A061479 -id:A061479 - cp's OEIS frontend

A061470 First (leftmost) digit - second digit + third digit - fourth digit .... = 1.

1, 10, 21, 32, 43, 54, 65, 76, 87, 98, 100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 243, 254, 265, 276, 287, 298, 320, 331, 342, 353, 364, 375, 386, 397, 430, 441, 452, 463, 474, 485, 496, 540, 551, 562, 573, 584, 595, 650, 661, 672, 683

Offset: 1

Amarnath Murthy, May 05 2001

			221 is in the sequence since 2-2+1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Cf. A225693.

a061470 n = a061470_list !! (n-1)
a061470_list = filter ((== 1) . a225693) [0..]
-- Reinhard Zumkeller, Aug 08 2014

d[n_]:=IntegerDigits[n]; a[n_]:=Differences[Reverse[Total/@{Take[d[n],{1,-1,2}],Take[d[n],{2,-1,2}]}]]; Select[Range[690],a[#]=={1} &] (* Jayanta Basu, May 18 2013 *)
Select[Range[1000],Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]]==1&] (* Harvey P. Dale, May 24 2021 *)

A225693(a(n)) = 1. - Reinhard Zumkeller, Aug 08 2014

More terms from Robert G. Wilson v, May 10 2001 and from Larry Reeves (larryr(AT)acm.org), May 14 2001

A061870 Numbers such that |first digit - second digit + third digit - fourth digit ...| = 1.

Original entry on oeis.org

1, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 100, 111, 120, 122, 131, 133, 142, 144, 153, 155, 164, 166, 175, 177, 186, 188, 197, 199, 210, 221, 230, 232, 241, 243, 252, 254, 263, 265, 274, 276, 285, 287, 296, 298, 320, 331, 340, 342

Offset: 1

Robert G. Wilson v, May 10 2001

Multiples of 11 plus or minus 1. If 11k+1 is a perfect square (see A219257) then a(n) is the square root of 11k+1. [Gary Detlefs, Feb 22 2010]

			120 is in the sequence since |1-2+0| = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Subsequence of A175885.

fQ[n_] := Abs[Plus @@ (((-1)^Range[Floor[Log[10, n] + 1]])*IntegerDigits@n)] == 1; Select[ Range@342, fQ@# &]

altsum(v)=sum(i=1,#v,v[i]*(-1)^i)
is(n)=abs(altsum(digits(n)))==1 \\ Charles R Greathouse IV, May 21 2014

def ok(n): return abs(sum(int(di)*(-1)**i for i, di in enumerate(str(n)))) == 1
print([k for k in range(343) if ok(k)]) # Michael S. Branicky, Jan 26 2023

A060978 |First digit - second digit + third digit - fourth digit ...| = 10.

Original entry on oeis.org

109, 208, 219, 307, 318, 329, 406, 417, 428, 439, 505, 516, 527, 538, 549, 604, 615, 626, 637, 648, 659, 703, 714, 725, 736, 747, 758, 769, 802, 813, 824, 835, 846, 857, 868, 879, 901, 912, 923, 934, 945, 956, 967, 978, 989, 1090, 1209, 1308, 1319, 1407

Offset: 1

Robert G. Wilson v, May 10 2001

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Do[ a = IntegerDigits[ n ]; l = Length[ a ]; e = o = {}; Do[ o = Append[ o, a[ [ 2k - 1 ] ] ], {k, 1, l/2 + .5} ]; Do[ e = Append[ e, a[ [ 2k ] ] ], {k, 1, l/2} ]; If[ Abs[ Apply[ Plus, o ] - Apply[ Plus, e ] ] == 10, Print[ n ] ], {n, 1, 2000} ]
Select[Range[1500],Abs[Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]]]==10&] (* Harvey P. Dale, Apr 13 2020 *)

A060980 |First digit - second digit + third digit - fourth digit ...| = 12.

