[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a010690 -id:a010690
Displaying 1-8 of 8 results found.
page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A168184
Characteristic function of numbers that are not multiples of 10.
+10
17
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).
FORMULA
a(n+10) = a(n);
a(n) = A000007(A010879(n));
a(A067251(n)) = 1; a(A008592(n)) = 0;
not the same as A168046: a(n)=A168046 for n<=100;
A033442(n) = Sum_{k=0..n} a(k)*(n-k).
Dirichlet g.f.: (1-1/10^s)*zeta(s). - R. J. Mathar, Feb 19 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
MATHEMATICA
Table[If[Mod[n, 10]==0, 0, 1], {n, 0, 110}] (* or *) PadRight[{}, 110, {0, 1, 1, 1, 1, 1, 1, 1, 1, 1}] (* Harvey P. Dale, Jun 03 2023 *)
PROG
(Haskell)
a168184 = (1 -) . (0 ^) . (`mod` 10)
a168184_list = cycle [0, 1, 1, 1, 1, 1, 1, 1, 1, 1]
-- Reinhard Zumkeller, Oct 10 2012
(PARI) a(n)=n%10>0 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
Cf. A168185, A145568, A168182, A168181, A109720, A097325, A011558, A166486, A011655, A000035, A010690, A033442.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Nov 30 2009
STATUS
approved
A318960
One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 1 (mod 4) case.
+10
7
1, 5, 5, 21, 53, 53, 181, 181, 181, 181, 181, 181, 181, 16565, 49333, 49333, 49333, 49333, 573621, 1622197, 1622197, 1622197, 10010805, 10010805, 10010805, 77119669, 211337397, 479772853, 479772853, 479772853, 2627256501, 6922223797, 15512158389, 15512158389
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
a(n) is the unique number k in [1, 2^n] and congruent to 1 (mod 4) such that k^2 + 7 is divisible by 2^(n+1).
The 2-adic integers are very different from p-adic ones where p is an odd prime. For example, provided that there is at least one solution, the number of solutions to x^n = a over p-adic integers is gcd(n, p-1) for odd primes p and gcd(n, 2) for p = 2. For odd primes p, x^2 = a is solvable iff a is a quadratic residue modulo p, while for p = 2 it's solvable iff a == 1 (mod 8). If gcd(n, p-1) > 1 and gcd(a, p) = 1, then the solutions to x^n = a differ starting at the rightmost digit for odd primes p, while for p = 2 they differ starting at the next-to-rightmost digit. As a result, the formulas and the program here are different from those in other entries related to p-adic integers.
LINKS
Jianing Song, Table of n, a(n) for n = 2..999 (offset corrected by Jianing Song)
G. P. Michon, Introduction to p-adic integers, Numericana.
FORMULA
a(2) = 1; for n >= 3, a(n) = a(n-1) if a(n-1)^2 + 7 is divisible by 2^(n+1), otherwise a(n-1) + 2^(n-1).
a(n) = 2^n - A318961(n).
a(n) = Sum_{i=0..n-1} A318962(i)*2^i.
EXAMPLE
The unique number k in [1, 4] and congruent to 1 modulo 4 such that k^2 + 7 is divisible by 8 is 1, so a(2) = 1.
a(2)^2 + 7 = 8 which is not divisible by 16, so a(3) = a(2) + 2^2 = 5.
a(3)^2 + 7 = 32 which is divisible by 32, so a(4) = a(3) = 5.
a(4)^2 + 7 = 32 which is divisible by 64, so a(5) = a(4) + 2^4 = 21.
a(5)^2 + 7 = 448 which is divisible by 128, so a(6) = a(5) + 2^5 = 53.
...
PROG
(PARI) a(n) = truncate(-sqrt(-7+O(2^(n+1))))
CROSSREFS
Cf. A318962.
Expansions of p-adic integers:
this sequence, A318961 (2-adic, sqrt(-7));
A268924, A271222 (3-adic, sqrt(-2));
A268922, A269590 (5-adic, sqrt(-4));
A048898, A048899 (5-adic, sqrt(-1));
A290567 (5-adic, 2^(1/3));
A290568 (5-adic, 3^(1/3));
A290800, A290802 (7-adic, sqrt(-6));
A290806, A290809 (7-adic, sqrt(-5));
A290803, A290804 (7-adic, sqrt(-3));
A210852, A212153 (7-adic, (1+sqrt(-3))/2);
A290557, A290559 (7-adic, sqrt(2));
A286840, A286841 (13-adic, sqrt(-1));
A286877, A286878 (17-adic, sqrt(-1)).
