A135499 -id:A135499 - cp's OEIS frontend

A008593 Multiples of 11.

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 297, 308, 319, 330, 341, 352, 363, 374, 385, 396, 407, 418, 429, 440, 451, 462, 473, 484, 495, 506, 517, 528, 539, 550, 561, 572, 583

Offset: 0

N. J. A. Sloane

Numbers for which the sum of "digits" in base 100 is divisible by 11. For instance, 193517302 gives 1 + 93 + 51 + 73 + 02 = 220, and 2 + 20 = 22 = 2 * 11. - Daniel Forgues, Feb 22 2016

Numbers in which the sum of the digits in the even positions equals the sum of the digits in the odd positions. - Stefano Spezia, Jan 05 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 323.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A008592, A008604, A059632; union of A135499 and A060979.

a008593 = (* 11)
a008593_list = [0, 11 ..]  -- Reinhard Zumkeller, Jul 05 2014

[11*n: n in [0..60]]; // Vincenzo Librandi, Sep 18 2011

g:=(1+10*z)/((1-z)): gser:=series(g, z=0, 88): seq((coeff(gser, z, n))*n, n=0..77); # Zerinvary Lajos, Feb 25 2009

Range[0, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

makelist(11*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=11*n \\ Charles R Greathouse IV, Nov 06 2014

a(n) = 11*n.
G.f.: 11*x/(1-x)^2. - David Wilding, Jun 21 2014
E.g.f.: 11*x*exp(x). - Stefano Spezia, Oct 08 2022
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008604(n)/2. (End)

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

A214527 Numbers k such that the alternating sum of decimal digits of k is zero.

Original entry on oeis.org

101, 112, 123, 134, 145, 156, 167, 178, 189, 202, 213, 224, 235, 246, 257, 268, 279, 303, 314, 325, 336, 347, 358, 369, 404, 415, 426, 437, 448, 459, 505, 516, 527, 538, 549, 606, 617, 628, 639, 707, 718, 729, 808, 819, 909, 1010, 1021, 1032, 1043, 1054, 1065, 1076

Offset: 1

Alex Ratushnyak, Aug 08 2012

Alternating sum starts with plus after the most significant digit, for example
1+0-1 = 0, so 101 is in the sequence, also
4+3-7 = 0, 1+0-7+6 = 0, so 437 and 1076 are in the sequence.

Cf. A135499: same definition except that the alternating sum starts with minus after the most significant digit.

a = []
for n in range(1, 2000):
    t = str(n)
    s = int(t[0])
    s += sum((-1)**i * int(d) for i, d in enumerate(t[1:]))
    if s == 0:
         a.append(n)
print(a)

A244212 Numbers n for which the alternating sum of the digits of n^n is 0.

Original entry on oeis.org

22, 55, 77, 99, 132, 187, 286, 1056, 1463, 1474, 1606, 1837, 2277, 2981, 4785, 4851, 5313, 5588, 5929, 7227, 8272, 8415, 8492, 8954, 11517, 12573, 12628, 13156, 14883, 15972, 17688, 22066, 23936, 24915, 25850, 27522, 34045, 36289, 36806, 38489, 40744, 43450, 46794, 48092

Offset: 1

Anthony Sand, Jun 23 2014

The result of alternately adding and subtracting the digits of n sometimes differs in sign when the procedure goes from left to right or right to left. For example, if n = 1234, 1 - 2 + 3 - 4 = -2, whereas 4 - 3 + 2 - 1 = +2. However, if the sum is zero when adding and subtracting from left to right, it will also be zero when adding and subtracting from right to left.
n such that A000312(n) is in A135499. - Robert Israel, Jul 13 2014
All terms are multiples of 11. This follows from the divisibility rule for 11. - Jens Kruse Andersen, Jul 13 2014
Number of terms less than 10^k: 0, 4, 7, 24, 55, 135, ..., . - Robert G. Wilson v, Jul 18 2014
Numbers for which the alternating sum of the digits of n^n are == 0 (Mod 10): 12, 22, 23, 35, 45, 46, 47, 55, 57, 77, 99, 117, 126, 132, 151, ..., . Obviously the members of A244212 are included here. - Robert G. Wilson v, Jul 20 2014

			22^22 = 341427877364219557396646723584, therefore the alternating sum = 4 - 8 + 5 - 3 + 2 - 7 + 6 - 4 + 6 - 6 + 9 - 3 + 7 - 5 + 5 - 9 + 1 - 2 + 4 - 6 + 3 - 7 + 7 - 8 + 7 - 2 + 4 - 1 + 4 - 3 = 0.

Jens Kruse Andersen and Robert G. Wilson v, Table of n, a(n) for n = 1..139 (a(48) to a(63) from Jens Kruse Andersen).

Cf. A000312, A065816, A135499, A244144.

filter:= proc(n) local x,j;
   x:= convert(n^n,base,10);
   evalb(add((-1)^j*x[j],j=1..nops(x)) = 0)
end proc;
select(filter, 11 * [$1..1000]); # Robert Israel, Jul 13 2014

fQ[n_] := Block[{id = IntegerDigits[ n^n]}, Sum[ id[[i]]*(-1)^i, {i, Length@ id}] == 0]; k = 11; lst = {}; While[k < 100001, If[ fQ@ k, AppendTo[ lst, k]; Print@ k]; k+= 11]; lst (* Robert G. Wilson v, Jul 13 2014 *)

isok(n) = d = digits(n^n) ; sum(i=1, #d, d[i]*(-1)^i) == 0; \\ Michel Marcus, Jun 25 2014

s = 0; m = 1; for digit[n,i=1..j] of n, s = s + digit[i] * m; m = -m; next i; if s = 0, print n;

a(9)-a(24) from Michel Marcus, Jun 23 2014

a(25)-a(44) from Robert G. Wilson v, Jul 13 2014

A384212 a(n) is the number of bases >= 2 in which the alternating sum of digits of n is equal to 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 3, 2, 2, 1, 6, 2, 2, 3, 4, 1, 6, 1, 4, 3, 2, 2, 7, 1, 2, 3, 6, 1, 5, 1, 4, 5, 2, 1, 8, 2, 3, 3, 4, 1, 5, 2, 6, 3, 2, 1, 9, 1, 2, 5, 5, 3, 6, 1, 4, 2, 6, 1, 10, 1, 2, 5, 4, 2, 5, 1, 8, 3, 2, 1, 8, 3, 2, 2

