------|-----------|-------------|
|   |   1   | A000012 |  1  |  1    |  1      |  1        |  1          |
|   |-------|---------|-----|-------|---------|-----------|-------------|
|   |   2   | A000034 |     |  1 2  |  1 2    |  1 2      |  1 2        |
|   |-------|---------|-----|-------|---------|-----------|-------------|
| D |   3   | A010684 |     |       |  1   3  |  1   3    |  1   3      |
| I |       | A000012 |     |       |  1      |  1        |  1          |
| V |-------|---------|-----|-------|---------|-----------|-------------|
| I |   4   | A069705 |     |       |         |  1 2   4  |  1 2   4    |
| S |       | A000034 |     |       |         |  1 2      |  1 2        |
| O |       | A000012 |     |       |         |  1        |  1          |
| R |-------|---------|-----|-------|---------|-----------|-------------|
| S |   5   | A010686 |     |       |         |           |  1       5  |
|   |       | A010684 |     |       |         |           |  1   3      |
|   |       | A000034 |     |       |         |           |  1 2        |
|   |       | A000012 |     |       |         |           |  1          |
|   |       | A000012 |     |       |         |           |  1          |
|---|-------|---------|-----|-------|---------|-----------|-------------|
.
In the above table both the zone of partitions and the "Link" zone are the same zones as in the table of the example section of A338156, but here in the lower zone the divisors are ordered in accordance with the sections of the set of partitions of n.
The number of rows in the j-th section of the lower zone is equal to A000041(j-1).
The divisors of the j-th section are also the parts of the j-th section of the set of partitions of n.

Another version of A338156.
Row n has length A006128(n).
The sum of row n is A066186(n).
The product of row n is A007870(n).
Row n lists the first n rows of A336812.
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
Cf. A000012, A000034, A000041, A000070, A002260, A010684, A010686, A027750, A066633, A069705, A135010, A138785, A181187, A221529, A221649, A237593, A302246, A302247, A336811, A340011, A340031, A340032, A340035, A340056, A340057.

A374434 Triangle read by rows: T(n, k) = Product_{p in PF(n) symmetric difference PF(k)} p, where PF(a) is the set of the prime factors of a.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 6, 1, 2, 2, 1, 6, 1, 5, 5, 10, 15, 10, 1, 6, 6, 3, 2, 3, 30, 1, 7, 7, 14, 21, 14, 35, 42, 1, 2, 2, 1, 6, 1, 10, 3, 14, 1, 3, 3, 6, 1, 6, 15, 2, 21, 6, 1, 10, 10, 5, 30, 5, 2, 15, 70, 5, 30, 1, 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1

Offset: 0

Peter Luschny, Jul 10 2024

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  1;
  [ 3]  3,  3,  6,  1;
  [ 4]  2,  2,  1,  6,  1;
  [ 5]  5,  5, 10, 15, 10,  1;
  [ 6]  6,  6,  3,  2,  3, 30,  1;
  [ 7]  7,  7, 14, 21, 14, 35, 42,  1;
  [ 8]  2,  2,  1,  6,  1, 10,  3, 14,  1;
  [ 9]  3,  3,  6,  1,  6, 15,  2, 21,  6,  1;
  [10] 10, 10,  5, 30,  5,  2, 15, 70,  5, 30,   1;
  [11] 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1;

Family: A374433 (intersection), this sequence (symmetric difference), A374435 (difference), A374436 (union).

Cf. A007947 (column 0), A000034 (central terms), A050873 (gcd).

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374434 := (n, k) -> mul(symmdiff(PF(n), PF(k))):
seq(print(seq(A374434(n, k), k = 0..n)), n = 0..11);

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Times @@ SymmetricDifference[s[k], s[n]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

# Function A374434 defined in A374433.
for n in range(11): print([A374434(n, k) for k in range(n + 1)])

From Michael De Vlieger, Jul 11 2024: (Start)
T(0,0) = T(n,0) = rad(n)/rad(0) = 1 where rad = A007947;
T(n,k) = rad(k*n)/rad(gcd(k,n))
       = A007947(k*n)/A007947(S(n,k)) where S = A050873
       = A374436(n,k)/A374433(n,k). (End)

A374435 Triangle read by rows: T(n, k) = Product_{p in PF(n) difference PF(k)} p, where PF(a) is the set of the prime factors of a.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 2, 2, 1, 2, 1, 5, 5, 5, 5, 5, 1, 6, 6, 3, 2, 3, 6, 1, 7, 7, 7, 7, 7, 7, 7, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 10, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1

Offset: 0

Peter Luschny, Jul 10 2024

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  1;
  [ 3]  3,  3,  3,  1;
  [ 4]  2,  2,  1,  2,  1;
  [ 5]  5,  5,  5,  5,  5,  1;
  [ 6]  6,  6,  3,  2,  3,  6,  1;
  [ 7]  7,  7,  7,  7,  7,  7,  7,  1;
  [ 8]  2,  2,  1,  2,  1,  2,  1,  2,  1;
  [ 9]  3,  3,  3,  1,  3,  3,  1,  3,  3,  1;
  [10] 10, 10,  5, 10,  5,  2,  5, 10,  5, 10,  1;
  [11] 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1;

Family: A374433 (intersection), A374434 (symmetric difference), this sequence (difference), A374436 (union).

Cf. A007947 (column 0), A000034 (central terms).

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374435 := (n, k) -> mul(PF(n) minus PF(k)):
seq(print(seq(A374435(n, k), k = 0..n)), n = 0..11);

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Apply[Times, Complement[s[n], s[k]]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

# Function A374435 defined in A374433.
for n in range(12): print([A374435(n, k) for k in range(n + 1)])

A010721 Period 2: repeat (5,10).

Original entry on oeis.org

5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10

Offset: 0

N. J. A. Sloane

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

Cf. A176325.

Table[5 + 5 Mod[n, 2], {n, 0, 50}] (* Jinyuan Wang, Feb 26 2020 *)
PadRight[{},80,{5,10}] (* Harvey P. Dale, Mar 18 2022 *)

a(n)=n%2*5+5 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = -5/2*(-1)^n + 15/2. - Paolo P. Lava, Oct 27 2006
From R. J. Mathar, Oct 20 2008: (Start)
a(n) = 5*A000034(n).
G.f.: 5(1+2x)/((1-x)(1+x)). (End)

cp's OEIS Frontend

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