Ron R. King - cp's OEIS frontend

A230445 Triangle read by rows: T(n,m) = 3^m*2^(n-m)-1, the number of neighbors in an n-dimensional cubic array.

0, 1, 2, 3, 5, 8, 7, 11, 17, 26, 15, 23, 35, 53, 80, 31, 47, 71, 107, 161, 242, 63, 95, 143, 215, 323, 485, 728, 127, 191, 287, 431, 647, 971, 1457, 2186, 255, 383, 575, 863, 1295, 1943, 2915, 4373, 6560, 511, 767, 1151, 1727, 2591, 3887, 5831, 8747, 13121

Offset: 0

Ron R. King, Oct 18 2013

Let n be the dimension of the cubic array.
Let m be the "placement depth" of the cell within the array. m = (number of horizontal or vertical neighbors)-n. 0 <= m <= n.
Let T(n,m) represent the number of neighbors (horizontally, vertically, or diagonally) a cell has in an n-dimensional cube that has at least 3^n cells.
The sequence forms a triangle structure similar to Pascal’s triangle: T(0,0) in row one, T(1,0), T(1,1) in row two, etc.
The triangle in A094615 is a subtriangle. - Philippe Deléham, Oct 31 2013
In a finite n-dimensional hypercube lattice, the sequence gives the number of nodes situated at a Chebyshev distance of 1 for a node, situated on an m-cube bound, which is not on an (m-1)-cube bound. The number of m-cube bounds for n-cube is given by A013609. In cellular automata theory, the cell surrounding with Chebyshev distance 1 is called Moore's neighborhood. For von Neumann neighborhood (with Manhattan distance 1), an analogous sequence is represented by A051162. - Dmitry Zaitsev, Oct 22 2015

			Triangle starts:
n \ m  0    1    2    3    4    5     6     7     8     9    10 ...
0:     0
1:     1    2
2:     3    5    8
3:     7   11   17   26
4:    15   23   35   53   80
5:    31   47   71  107  161  242
6:    63   95  143  215  323  485   728
7:   127  191  287  431  647  971  1457  2186
8:   255  383  575  863 1295 1943  2915  4373  6560
9:   511  767 1151 1727 2591 3887  5831  8747 13121 19682
10: 1023 1535 2303 3455 5183 7775 11663 17495 26243 39365 59048
... (reformatted (and extended) by _Wolfdieter Lang_, May 04 2022)
For a 3-d cube, at a corner, the number of horizontal and vertical neighbors is 3, hence m = 3-3 = 0.
Along the edge, the number of horizontal and vertical neighbors is 4, hence m = 4-3 = 1.
In a face, the number of horizontal and vertical neighbors is 5, hence m = 5-3 = 2.
In the interior, the number of horizontal and vertical neighbors is 6, hence m = 6-3 = 3.
T(3,2) = 17 because a cell on the face of a 3-d cube has 17 neighbors.

Dmitry A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35.

Cf. A000225, A024023, A051162, A013609.
Sequence numbers are 1 less than A036561.
Cf. A048473, A094615.

void a10(){int p3[10], p2[10], n, m, a; p3[0]=1; p2[0]=1;
for(n=1;n<10;n++){ p2[n]=p2[n-1]*2; p3[n]=p3[n-1]*3;
  for(m=0;m<=d;m++){ a=p3[m]*p2[n-m]-1; printf("%d ",a); }
  printf("\n"); } } /* Dmitry Zaitsev, Oct 23 2015 */

Table[3^m 2^(n - m) - 1, {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Oct 23 2015 *)

T(n,m) = 3^m*2^(n-m)-1, 0 <= m <= n.
T(n,0) = 2^n-1. (A000225)
T(n,n) = 3^n-1. (A024023)
T(n,m) = (3*T(n,m-1)+1)/2, first part of the Collatz sequence for the number 2^n-1, for n >= 1.
T(n,m) = (T(n-1,m) + T(n,m+1))/2, 0 <= m <= n-1.
T(n,m) = 1 + T(n-1,m-1) + T(n,m-1), 1 <= m <= n.
m = T2(n,k)-n, where T2(n,k) is A051162.
From Wolfdieter Lang, May 04 2022: (Start)
G.f. for column m: G(m, x) = x^m*(3^m - 1 - (3^m - 2)*x)/((1 - 2*x)*(1 - x)).
G.f. for row polynomials R(n, x) = Sum_{m=1..n} T(n, m)*x^m, for n >= 0: G(z, x) = z*(1 + (2 - 5*z)*x)/((1 - 2*z)*(1 - z)*(1 - 3*x*z)*(1 - x*z)).
(End)