Also expansions of 10-adic integers:
A007185, A010690 (nontrivial roots to x^2-x);
A216092, A216093, A224473, A224474 (nontrivial roots to x^3-x).
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 06 2018
EXTENSIONS
Offset corrected by Jianing Song, Aug 28 2019
STATUS
approved
A318961
One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 3 (mod 4) case.
+10
7
3, 3, 11, 11, 11, 75, 75, 331, 843, 1867, 3915, 8011, 16203, 16203, 16203, 81739, 212811, 474955, 474955, 474955, 2572107, 6766411, 6766411, 23543627, 57098059, 57098059, 57098059, 57098059, 593968971, 1667710795, 1667710795, 1667710795, 1667710795, 18847579979
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
a(n) is the unique number k in [1, 2^n] and congruent to 3 (mod 4) such that k^2 + 7 is divisible by 2^(n+1).
The 2-adic integers are very different from p-adic ones where p is an odd prime. For example, provided that there is at least one solution, the number of solutions to x^n = a over p-adic integers is gcd(n, p-1) for odd primes p and gcd(n, 2) for p = 2. For odd primes p, x^2 = a is solvable iff a is a quadratic residue modulo p, while for p = 2 it's solvable iff a == 1 (mod 8). If gcd(n, p-1) > 1 and gcd(a, p) = 1, then the solutions to x^n = a differ starting at the rightmost digit for odd primes p, while for p = 2 they differ starting at the next-to-rightmost digit. As a result, the formulas and the program here are different from those in other entries related to p-adic integers.
LINKS
Jianing Song, Table of n, a(n) for n = 2..999 (offset corrected by Jianing Song)
G. P. Michon, Introduction to p-adic integers, Numericana.
FORMULA
a(2) = 3; for n >= 3, a(n) = a(n-1) if a(n-1)^2 + 7 is divisible by 2^(n+1), otherwise a(n-1) + 2^(n-1).
a(n) = 2^n - A318960(n).
a(n) = Sum_{i=0..n-1} A318963(i)*2^i.
EXAMPLE
The unique number k in [1, 4] and congruent to 3 modulo 4 such that k^2 + 7 is divisible by 8 is 3, so a(2) = 3.
a(2)^2 + 7 = 16 which is divisible by 16, so a(3) = a(2) = 3.
a(3)^2 + 7 = 16 which is not divisible by 32, so a(4) = a(3) + 2^3 = 11.
a(4)^2 + 7 = 128 which is divisible by 64, so a(5) = a(4) = 11.
a(5)^2 + 7 = 128 which is divisible by 128, so a(6) = a(5) = 11.
...
PROG
(PARI) a(n) = if(n==2, 3, truncate(sqrt(-7+O(2^(n+1)))))
CROSSREFS
Cf. A318963.
Expansions of p-adic integers:
A318960, this sequence (2-adic, sqrt(-7));
A268924, A271222 (3-adic, sqrt(-2));
A268922, A269590 (5-adic, sqrt(-4));
A048898, A048899 (5-adic, sqrt(-1));
A290567 (5-adic, 2^(1/3));
A290568 (5-adic, 3^(1/3));
A290800, A290802 (7-adic, sqrt(-6));
A290806, A290809 (7-adic, sqrt(-5));
A290803, A290804 (7-adic, sqrt(-3));
A210852, A212153 (7-adic, (1+sqrt(-3))/2);
A290557, A290559 (7-adic, sqrt(2));
A286840, A286841 (13-adic, sqrt(-1));
A286877, A286878 (17-adic, sqrt(-1)).
Also expansions of 10-adic integers:
A007185, A010690 (nontrivial roots to x^2-x);
A216092, A216093, A224473, A224474 (nontrivial roots to x^3-x).
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 06 2018
EXTENSIONS
Offset corrected by Jianing Song, Aug 28 2019
STATUS
approved
A014393
Final 2 digits of 9^n.
+10
2
1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Period is 10, i.e., a(n+10) = a(n). - Martin Renner, Jun 11 2020
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
FORMULA
a(n) = 9^n mod 100. - Martin Renner, Jun 11 2020
MAPLE
seq(9^n mod 100, n=0..80); # Martin Renner, Jun 11 2020
MATHEMATICA
Flatten[Prepend[FromDigits[Take[IntegerDigits[#], -2]]&/@(9^Range[2, 60]), {1, 9}]] (* Harvey P. Dale, Jan 22 2011 *)
PowerMod[9, Range[0, 80], 100] (* Vincenzo Librandi, Aug 16 2016 *)
PROG
(Magma) [Modexp(9, n, 100): n in [0..110]]; // Vincenzo Librandi, Aug 16 2016
(PARI) a(n) = lift(Mod(9, 100)^n); \\ Michel Marcus, Aug 16 2016
CROSSREFS
Cf. A001019 (9^n), A010690 (final digit of 9^n).
