Christopher J. Shore - cp's OEIS frontend

User: Christopher J. Shore

Christopher J. Shore has authored 3 sequences.

A329267 a(n) is the absolute difference between n and its nearest palindromic neighbor.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2

Offset: 0

Christopher J. Shore, Nov 09 2019

Empirical observation: this sequence is similar to A261424 but yields the absolute difference between n and its nearest palindromic neighbor. It answers the question "How far from this number is the nearest palindrome?"

			For 0 <= n <= 9, n is palindromic so a(n) = 0.
a(10) = 10-9 = 11-10 = 1 (10 is equidistant from its two nearest palindromes).
a(11) = 0 because 11 is palindromic.
For 12 <= n <= 16, a(n) = n-11 because 11 is the nearest palindromic number.
For 17 <= n <= 22, a(n) = 22-n because 22 is the nearest palindromic number.
.
   n  nearest palindrome  difference
  --  ------------------  ----------
   1           1            1-1 = 0
   2           2            2-2 = 0
   3           3            3-3 = 0
   4           4            4-4 = 0
   5           5            5-5 = 0
   6           6            6-6 = 0
   7           7            7-7 = 0
   8           8            8-8 = 0
   9           9            9-9 = 0
  10        9 or 11     10-9 = 11-10 = 1
  11          11           11-11 = 0
  12          11           12-11 = 1
  13          11           13-11 = 2
  14          11           14-11 = 3
  15          11           15-11 = 4
  16          11           16-11 = 5
  17          22           22-17 = 5
  18          22           22-18 = 4
  19          22           22-19 = 3
  20          22           22-20 = 2
  21          22           22-21 = 1
  22          22           22-22 = 0
  23          22           23-22 = 1

Index entries for sequences related to distance to nearest element of some set

Cf. A261424, A261423.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse@ d]; a[n_] := Block[{k=0}, While[! palQ[n+k] && ! palQ[n-k], k++]; k]; Array[a, 121] (* Giovanni Resta, Nov 12 2019 *)

ispal(n) = my (d=digits(n)); d==Vecrev(d)
a(n) = for (k=0, oo, if (ispal(n-k) || ispal(n+k), return (k))) \\ Rémy Sigrist, Dec 03 2019

A272459 The total number of different isosceles trapezoids, excluding squares, that can be drawn on an n X n square grid where the corners of each individual trapezoid lie on a lattice point.

Original entry on oeis.org

0, 1, 7, 18, 40, 71, 119, 180, 264, 365, 495, 646, 832, 1043, 1295, 1576, 1904, 2265, 2679, 3130, 3640, 4191, 4807, 5468, 6200, 6981, 7839, 8750, 9744, 10795, 11935, 13136, 14432, 15793, 17255, 18786, 20424, 22135, 23959, 25860, 27880, 29981, 32207, 34518

Offset: 1

Christopher J. Shore, Apr 29 2016

This is an observation from a high school mathematics investigation: How many different isosceles trapezoids can be drawn on an n X n grid such that the corners of each individual trapezoid lie on a lattice point? The sequence gives the total number of different trapezoids that can be drawn.
There are two "families" or types of trapezoids that can be drawn on a grid. The first is where the parallel sides are drawn horizontally on the grid. The second is where the parallel sides are drawn diagonally with a gradient of 1. The number in each type follow a pattern.
X 1 grid: No trapezoids of either type can be drawn.
X 2 grid: 1 trapezoid of type 2. One parallel side is drawn diagonally through 1 square (having length sqrt(2)) and the other is drawn diagonally through two squares (length 2*sqrt(2)). Thus, the non-parallel sides are drawn horizontally or vertically to join between the parallel sides (each length 1).
X 3 grid: 3 trapezoids of type 1 and 4 trapezoids of type 2. The 3 trapezoids of type 1 are constructed by one parallel line drawn horizontally with length 3, the other parallel line drawn with length 1 and the perpendicular heights being successively 1, 2 and 3. Type-2 trapezoids are constructed in the same way as outlined above but with varying lengths and heights.
X 4 grid: 8 type-1 trapezoids and 10 type-2 trapezoids.
X 5 grid: 20 type-1 trapezoids and 20 type-2 trapezoids.
Hence the pattern is as follows:
              Type 1    Type 2    Total
X 1 grid       0         0        0
X 2 grid       0         1        1
X 3 grid       3         4        7
X 4 grid       8        10       18
X 5 grid      20        20       40
X 6 grid      36        35       71
X 7 grid      63        56      119

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000292, A032438, A143785.

