A083420 -id:A083420 - cp's OEIS frontend

A267938 Decimal representation of the n-th iteration of the "Rule 251" elementary cellular automaton starting with a single ON (black) cell.

1, 5, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Robert Price, Jan 22 2016

Seems to differ from A267890, A267888 and A083420 only at n=1. - R. J. Mathar, Jun 21 2025

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267936.

rule=251; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 23 2016 and Apr 16 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>3.
G.f.: (1+10*x^2-8*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 2^(2*n+1) - 1 for n>1. - Colin Barker, Nov 25 2016

A331893 Positive numbers k such that both k and -k are a palindromes in negabinary representation.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 57, 65, 85, 127, 155, 217, 257, 273, 325, 341, 455, 511, 635, 857, 889, 993, 1025, 1105, 1253, 1285, 1365, 1799, 2047, 2159, 2555, 2667, 3417, 3577, 3641, 3937, 4097, 4161, 4369, 4433, 4965, 5125, 5189, 5397, 5461, 6951, 7175, 7967, 8191

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			5 is a term since the negabinary representation of 5, 101, and the negabinary representation of -5, 1111, are both palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A331891 and A331892.

Cf. A005352, A039724, A212529.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := And @@ (PalindromeQ @ negabin[#] & /@ {n, -n}); Select[Range[2^13], nbPalinQ]

A084181 2^n+(-2)^n-(-1)^n.

Original entry on oeis.org

1, 1, 7, 1, 31, 1, 127, 1, 511, 1, 2047, 1, 8191, 1, 32767, 1, 131071, 1, 524287, 1, 2097151, 1, 8388607, 1, 33554431, 1, 134217727, 1, 536870911, 1, 2147483647, 1, 8589934591, 1, 34359738367, 1, 137438953471, 1, 549755813887, 1, 2199023255551

Offset: 0

Paul Barry, May 19 2003

Binomial transform is A084182.

Index entries for linear recurrences with constant coefficients, signature (-1,4,4).

Cf. A083420.

LinearRecurrence[{-1,4,4},{1,1,7},50] (* or *) Riffle[ LinearRecurrence[ {5,-4},{1,7},30],1] (* Harvey P. Dale, Jan 02 2019 *)

a(n)=2^n+(-2)^n-(-1)^n;
G.f.: (1+2x+4x^2)/((1+x)(1+2x)(1-2x));
E.g.f.: exp(2x)-exp(-x)+exp(-2x).

A084184 Partial sums of a Jacobsthal related sequence.

Original entry on oeis.org

0, 1, 2, 7, 10, 31, 42, 127, 170, 511, 682, 2047, 2730, 8191, 10922, 32767, 43690, 131071, 174762, 524287, 699050, 2097151, 2796202, 8388607, 11184810, 33554431, 44739242, 134217727, 178956970, 536870911, 715827882, 2147483647, 2863311530, 8589934591

Offset: 0

Paul Barry, May 19 2003

Partial sums of A084183.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A001045, A002450, A083420, A084183.

A084184:=n->(5-(-1)^n)*(2^n-1)/6: seq(A084184(n), n=0..50); # Wesley Ivan Hurt, Jan 28 2017

LinearRecurrence[{0,5,0,-4},{0,1,2,7},40] (* Harvey P. Dale, Jan 18 2015 *)

concat(0, Vec(x*(1+2*x+2*x^2)/((1-x^2)*(1-4*x^2)) + O(x^40))) \\ Colin Barker, Sep 09 2016

G.f.: x*(1+2*x+2*x^2) / ((1-x^2)*(1-4*x^2)). - typo fixed by Colin Barker, Sep 09 2016
a(2*n+1) = A083420(n). a(2*n) = 2*A002450(n) = 2*A001045(2*n).
From Colin Barker, Sep 09 2016: (Start)
a(n) = (5-(-1)^n)*(2^n-1)/6.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
(End)

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A173209 Partial sums of A000069.