A157806 Absolute value of the difference between numerator and denominator of fractions arranged by Cantor's ordering (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, ...) with equivalent fractions removed.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 1, 3, 4, 4, 5, 3, 1, 1, 3, 5, 6, 2, 2, 6, 7, 5, 1, 1, 5, 7, 8, 4, 4, 8, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 10, 2, 2, 10, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 12, 8, 4, 4, 8, 12, 13, 11, 7, 1, 1, 7, 11, 13, 14, 10, 6, 2, 2, 6, 10, 14, 15, 13, 11

Offset: 0

Ron R. King, Mar 07 2009

Andrew Howroyd, Table of n, a(n) for n = 0..10000

Sum of numerator and denominator is given by A038567.

seq(n)={my(L=List(), r=1, k=1); while(#L<=n, if(k==r, r++; k=0); k++; if(gcd(k,r-k)==1, listput(L, abs(r-2*k)))); Vec(L)} \\ Andrew Howroyd, Feb 03 2021

Terms a(56) and beyond from Andrew Howroyd, Feb 03 2021

A157807 Numerators of fractions arranged in "antidiagonal boustrophedon" ordering with equivalent fractions removed: (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, 5/2, ...).

Original entry on oeis.org

1, 2, 1, 1, 3, 4, 3, 2, 1, 1, 5, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 8, 7, 5, 4, 2, 1, 1, 3, 7, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 5, 7, 11, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 9, 11, 13, 14, 13, 11, 8, 7, 4, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 16, 15, 14

Offset: 1

Ron R. King, Mar 07 2009

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A157813 (denominators), A038566.

With Cantor's ordering: A020652, A020653, A352911.

R:= NULL: count:= 0:
for m from 2 while count < 100 do
  S:= select(t -> igcd(t,m-t)=1, [$1..m-1]);
  count:= count+nops(S);
  if m::even then R:= R, op(S) else R:= R, seq(m-t,t=S) fi;
od:
R; # Robert Israel, Oct 09 2023

from math import gcd
for s in range(2, 100, 2):
  for i in range(1, s):
    if gcd(i, s - i) != 1: continue
    print(i)
  for i in range(s, 0, -1):
    if gcd(i, s + 1 - i) != 1: continue
    print(i)
# Hiroaki Yamanouchi, Oct 06 2014

A-number in cross-reference corrected by R. J. Mathar, Sep 23 2009
a(19)-a(20) corrected and a(58)-a(82) added by Hiroaki Yamanouchi, Oct 06 2014
Name corrected by Andrey Zabolotskiy, Oct 10 2023

A157813 Denominators of fractions arranged in "antidiagonal boustrophedon" ordering with equivalent fractions removed: (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, 5/2, ...).

Original entry on oeis.org

1, 1, 2, 3, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 6, 7, 5, 3, 1, 1, 2, 4, 5, 7, 8, 9, 7, 3, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 7, 5, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 11, 9, 5, 3, 1, 1, 2, 4, 7, 8, 11, 13, 14, 15, 13, 11, 9, 7, 5, 3, 1, 1, 2, 3, 4

Offset: 1

Ron R. King, Mar 07 2009

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A157807 (numerators), A038567.

With Cantor's ordering: A020652, A020653, A352911.

R:= NULL: count:= 0:
for m from 2 while count < 100 do
  S:= select(t -> igcd(t,m-t)=1, [$1..m-1]);
  count:= count+nops(S);
  if m::odd then R:= R, op(S) else R:= R, seq(m-t,t=S) fi;
od:
R; # Robert Israel, Oct 09 2023

from math import gcd
for s in range(2, 100, 2):
  for i in range(1, s):
    if gcd(i, s - i) != 1: continue
    print(s - i)
  for i in range(s, 0, -1):
    if gcd(i, s + 1 - i) != 1: continue
    print(s + 1 - i)
# Hiroaki Yamanouchi, Oct 06 2014

a(58)-a(83) from Hiroaki Yamanouchi, Oct 06 2014

Name corrected by Andrey Zabolotskiy, Oct 10 2023

A123522 Not of the form n + [log_10 n].