Offset: 1

Felix Huber, May 24 2025

The alternating sum of digits of n is equal to 1 for base n and equal to n for bases > n.

			a(72) = 10 because the alternating sum of digits of n is equal to 0 in the 10 bases 2 [1, 0, 0, 1, 0, 0, 0], 3 [2, 2, 0, 0], 5 [2, 4, 2], 7 [1, 3, 2], 8 [1, 1, 0], 11 [6, 6], 17 [4, 4], 23 [3, 3], 35 [2, 2] and 71 [1, 1].

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A055240, A061845, A225693, A135499, A135551, A384211.

A384212:=proc(n)
    local a,b,c,i;
    a:=0;
    for b from 2 to n-1 do
        c:=convert(n,'base',b);
    	   if add(c[i]*(-1)^i,i=1..nops(c))=0 then
           a:=a+1
        fi
    od;
    return a
end proc;
seq(A384212(n),n=1..87);
A384212bases:=proc(n)
    local L,b,c,i;
    L:=[];
    for b from 2 to n-1 do
        c:=convert(n,'base',b);
    	if add(c[i]*(-1)^i,i=1..nops(c))=0 then
            L:=[op(L),b,ListTools:-Reverse(c)]
        fi
    od;
    return op(L)
end proc;
A384212bases(72);

q[n_, b_] := Module[{d = IntegerDigits[n, b]}, Sum[(-1)^k*d[[k]], {k, 1, Length[d]}] == 0 ]; a[n_] := Count[Range[2, n-1], ?(q[n, #] &)]; Array[a, 100] (* _Amiram Eldar, May 24 2025 *)

a(n) = sum(b=2, n-1, my(d=digits(n, b)); sum(k=1, #d, (-1)^k*d[k]) == 0); \\ Michel Marcus, May 24 2025

from sympy.ntheory import digits
def s(v): return sum(v[::2]) - sum(v[1::2])
def a(n): return sum(1 for b in range(2, n) if s(digits(n, b)[1:]) == 0)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, May 24 2025

Trivial bounds: 1 <= a(n) <= n - 2 for n >= 3 because the representation of n in base n-1 is [1,1] and the alternating sum of digits of n is > 0 for bases >= n.

A309489 Numbers k such that the sum of digits in odd places is equal to the sum of digits in even places in k!.

Original entry on oeis.org

11, 18, 20, 25, 27, 28, 32, 35, 41, 43, 51, 57, 60, 68, 72, 74, 80, 102, 105, 107, 113, 121, 122, 138, 140, 145, 156, 161, 166, 171, 228, 233, 245, 282, 301, 307, 308, 311, 315, 329, 333, 335, 340, 347, 349, 351, 353, 366, 386, 412, 420, 454, 469, 478

Offset: 1

Daniel Starodubtsev, Aug 04 2019

Numbers k such that A000142(k) is a term of A135499. - Michel Marcus, Aug 04 2019

Daniel Starodubtsev, Table of n, a(n) for n = 1..300

Cf. A000142 (n!), A135499.

aQ[n_] := Total[(d = IntegerDigits[n!])[[1;;-1;;2]]] == Total[d[[2;;-1;;2]]]; Select[Range[500], aQ] (* Amiram Eldar, Aug 04 2019 *)
sdoeQ[n_]:=With[{nf=IntegerDigits[n!]},Total[Take[nf,{1,-1,2}]]==Total[Take[nf,{2,-1,2}]]]; Select[Range[500],sdoeQ] (* Harvey P. Dale, Jun 04 2025 *)

isok(k) = {my(d = digits(k!), so=0, se=0); for (i=1, #d, if (i%2, so += d[i], se += d[i])); so == se; } \\ Michel Marcus, Aug 04 2019

def c(n): s = str(n); return sum(map(int, s[::2])) == sum(map(int, s[1::2]))
def afind(limit):
    k, factk = 1, 1
    while k <= limit:
        if c(factk): print(k, end=", ")
        k += 1; factk *= k
afind(500) # Michael S. Branicky, Nov 18 2021

A382337 Palindromes in base 10 in which the difference between the sums of the digits in the even and odd positions is zero.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 242, 363, 484, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 4004, 4114, 4224, 4334, 4444, 4554, 4664, 4774, 4884, 4994

Offset: 1

Alexander Yutkin, Mar 22 2025

Index entries for sequences related to palindromes.

Intersection of A002113 and A135499.

Select[11 * Range[0, 500], PalindromeQ[#] && Total[IntegerDigits[#][[1 ;; -1 ;; 2]]] == Total[IntegerDigits[#][[2 ;; -1 ;; 2]]] &] (* Amiram Eldar, Mar 22 2025 *)

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A008593 Multiples of 11.

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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.

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A214527 Numbers k such that the alternating sum of decimal digits of k is zero.

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A244212 Numbers n for which the alternating sum of the digits of n^n is 0.

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A384212 a(n) is the number of bases >= 2 in which the alternating sum of digits of n is equal to 0.

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