Original entry on oeis.org

1, 3, 7, 14, 22, 33, 46, 60, 76, 95, 116, 138, 163, 189, 217, 248, 280, 315, 352, 390, 431, 473, 517, 564, 613, 663, 715, 770, 826, 885, 946, 1008, 1072, 1139, 1208, 1278, 1351, 1425, 1501, 1580, 1661, 1743, 1827, 1914, 2002, 2093, 2186, 2280, 2377, 2475, 2575

Offset: 1

Jonathan Vos Post, Feb 12 2010

Partial sums of odious numbers. Second differences give A007413. The subsequence of prime partial sums of odious numbers begins: 3, 7, 163, 431, 613, 2377, 3691, which is a subsequence of A027697. The subsequence of odious partial sums of odious numbers begins: 1, 7, 14, 22, 76, 138, 217, 280, 352, 431, 517, 613, 770, 885.

			a(65) = 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 19 + 21 + 22 + 25 + 26 + 28 + 31 + 32 + 35 + 37 + 38 + 41 + 42 + 44 + 47 + 49 + 50 + 52 + 55 + 56 + 59 + 61 + 62 + 64 + 67 + 69 + 70 + 73 + 74 + 76 + 79 + 81 + 82 + 84 + 87 + 88 + 91 + 93 + 94 + 97 + 98 + 100 + 103 + 104 + 107 + 109 + 110 + 112 + 115 + 117 + 118 + 121 + 122 + 124 + 127 + 128.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.

Cf. A000069, A000079, A000120, A000773, A000788, A001969, A002042, A002089, A007413, A019568, A023416, A027697, A059015, A083420, A010060, A132679, A133009.

Accumulate@ Select[Range[100], OddQ@ DigitCount[#, 2, 1] &] (* Michael De Vlieger, Nov 01 2022 *)

a(n)=n^2-(n+1)\2+(n%2&&hammingweight(n)%2) \\ Charles R Greathouse IV, Mar 22 2013

a(n) = SUM[i=1..n] A000069(i) = SUM[i=1..n] {i such that A010060(i)=1}.

a(n) = n^2 - n/2 + O(1). - Charles R Greathouse IV, Mar 22 2013

A261349 T(n,k) is the decimal equivalent of a code for k that maximizes the sum of the Hamming distances between (cyclical) adjacent code words; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 2, 0, 7, 1, 6, 3, 4, 2, 5, 0, 15, 1, 14, 3, 12, 2, 13, 6, 9, 7, 8, 5, 10, 4, 11, 0, 31, 1, 30, 3, 28, 2, 29, 6, 25, 7, 24, 5, 26, 4, 27, 12, 19, 13, 18, 15, 16, 14, 17, 10, 21, 11, 20, 9, 22, 8, 23, 0, 63, 1, 62, 3, 60, 2, 61, 6, 57, 7, 56, 5

Offset: 0

Alois P. Heinz, Nov 18 2015

This code might be called "Anti-Gray code".

The sum of the Hamming distances between (cyclical) adjacent code words of row n gives 0, 2, 6, 20, 56, 144, 352, ... = A014480(n-1) for n>1.

			Triangle T(n,k) begins:
  0;
  0,  1;
  0,  3, 1,  2;
  0,  7, 1,  6, 3,  4, 2,  5;
  0, 15, 1, 14, 3, 12, 2, 13, 6,  9, 7,  8, 5, 10, 4, 11;
  0, 31, 1, 30, 3, 28, 2, 29, 6, 25, 7, 24, 5, 26, 4, 27, 12, 19, ... ;
  0, 63, 1, 62, 3, 60, 2, 61, 6, 57, 7, 56, 5, 58, 4, 59, 12, 51, ... ;

Alois P. Heinz, Rows n = 0..13, flattened
Wikipedia, Gray code
Wikipedia, Hamming distance

Columns k=0-3 give: A000004, A000225, A000012 (for n>1), A000918 (for n>1).
Row lengths give A000079.
Row sums give A006516.
Cf. A003188, A014480, A083329, A083420, A101080.

g:= n-> Bits[Xor](n, iquo(n, 2)):
T:= (n, k)-> (t-> `if`(m=0, t, 2^n-1-t))(g(iquo(k, 2, 'm'))):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);

T(n,k) = A003188(k/2) if k even, T(n,k) = 2^n-1-A003188((k-1)/2) else.
A101080(T(n,2k),T(n,2k+1)) = n, A101080(T(n,2k),T(n,2k-1)) = n-1.
T(n,2^n-1) = A083329(n-1) for n>0.
T(n,2^n-2) = A000079(n-2) for n>1.
T(2n,2n) = A003188(n).
T(2n+1,2n+1) = 2*4^n - 1 - A003188(n) = A083420(n) - A003188(n).

A267941 Decimal representation of the n-th iteration of the "Rule 253" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Robert Price, Jan 22 2016

With the exception of a(1) the same as A267938, A267890, A267888 and A083420. - R. J. Mathar, Jan 24 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A060576.

rule=253; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 23 2016 and Apr 16 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>3.
G.f.: (1-2*x+20*x^2-16*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 2^(2*n+1) - 1 for n>1. - Colin Barker, Nov 26 2016

A269255 a(n) = (2^(2n+1) - 1)(3^(n+1) - 1)/2.