Original entry on oeis.org

309, 408, 419, 507, 518, 529, 606, 617, 628, 639, 705, 716, 727, 738, 749, 804, 815, 826, 837, 848, 859, 903, 914, 925, 936, 947, 958, 969, 1409, 1508, 1519, 1607, 1618, 1629, 1706, 1717, 1728, 1739, 1805, 1816, 1827, 1838, 1849, 1904, 1915, 1926, 1937

Offset: 1

Robert G. Wilson v, May 10 2001

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Do[ a = IntegerDigits[ n ]; l = Length[ a ]; e = o = {}; Do[ o = Append[ o, a[ [ 2k - 1 ] ] ], {k, 1, l/2 + .5} ]; Do[ e = Append[ e, a[ [ 2k ] ] ], {k, 1, l/2} ]; If[ Abs[ Apply[ Plus, o ] - Apply[ Plus, e ] ] == 12, Print[ n ] ], {n, 1, 2000} ]
Select[Range[2000],Abs[Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]]]==12&] (* Harvey P. Dale, Sep 12 2017 *)

A060982 a(n) = Smallest nontrivial number k > 9 such that |first (leftmost) decimal digit of k - second digit + third digit - fourth digit ...| = n.

Original entry on oeis.org

11, 10, 13, 14, 15, 16, 17, 18, 19, 90, 109, 209, 309, 409, 509, 609, 709, 809, 909, 10909, 20909, 30909, 40909, 50909, 60909, 70909, 80909, 90909, 1090909, 2090909, 3090909, 4090909, 5090909, 6090909, 7090909, 8090909, 9090909, 109090909, 209090909, 309090909

Offset: 0

Robert G. Wilson v, May 10 2001

Starting with 109, this sequence has the same terms as A061479 and A061882. - Georg Fischer, May 24 2022

Michael S. Branicky, Table of n, a(n) for n = 0..4500

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

m = 2; Do[ While[ a = IntegerDigits[ m ]; l = Length[ a ]; e = o = {}; Do[ o = Append[ o, a[ [ 2k - 1 ] ] ], {k, 1, l/2 + .5} ]; Do[ e = Append[ e, a[ [ 2k ] ] ], {k, 1, l/2} ]; Abs[ Apply[ Plus, o ] - Apply[ Plus, e ] ] != n, m++ ]; Print[ m ], {n, 1, 50} ]

def f(m): return abs(sum((-1)**i*int(d) for i, d in enumerate(str(m))))
def a(n):
    m = 10
    while f(m) != n: m += 1
    return m
print([a(n) for n in range(28)]) # Michael S. Branicky, Nov 10 2021

# faster version based on formula
def a(n):
    if n < 10: return [11, 10, 13, 14, 15, 16, 17, 18, 19, 90][n]
    q, r = divmod(n, 9)
    return int(str(r if r else 9) + "09"*(q if r else q-1))
print([a(n) for n in range(40)]) # Michael S. Branicky, Nov 10 2021

For n > 8, if r = 0, a(n) = 90..90, else a(n) = r09..09, where r = n mod 9 and 90 and 09, resp., occur ceiling(n/9) times. - Michael S. Branicky, Nov 10 2021

a(39) and beyond from Michael S. Branicky, Nov 10 2021

Definition amended by Georg Fischer, May 24 2022

A061882 a(n) = Smallest nontrivial number k > 9 such that first (leftmost) digit - second digit + third digit - fourth digit ... of k = n.

Original entry on oeis.org

11, 10, 20, 30, 40, 50, 60, 70, 80, 90, 109, 209, 309, 409, 509, 609, 709, 809, 909, 10909, 20909, 30909, 40909, 50909, 60909, 70909, 80909, 90909, 1090909, 2090909, 3090909, 4090909, 5090909, 6090909, 7090909, 8090909, 9090909, 109090909

Offset: 0

Larry Reeves (larryr(AT)acm.org), May 15 2001

Starting with 109, this sequence has the same terms as A060982 and A061479. - Georg Fischer, May 24 2022

Cf. A008593, A051885, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

a(n) = my(k=1,d=digits(k)); while (sum(k=1, #d, (-1)^(k+1)*d[k]) != n, k++; d=digits(k)); k; \\ Michel Marcus, May 24 2022

Definition amended by Georg Fischer, May 24 2022

A225693 Alternating sum of digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, 8, 7, 6, 5, 4, 3, 2

Offset: 0

N. J. A. Sloane, May 27 2013

A number n is divisible by 11 if and only if a(n) is divisible by 11. For generalizations see Sharpe and Webster, or the links below.
The primes p for which the absolute value of the alternating sum of digits of p is also a prime begin: 2, 3, 5, 7, 13, 29, 31, 41, 47, 53, 61, 79, 83, 97, 101, 113, 137, 139, 151. - Jonathan Vos Post, May 27 2013
The above prime sequence is A115261. - Jens Kruse Andersen, Jul 13 2014
Digital sum with alternating signs starting with a positive sign for the most significant digit. - Hieronymus Fischer, Mar 23 2014

Hieronymus Fischer, Table of n, a(n) for n = 0..10000
Jim Loy, Divisibility Tests
Stu Savory, Divisibility by prime numbers under 50
D. Sharpe and R. Webster, Reversing digits: divisibility by 27, 81, and 121, Mathematical Spectrum, 45 (2012/2013), 69-71.

