A135600 -id:A135600 - cp's OEIS frontend

A135643 Straight-line numbers > 99.

111, 123, 135, 147, 159, 210, 222, 234, 246, 258, 321, 333, 345, 357, 369, 420, 432, 444, 456, 468, 531, 543, 555, 567, 579, 630, 642, 654, 666, 678, 741, 753, 765, 777, 789, 840, 852, 864, 876, 888, 951, 963, 975, 987, 999, 1111, 1234

Offset: 1

Omar E. Pol, Nov 30 2007, Dec 09 2008, Nov 14 2009

Numbers with more than two digits whose digits are in arithmetic progression. The structure of digits represents a straight line. In the graphic representation the points are connected by imaginary line segments. For a(1) to a(45) this sequence is equal to A034840. Each term of this sequence that is greater than 9876543210 is a repdigit number (A010785).

Note that the sequence of straight-line numbers starts: 10, 11, 12, ..., 98, 99, 111, 123, ... All 2-digit numbers are straight-line numbers, but here the numbers < 100 are omitted. - Omar E. Pol, Nov 14 2009

			The number 3579 is a straight-line number:
  . . . 9
  . . . .
  . . 7 .
  . . . .
  . 5 . .
  . . . .
  3 . . .
  . . . .
  . . . .
  . . . .

Jens Kruse Andersen, Table of n, a(n) for n = 1..168

Cf. A010785, A034840.
Cf. A135600, A135601, A135602, A135603, A135641, A135642, A163278, A167847. - Omar E. Pol, Nov 14 2009
Cf. A247616 (subsequence).

a135643 n = a135643_list !! (n-1)
a135643_list = filter f [100..] where
   f x = all (== 0) ws where
         ws = zipWith (-) (tail vs) vs
         vs = zipWith (-) (tail us) us
         us = map (read . return) $ show x
-- Reinhard Zumkeller, Sep 21 2014

Select[Range[100,1300],Length[Union[Differences[IntegerDigits[#]]]]==1&] (* Harvey P. Dale, May 09 2012 *)

is(n) = my (d=digits(n), cvx=0, ccv=0, str=0); for (i=1, #d-2, my (x=d[i]+d[i+2]-2*d[i+1]); if (x>0, cvx++, x<0, ccv++, str++)); return (cvx==0 && ccv==0 && str>0) \\ Rémy Sigrist, Aug 09 2017

from itertools import count, islice
def agen():
    progressions = ["".join(map(str, range(i, j+1, d))) for i in range(10) for d in range(1, 10-i) for j in range(i+2*d, 10)]
    s =  [p for p in progressions if p[0] != "0"]          # up
    s += [p[::-1] for p in progressions]                   # down
    s += [d*i for d in "123456789" for i in range(3, 11)]  # flat
    yield from sorted(set(int(w) for w in s))
    yield from (int(f*d) for d in count(11) for f in "123456789")
print(list(islice(agen(), 178))) # Michael S. Branicky, Aug 03 2022

A135642 Concave numbers.

Original entry on oeis.org

110, 120, 121, 122, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 145, 146, 150, 151, 152, 153, 154, 155, 156, 157, 158, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182

Offset: 1

Omar E. Pol, Nov 30 2007

The structure of digits represents a concave function or a concave object. In the graphic representation the points are connected by imaginary line segments from left to right.
Only strictly concave numbers are included in this sequence; the interpolation between at least one pair of digits must be strictly less than some intermediate digit. - Franklin T. Adams-Watters, Jan 26 2014
Also numbers where the second difference of consecutive digits is at most 0 and at least one of the second differences is negative. - David A. Corneth, Aug 02 2022

			The number 12221 is a concave number. Note that the number of this sequence (A135642) is also a concave number as shown below:
.
9     . . . . . .      . . . . . .
8     . . . . . .      . . . . . .
7     . . . . . .      . . . . . .
6     . . . x . .      . . . 6 . .
5     . . x . . .      . . 5 . . .
4     . . . . x .      . . . . 4 .
3     . x . . . .      . 3 . . . .
2     . . . . . x      . . . . . 2
1     x . . . . .      1 . . . . .
0     . . . . . .      . . . . . .
.
Another example is 1342. On the other hand, 3124 is not in the sequence, it's in A135641. 1234 is not in the sequence, it's in A135643. 1243 is not in the sequence, it's in A163278. - _Omar E. Pol_, Jan 29 2014

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A135600, A135601, A135602, A135603, A135641, A135643, A163278.

concaveQ[n_] := With[{dd = IntegerDigits[n]}, AllTrue[SequencePosition[dd, {, , _}][[All, 1]], dd[[#]] + dd[[#+2]] < 2 dd[[#+1]]&]];
Select[Range[100, 200], concaveQ] (* Jean-François Alcover, Nov 01 2018 *)