Original entry on oeis.org

1, 28, 403, 5080, 61831, 745108, 8952763, 107475760, 1289869711, 15479049388, 185750955523, 2229020652040, 26748283770391, 320979546636868, 3851755118036683, 46221063628493920, 554652772325571871, 6655833302847731548, 79869999773355124243, 958439997835247481400, 11501279976237683562151

Offset: 0

Ilya Gutkovskiy, Jul 13 2016

The sum of divisors of powers of 12 (A001021).

The sum of divisors of powers of prime p are sigma_1(p^n) = Sum_{m=0}^n p^m = (p^(n+1) - 1)/(p - 1) (see examples in the links section).

			a(1) = 28, because 12^1 = 12 and 12 has 6 divisors (1, 2, 3, 4, 6, 12) -> 1 + 2 + 3 + 4 + 6 + 12 = 28.

Ilya Gutkovskiy, The sum of the divisors of k^n
Index entries for linear recurrences with constant coefficients, signature (20,-115,240,-144)

Cf. A000203, A000244, A000302, A001021, A003462, A021016, A083420.

LinearRecurrence[{20, -115, 240, -144}, {1, 28, 403, 5080}, 21]
Table[(2^(2 n + 1) - 1) ((3^(n + 1) - 1)/2), {n, 0, 20}]
Table[DivisorSigma[1, 12^n], {n, 0, 20}]

a(n)=(2^(2*n+1)-1)*(3^(n+1)-1)/2 \\ Charles R Greathouse IV, Jul 26 2016

O.g.f.: (1 + 8*x - 42*x^2)/((1 - x)*(1 - 3*x)*(1 - 4*x)*(1 - 12*x)).
E.g.f.: (1 - 3*exp(2*x) - 2*exp(3*x) + 6*exp(11*x))*exp(x)/2.
a(n) = 20*a(n-1) - 115*a(n-2) + 240*a(n-3) - 144*a(n-4).
a(n) = A000203(A001021(n)).
a(n) = A000203(A000244(n))*A000203(A000302(n)).
a(n) = A083420(n)*A003462(n+1).
Sum_{n>=0} (-1)^n*a(n)/n! = (6 - 2*exp(8) - 3*exp(9) + exp(11))/(2*exp(12)) = 0.0909619117822510506...
Lim_{n->infinity} a(n)/a(n+1) = 1/12 = A021016.

A280293 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [-5, 4].

Original entry on oeis.org

3, 1, 6, 7, 18, 31, 66, 127, 258, 511, 1026, 2047, 4098, 8191, 16386, 32767, 65538, 131071, 262146, 524287, 1048578, 2097151, 4194306, 8388607, 16777218, 33554431, 67108866, 134217727, 268435458, 536870911, 1073741826, 2147483647, 4294967298, 8589934591

Offset: 0

Paul Curtz, Dec 31 2016

From 1, the last digit is a periodic sequence of length 4:repeat [1, 6, 7, 8].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A007283, A083420, A178789, A280173.

LinearRecurrence[{2,1,-2},{3,1,6},50] (* Paolo Xausa, Nov 13 2023 *)

Vec((3-5*x+x^2) / ((1-x)*(1+x)*(1-2*x)) + O(x^40)) \\ Colin Barker, Dec 31 2016

a(2n) = 4^n + 2. a(2n+1) = 2*4^n - 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
a(n) = 2^n + periodic sequence of length 2: repeat [2, -1].
a(n) = 2^(n+2) - A280173(n).
a(n+2) = a(n) + 3*2^n, a(0) = 3, a(1) = 1.
G.f.: (3-5*x+x^2) / ((1-x)*(1+x)*(1-2*x)). - Colin Barker, Dec 31 2016

More terms from Colin Barker, Dec 31 2016

cp's OEIS Frontend

A267938 Decimal representation of the n-th iteration of the "Rule 251" elementary cellular automaton starting with a single ON (black) cell.

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A331893 Positive numbers k such that both k and -k are a palindromes in negabinary representation.

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A084181 2^n+(-2)^n-(-1)^n.

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A084184 Partial sums of a Jacobsthal related sequence.

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A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

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A173209 Partial sums of A000069.

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A261349 T(n,k) is the decimal equivalent of a code for k that maximizes the sum of the Hamming distances between (cyclical) adjacent code words; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

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A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

A269255 a(n) = (2^(2n+1) - 1)(3^(n+1) - 1)/2.