A055017 is closely related (but less natural).
Cf. A061479.
Cf. A004086.
Indices of 0..3: A135499, A061470, A061471, A061472.
Cf. A007953, A257588.

a225693 = f 1 0 where
   f _ a 0 = a
   f s a x = f (negate s) (s * a + d) x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 11 2015, Aug 08 2014

A225693 :=proc(n) local t1,i;
t1:=convert(n,base,10);
add((-1)^(i+nops(t1))*t1[i],i=1..nops(t1));
end;
[seq(A225693(n),n=0..120)];

Table[Total[Times@@@Partition[Riffle[IntegerDigits[n],{1,-1},{2,-1,2}],2]],{n,0,90}] (* Harvey P. Dale, Nov 27 2015 *)

a(n) = my(d=digits(n)); sum(k=1, #d, (-1)^(k+1)*d[k]); \\ Michel Marcus, Jul 15 2022

def a(n): return sum(int(d)*(-1)**i for i, d in enumerate(str(n)))
print([a(n) for n in range(87)]) # Michael S. Branicky, Jul 14 2022

"Version for general bases"
"Set base = 10 for this sequence"
altDigitalSumLeft: base
base > 1 ifTrue:  [m:= self integerFloorLog: base]
         ifFalse: [^self \\ 2].
p:=1.
s:=0.
1 to: m by: 2 do: [ :k |
    p := p*base.
    s := s - (self // p) .
    p := p*base.
    s := s + (self // p) ].
^(self + ((base + 1)*s)) * (m alternate)
"Version for base 10 using altDigitalSumRight from A055017"
A225693
^(self A004086) altDigitalSumLeft: 10
[by Hieronymus Fischer, Mar 23 2014]

If n has decimal expansion abc..xyz with least significant digit z, a(n) = a - b + c - d + ...
From Hieronymus Fischer, Mar 23 2014: (Start)
Formulas for general bases b > 1 (b = 10 for this sequence). Always m := floor(log_b(n)).
a(n) = Sum_{k>=0} (-1)^k*(floor(n*b^(k-m)) mod b). The sum is finite with floor(log_b(n)) as the highest index.
a(n) = (-1)^m*n - (b+1)*Sum_{k=1..m} (-1)^k*floor(n*b^(k-m-1)).
a(n) = (-1)^m*(n + (b+1)*Sum_{k>=1} (-1)^k*floor(n/b^k)).
a(n) = -(-1)^(m-k)*a(n mod b^k) + a(floor(n/b^k)), for 0 <= k <= m+1.
a(n) = (-1)^m*a(n mod b) + a(floor(n/b)).
a(n) = -(-1)^m*a(n mod b^2) + a(floor(n/b^2)).
a(n) = (-1)^m*A055017(n).
a(n) = A055017(A004086(n)).
a(A004086(A004086(n))) = a(n).
(End)
a(A135499(n)) = 0; a(A061470(n)) = 1. - Reinhard Zumkeller, Aug 08 2014
a(A061471(n)) = 2; a(A061472(n)) = 3. - Bernard Schott, Jul 14 2022

Comment corrected by Jens Kruse Andersen, Jul 13 2014

A060979 |First digit - second digit + third digit - fourth digit ...| = 11.