{ isconcave(n) = my(t,r); t=eval(Vec(Str(n))); r=0; for(i=1, #t, for(j=i+2, #t, for(k=i+1, j-1, if( t[k]*(j-i) < t[i]*(j-k) + t[j]*(k-i), return(0)); if( t[k]*(j-i) > t[i]*(j-k) + t[j]*(k-i), r=1); ))); r } /* Franklin T. Adams-Watters and Max Alekseyev, Jan 30 2014 */

is(n) = if(n<100, return(0)); my(d=digits(n), v=vector(#d-2, i, d[i+2] - 2*d[i+1] + d[i])); v=Set(v); v[1] < 0 && v[#v] <= 0 \\ David A. Corneth, Aug 02 2022

A135602 Right-angled numbers with an internal digit as the vertex.

Original entry on oeis.org

101, 121, 212, 232, 323, 343, 434, 454, 545, 565, 656, 676, 767, 787, 878, 898, 989, 1012, 1210, 1232, 2101, 2123, 2321, 2343, 3212, 3234, 3432, 3454, 4323, 4345, 4543, 4565, 5434, 5456, 5654, 5676, 6545, 6567, 6765, 6787, 7656, 7678, 7876, 7898, 8767, 8789, 8987

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents a right angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite. The final term has 19 digits: 9876543210123456789.

			Illustration using the final term of this sequence:
  9 . . . . . . . . . . . . . . . . . 9
  . 8 . . . . . . . . . . . . . . . 8 .
  . . 7 . . . . . . . . . . . . . 7 . .
  . . . 6 . . . . . . . . . . . 6 . . .
  . . . . 5 . . . . . . . . . 5 . . . .
  . . . . . 4 . . . . . . . 4 . . . . .
  . . . . . . 3 . . . . . 3 . . . . . .
  . . . . . . . 2 . . . 2 . . . . . . .
  . . . . . . . . 1 . 1 . . . . . . . .
  . . . . . . . . . 0 . . . . . . . . .

Jens Kruse Andersen, Table of n, a(n) for n = 1..525

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135601, A135603.

ups = list(tuple(range(i, j)) for i in range(9) for j in range(i+2, 11))
s = set(L[::-1] + R[1:] for L in ups for R in ups if L[0] == R[0])
s |= set(L[:-1] + R[::-1] for L in ups for R in ups if L[-1] == R[-1])
afull = sorted(int("".join(map(str, t))) for t in s if t[0] != 0)
print(afull[:47]) # Michael S. Branicky, Aug 02 2022

A135601 Acute-angled numbers with an internal digit as the vertex.

Original entry on oeis.org

102, 103, 104, 105, 106, 107, 108, 109, 120, 130, 131, 132, 140, 141, 142, 143, 150, 151, 152, 153, 154, 160, 161, 162, 163, 164, 165, 170, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 186, 187, 190, 191, 192, 193, 194, 195

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents an acute angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite. The final term has 14 digits: 98765432102468.

			Illustration using the final term of this sequence:
  9 . . . . . . . . . . . . .
  . 8 . . . . . . . . . . . 8
  . . 7 . . . . . . . . . . .
  . . . 6 . . . . . . . . 6 .
  . . . . 5 . . . . . . . . .
  . . . . . 4 . . . . . 4 . .
  . . . . . . 3 . . . . . . .
  . . . . . . . 2 . . 2 . . .
  . . . . . . . . 1 . . . . .
  . . . . . . . . . 0 . . . .

Micah Manary, Table of n, a(n) for n = 1..2337
Micah Manary, Python program

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135602, A135603.

progressions = set(tuple(range(i, j+1, d)) for i in range(10) for d in range(1, 10-i) for j in range(i+d, 10))
s = set()
for left in progressions:
    for right in progressions:
        dl, dr = left[1] - left[0], right[1] - right[0]
        if dl + dr > 2:
            if left[-1] == right[-1]: s.add(left[:-1] + right[::-1])
            if left[0] == right[0]: s.add(left[::-1] + right[1:])
afull = sorted(int("".join(map(str, t))) for t in s if t[0] != 0)
print(afull[:53]) # Michael S. Branicky, Aug 02 2022

If a(n) does not end in 0, then A004086(a(n)) is a term; if a(n) does not start with 9, then A061601(a(n)) is a term. - Michael S. Branicky, Aug 02 2022

A135603 Obtuse-angled numbers with an internal digit as the vertex.

Original entry on oeis.org

100, 110, 112, 113, 114, 115, 116, 117, 118, 119, 122, 124, 125, 126, 127, 128, 129, 133, 134, 136, 137, 138, 139, 144, 145, 146, 148, 149, 155, 156, 157, 158, 166, 167, 168, 169, 177, 178, 179, 188, 189, 199, 200, 211, 220, 221, 223, 224, 225, 226, 227, 228, 229, 233

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents an obtuse angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right.