Original entry on oeis.org

0, 10, 101, 1002, 10003, 100004, 1000005, 10000006, 100000007, 1000000008, 10000000009, 100000000010, 1000000000011, 10000000000012, 100000000000013, 1000000000000014, 10000000000000015, 100000000000000016

Offset: 0

Ron R. King, Nov 10 2006

Smallest number such that (sum of digits) mod (number of digits) = n in decimal representation. - Reinhard Zumkeller, Aug 15 2010

Consider a word on the alphabet {0,1,2,...,9} that has length of 10^a(n). The expected number of occurrences of a pattern (contiguous subsequence) p_1,p_2,...p_n for all such words is 1. - Geoffrey Critzer, Feb 03 2012

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (12, -21, 10).

[10^n + n -1: n in [0..30]]; // G. C. Greubel, Nov 30 2017

Table[10^n + n - 1, {n, 0, 20}]  (* Geoffrey Critzer, Feb 03 2012 *)
LinearRecurrence[{12,-21,10},{0,10,101},20] (* Harvey P. Dale, Jul 20 2021 *)

for(n=0,30, print1(10^n + n -1, ", ")) \\ G. C. Greubel, Nov 30 2017

a(n) = 10^n + n - 1.
From Reinhard Zumkeller, Aug 15 2010: (Start)
A180160(a(n)) = n and A180160(m) <> n for m < a(n).
A007953(a(n)) = n; A055642(a(n)) = n + 1. (End)
From R. J. Mathar, Aug 15 2010: (Start)
G.f.: x*(-10+19*x) / ( (10*x-1)*(x-1)^2 ).
a(n) = 12*a(n-1) -21*a(n-2) +10*a(n-3). (End)

a(0) and terms beyond a(9) from Reinhard Zumkeller, Aug 15 2010

A118656 If C represents the minimum number of segments required to construct n unit cubes, then this sequence represents the number of additional segments required to construct n+1 unit cubes. First differences of sequence A117227.

Original entry on oeis.org

12, 8, 8, 5, 8, 5, 5, 3, 8, 5, 5, 3, 8, 5, 5, 3, 5, 3, 8, 5, 5, 3, 5, 3, 5, 3, 3

Offset: 1

Ron R. King, May 18 2006

			a(1)=12 because 12 segments are required to construct the first unit cube. a(2)=8 because 8 additional unit segments are required to construct two unit cubes.

Cf. A117227.

A118659 Minimum number of unit faces required to construct n unit cubes.

Original entry on oeis.org

6, 11, 16, 20, 25, 29, 33, 36, 41, 45, 49, 52, 57, 61, 65, 68, 72, 75, 80, 84, 88, 91, 95, 98, 102, 105, 108, 113, 117, 121, 124, 128, 131, 135, 138, 141, 146, 150, 154, 157, 161, 164, 168, 171, 174, 178, 181, 184, 189, 193, 197, 200, 204, 207, 211, 214, 217, 221, 224

Offset: 1

Ron R. King, May 18 2006

			a(2)=11 because 6 unit faces are required to construct each cube but 1 face is shared. I.e., a(2)=6+5=11.

Cf. A117227, A270205.

a(n^3) = 3*(n^2)*(n+1) = A270205(n+1). - Mohammed Yaseen, Aug 22 2021

Missing term a(56)=214 inserted by Mohammed Yaseen, Aug 22 2021

A117227 Minimum number of unit segments required to construct n unit cubes.

Original entry on oeis.org

12, 20, 28, 33, 41, 46, 51, 54, 62, 67, 72, 75, 83, 88, 93, 96, 101, 104, 112, 117, 122, 125, 130, 133, 138, 141, 144

Offset: 1

Ron R. King, Apr 22 2006

This generalizes the 2-dimensional version which is A078633.

			a(2)=20 because 12 unit segments are required to construct each cube but 4 units are shared. I.e., a(2)=12+8=20.

Cf. A078633.

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User: Ron R. King

A230445 Triangle read by rows: T(n,m) = 3^m*2^(n-m)-1, the number of neighbors in an n-dimensional cubic array.

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A157806 Absolute value of the difference between numerator and denominator of fractions arranged by Cantor's ordering (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, ...) with equivalent fractions removed.

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A157807 Numerators of fractions arranged in "antidiagonal boustrophedon" ordering with equivalent fractions removed: (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, 5/2, ...).

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A157813 Denominators of fractions arranged in "antidiagonal boustrophedon" ordering with equivalent fractions removed: (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, 5/2, ...).

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A123522 Not of the form n + [log_10 n].

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A118656 If C represents the minimum number of segments required to construct n unit cubes, then this sequence represents the number of additional segments required to construct n+1 unit cubes. First differences of sequence A117227.

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A118659 Minimum number of unit faces required to construct n unit cubes.

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Examples

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Extensions

A117227 Minimum number of unit segments required to construct n unit cubes.

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