Original entry on oeis.org

209, 308, 319, 407, 418, 429, 506, 517, 528, 539, 605, 616, 627, 638, 649, 704, 715, 726, 737, 748, 759, 803, 814, 825, 836, 847, 858, 869, 902, 913, 924, 935, 946, 957, 968, 979, 1309, 1408, 1419, 1507, 1518, 1529, 1606, 1617, 1628, 1639, 1705, 1716

Offset: 1

Robert G. Wilson v, May 10 2001

Note that all terms are divisible by eleven.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Cf. A135499.

a060979 n = a060979_list !! (n-1)
a060979_list = filter (\x -> let digs = map (read . return) $ show x in
                             evens digs /= odds digs) [11, 22 ..]
   where evens [] = 0; evens [x] = x; evens (x:_:xs) = x + evens xs
         odds [] = 0; odds [x] = 0; odds (_:x:xs) = x + odds xs
-- Reinhard Zumkeller, Jul 05 2014

filter:= proc(n) local L,i;
  L:= convert(n,base,10);
  abs(add(L[i]*(-1)^i,i=1..nops(L))) = 11
end proc:
select(filter, [$1..1000] *~ 11); # Robert Israel, Jun 02 2023

Do[ a = IntegerDigits[ n ]; l = Length[ a ]; e = o = {}; Do[ o = Append[ o, a[ [ 2k - 1 ] ] ], {k, 1, l/2 + .5} ]; Do[ e = Append[ e, a[ [ 2k ] ] ], {k, 1, l/2} ]; If[ Abs[ Apply[ Plus, o ] - Apply[ Plus, e ] ] == 11, Print[ n ] ], {n, 1, 2000} ]
d11Q[n_]:=Module[{idn=IntegerDigits[n]},Abs[Total[Table[(-1)^(i+1) idn[[i]],{i,Length[idn]}]]]==11]; Select[Range[1800],d11Q] (* Harvey P. Dale, Aug 26 2012 *)

Erroneous comment deleted by Robert Israel, Jun 02 2023

A061478 First (leftmost) digit - second digit + third digit - fourth digit .... = 9.

Original entry on oeis.org

9, 90, 108, 119, 207, 218, 229, 306, 317, 328, 339, 405, 416, 427, 438, 449, 504, 515, 526, 537, 548, 559, 603, 614, 625, 636, 647, 658, 669, 702, 713, 724, 735, 746, 757, 768, 779, 801, 812, 823, 834, 845, 856, 867, 878, 889, 900, 911, 922, 933, 944, 955

Offset: 1

Amarnath Murthy, May 05 2001

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Select[Range[1000],Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]]==9&] (* Harvey P. Dale, Jul 16 2017 *)

More terms from Robert G. Wilson v, May 10 2001 and from Larry Reeves (larryr(AT)acm.org), May 15 2001

A061471 First (leftmost) digit - second digit + third digit - fourth digit .... = 2.

Original entry on oeis.org

2, 20, 31, 42, 53, 64, 75, 86, 97, 101, 112, 123, 134, 145, 156, 167, 178, 189, 200, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 365, 376, 387, 398, 420, 431, 442, 453, 464, 475, 486, 497, 530, 541, 552, 563, 574, 585, 596, 640, 651

Offset: 1

Amarnath Murthy, May 05 2001

a(n) == 9*(-1)^d (mod 11) if a(n) has d digits. - Robert Israel, Aug 05 2020

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

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

filter:= proc(n) local d,L,j;
  L:= convert(n,base,10);
  d:= nops(L);
  add(L[j]*(-1)^(d-j),j=1..d)=2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 05 2020

okQ[n_] := With[{id = IntegerDigits[n]}, id.Array[2 Mod[#, 2] - 1&, Length[id]] == 2]; Select[Range[1000], okQ] (* Jean-François Alcover, Nov 17 2016 *)
Select[Range[700],Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]] == 2&] (* Harvey P. Dale, Mar 01 2023 *)

More terms from Robert G. Wilson v, May 10 2001 and from Larry Reeves (larryr(AT)acm.org), May 14 2001

cp's OEIS Frontend

A061470 First (leftmost) digit - second digit + third digit - fourth digit .... = 1.

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A061870 Numbers such that |first digit - second digit + third digit - fourth digit ...| = 1.

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A060978 |First digit - second digit + third digit - fourth digit ...| = 10.

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A060980 |First digit - second digit + third digit - fourth digit ...| = 12.

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A060982 a(n) = Smallest nontrivial number k > 9 such that |first (leftmost) decimal digit of k - second digit + third digit - fourth digit ...| = n.

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A061882 a(n) = Smallest nontrivial number k > 9 such that first (leftmost) digit - second digit + third digit - fourth digit ... of k = n.

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A225693 Alternating sum of digits of n.

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A060979 |First digit - second digit + third digit - fourth digit ...| = 11.