For each k >= 11, there are 354 k-digit terms. - Michael S. Branicky, Aug 03 2022

			Illustration of the number 9753111:
  9 . . . . . .
  . . . . . . .
  . 7 . . . . .
  . . . . . . .
  . . 5 . . . .
  . . . . . . .
  . . . 3 . . .
  . . . . . . .
  . . . . 1 1 1
  . . . . . . .

Michael S. Branicky, Table of n, a(n) for n = 1..10232

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135601, A135602.

from itertools import count, islice
def ok3(n):
    if n < 100: return False
    d = list(map(int, str(n)))
    m1, m2 = (d[1]-d[0], d[-1]-d[-2])
    return len({m1, m2}) == 2 and m1*m2 >= 0
def agen():
    seeds = [k for k in range(100, 1000) if ok3(k)]
    for digits in count(4):
        yield from sorted(seeds)
        new, pow10 = set(), 10**(digits-1)
        for q in seeds:
            d = list(map(int, str(q)))
            e1, e2 = d[0] - (d[1]-d[0]), d[-1] + (d[-1]-d[-2])
            if 9 >= e1 > 0: new.add(e1*pow10 + q)
            if 9 >= e2 >= 0: new.add(10*q + e2)
        seeds = new
print(list(islice(agen(), 54))) # Michael S. Branicky, Aug 03 2022

a(49) and beyond from Michael S. Branicky, Aug 03 2022

A163278 Concave-convex numbers.

Original entry on oeis.org

1010, 1011, 1020, 1021, 1022, 1023, 1030, 1031, 1032, 1033, 1034, 1035, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071

Offset: 1

Omar E. Pol, Oct 16 2009

Numbers with more than three digits that are not straight-line numbers (A135643), concave numbers (A135642) or convex numbers (A135641).

			The number of this sequence (A163278) is a concave-convex number:
. . . . . .
. . . . . 8
. . . . 7 .
. 6 . . . .
. . . . . .
. . . . . .
. . 3 . . .
. . . 2 . .
1 . . . . .
. . . . . .

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A135600, A135601, A135602, A135603, A135641, A135642, A135643.

is(n) = my (d=digits(n), cvx=0, ccv=0, str=0); for (i=1, #d-2, my (x=d[i]+d[i+2]-2*d[i+1]); if (x>0, cvx++, x<0, ccv++, str++)); return (cvx>0 && ccv>0) \\ Rémy Sigrist, Aug 09 2017

A134999 Triangle-shaped numbers.

Original entry on oeis.org

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 148, 149, 150, 151

Offset: 1

Omar E. Pol, Dec 05 2007

The structure of digits represents a triangle. Three digits are the vertex. In the graphic representation the points are connected by imaginary line segments. All 3-digit numbers are terms of this sequence, except for the straight-line numbers A135643. For 4 or more digits the terms are 1000, 1011, 1012, 1024, 1034, 1036, ...

			The triangle number 1024:
  . . . 4
  . . . .
  . . 2 .
  1 . . .
  . 0 . .

Cf. A134810, A135600, A135601, A135602, A135603, A135643.

Not to be confused with the triangular numbers, A000217!

A135999 Numbers that are both triangular numbers A000217 and triangle-shaped numbers A134999.

Original entry on oeis.org

105, 120, 136, 153, 171, 190, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 703, 780, 820, 861, 903, 946, 990, 1596, 1711, 2145, 2346, 3240, 3321, 3486, 3570, 3741, 4371, 4560, 4753, 5565

Offset: 1

Omar E. Pol, Dec 10 2007

The number 496 appears to be the unique perfect number A000396 of this sequence. The interesting numbers 120 and 153 are supertriangular numbers.

			The numbers 120 as a triangle-shaped number A134999:
. 2 .
1 . .
. . 0
The number 120 as a triangular number A000217:
. . . . . . . O
. . . . . . .O O
. . . . . . O O O
. . . . . .O O O O
. . . . . O O O O O
. . . . .O O O O O O
. . . . O O O O O O O
. . . .O O O O O O O O
. . . O O O O O O O O O
. . .O O O O O O O O O O
. . O O O O O O O O O O O
. .O O O O O O O O O O O O
. O O O O O O O O O O O O O
.O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O <--- A000217(15)=120.

Cf. A000217, A000396, A134999, A135600, A135601, A135602, A135603, A135643.

cp's OEIS Frontend

A135643 Straight-line numbers > 99.

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A135642 Concave numbers.

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A135602 Right-angled numbers with an internal digit as the vertex.

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A135601 Acute-angled numbers with an internal digit as the vertex.

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A135603 Obtuse-angled numbers with an internal digit as the vertex.

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A163278 Concave-convex numbers.

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A134999 Triangle-shaped numbers.

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A135999 Numbers that are both triangular numbers A000217 and triangle-shaped numbers A134999.

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