[(n*(-1-3*(-1)^n-12*n+10*n^2))/24 : n in [1..60]]; // Wesley Ivan Hurt, Sep 12 2016

A272459:=n->(n*(-1-3*(-1)^n-12*n+10*n^2))/24: seq(A272459(n), n=1..60); # Wesley Ivan Hurt, Sep 12 2016

CoefficientList[Series[x^2 (1 + 5 x + 3 x^2 + x^3)/((1 - x)^4 (1 + x)^2), {x, 0, 44}], x] (* Michael De Vlieger, May 08 2016 *)

concat(0, Vec(x^2*(1+5*x+3*x^2+x^3)/((1-x)^4*(1+x)^2) + O(x^50))) \\ Colin Barker, May 07 2016

a(n) = Sum_{k=0..n} A032438(k) + A000292(n-1). (conjectured)
a(n) = A143785(n-2) + A000292(n-1). (conjectured)
From Colin Barker, May 07 2016: (Start)
a(n) = (n*(-1 - 3*(-1)^n - 12*n + 10*n^2))/24.
a(n) = (5*n^3 - 6*n^2 - 2*n)/12 for n even.
a(n) = (5*n^3 - 6*n^2 + n)/12 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.
G.f.: x^2*(1+5*x+3*x^2+x^3) / ((1-x)^4*(1+x)^2).
(End)

A265384 Toothpick sequence starting at the vertex of y=3*abs(x).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 21, 23, 25, 27, 31, 35, 39, 43, 47, 55, 63, 65, 67, 69, 73, 77, 81, 85, 89, 97, 105, 109, 113, 117, 125, 133, 141, 149, 157, 173, 189, 191, 193, 195, 199, 203, 207, 211, 215, 223, 231, 235, 239, 243, 251, 259, 267, 275, 283, 299, 315, 319, 323, 327, 335, 343, 351, 359, 367, 383, 399, 407, 415, 423, 439, 455, 471, 487, 503, 535, 567

Offset: 1

Christopher J. Shore, Dec 07 2015

nonn

Consider the graph y=3*abs(x). The first toothpick extends vertically from (0,0) to (0,2). Each toothpick is of length 2 and is laid either horizontally or vertically.
Subsequent toothpicks are placed in a similar rule as A139250. Place toothpicks by the following rules:
- Toothpicks must always stay inside the graph of y=3*abs(x).
- Call the end of a toothpick exposed if it does not touch another toothpick, or the line y=3*abs(x)
- Each horizontal toothpick has its midpoint touching an exposed vertical toothpick
- If no horizontal toothpick can be laid, then a vertical toothpick should be laid on any exposed ends, from its end.
The sequence is the number of toothpicks laid after n rounds.
The structure is essentially the same as the Sierpinski's triangle but here every equilateral triangle is replaced with an isosceles triangle and then every isosceles triangle is replaced with seven toothpicks. There are infinitely many sequences of this type. - Omar E. Pol, Mar 12 2016

			The pattern is the total number of toothpicks laid after n rounds.
Following the rules above, the first round has 1 toothpick, the second and third rounds also have 1 toothpick, but the fourth and fifth round both have 2 toothpicks. Finding the total toothpicks placed in this pattern (1,1,1,2,2) gives 1,2,3,5,7. Subsequent rounds have this same pattern repeated from the emerging branches thus:
(1,1,1,2,2) ---> 1,2,3,5,7
2*(1,1,1,2,2) ---> 9,11,13,17,21
2*((1,1,1,2,2),2*(1,1,1,2,2)) ---> 23,25,27,31,35,39,43,47,55,63
2*((1,1,1,2,2),2*(1,1,1,2,2),2*((1,1,1,2,2),2*(1,1,1,2,2))) ---> 65,67,69,73,77,81,85,89,97,105,109,113,117,125,133,141,149,157,173,189
Summation of 1*the sequence 1,1,1,2,2
  (1)=1
1+(1)=2
2+(1)=3
3+(2)=5
5+(2)=7
Summation of 2*the sequence 1,1,1,2,2
7+2(1)=9
9+2(1)=11
11+2(1)=13
13+2(2)=17
17+2(2)=21
Summation of 3*the sequence 1,1,1,2,2
21+2(1)=23
23+2(1)=25
25+2(1)=27
27+2(2)=31
31+2(2)=35

Christopher J. Shore, Geogebra image of the toothpick sequence.
Index entries for sequences related to toothpick sequences

Cf. A047999, A151566, A139250.

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User: Christopher J. Shore

A329267 a(n) is the absolute difference between n and its nearest palindromic neighbor.

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A272459 The total number of different isosceles trapezoids, excluding squares, that can be drawn on an n X n square grid where the corners of each individual trapezoid lie on a lattice point.

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A265384 Toothpick sequence starting at the vertex of y=3*abs(x).

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