KEYWORD
nonn,base,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A176019
Decimal expansion of (3+sqrt(13))/6.
+10
2
1, 1, 0, 0, 9, 2, 5, 2, 1, 2, 5, 7, 7, 3, 3, 1, 5, 4, 8, 8, 5, 3, 2, 0, 3, 5, 4, 4, 5, 7, 8, 4, 1, 5, 9, 9, 1, 0, 4, 1, 8, 8, 2, 7, 6, 2, 3, 0, 7, 5, 4, 1, 0, 3, 5, 4, 5, 1, 7, 4, 2, 1, 7, 6, 0, 3, 7, 8, 6, 1, 1, 5, 8, 0, 4, 8, 8, 3, 5, 0, 7, 4, 2, 0, 0, 7, 6, 9, 8, 4, 7, 0, 0, 3, 0, 8, 1, 7, 8, 6, 2, 7, 8, 9, 1
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Continued fraction expansion of (3+sqrt(13))/6 is A010690.
LINKS
Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
EXAMPLE
(3+sqrt(13))/6 = 1.10092521257733154885...
CROSSREFS
Cf. A010470 (decimal expansion of sqrt(13)), A010690 (repeat 1, 9).
KEYWORD
cons,nonn
AUTHOR
Klaus Brockhaus, Apr 06 2010
STATUS
approved
A321643
a(n) = 5*2^n - (-1)^n.
+10
2
4, 11, 19, 41, 79, 161, 319, 641, 1279, 2561, 5119, 10241, 20479, 40961, 81919, 163841, 327679, 655361, 1310719, 2621441, 5242879, 10485761, 20971519, 41943041, 83886079, 167772161, 335544319, 671088641, 1342177279, 2684354561, 5368709119, 10737418241, 21474836479
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).
FORMULA
a(n+2) - a(n) = a(n+1) + a(n) = 15*2^n, n >= 0.
a(n) - 2*a(n-1) = period 2: repeat [3, -3], n > 0, a(0)=4, a(1)=11.
a(n+1) = 10*A051049(n) + period 2: repeat [1, 9].
a(n) = 12*2^n - A321483(n), n >= 0.
a(n) = 2^(n+2) + 3*A001045(n), n >= 0.
a(n) == A070366(n+4) (mod 9).
From Colin Barker, Dec 04 2018: (Start)
G.f.: (4 + 7*x) / ((1 + x)*(1 - 2*x)).
a(n) = a(n-1) + 2*a(n-2) for n > 1. (End)
E.g.f.: exp(-x)*(5*exp(3*x) - 1). - Elmo R. Oliveira, Aug 17 2024
MAPLE
[5*2^n-(-1)^n$n=0..30]; # Muniru A Asiru, Dec 05 2018
MATHEMATICA
a[n_] := 5*2^n - (-1)^n; Array[a, 30, 0] (* Amiram Eldar, Dec 03 2018 *)
PROG
(PARI) Vec((4 + 7*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 04 2018
(GAP) List([0..30], n->5*2^n-(-1)^n); # Muniru A Asiru, Dec 05 2018
(Python) for n in range(0, 30): print(5*2**n - (-1)**n) # Stefano Spezia, Dec 05 2018
CROSSREFS
Cf. A010690, A010701, A020714, A051049, A070366, A110286.
Cf. A001045, A081808, A321483.
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Dec 03 2018
STATUS
approved
A173261
Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.
+10
1
1, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 5, 1, 3, 1, 1, 6, 1, 4, 1, 2, 1, 7, 1, 5, 1, 3, 1, 1, 8, 1, 6, 1, 4, 1, 2, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
2,3
COMMENTS
One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:
-1, -1, 0, 1, 4, 9, 20, 41, 84, 169, 340, 681, 1364 ...: A084639;
-1, -1, 1, 3, 9, 19, 41, 83, 169, 339, 681, 1363, 2729;
-1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
-1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
B(n,k) = (n-1)*A001045(k) - T(n,k).
First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).
LINKS
G. C. Greubel, Antidiagonals n = 0..50 of the array, flattened
FORMULA
From G. C. Greubel, Dec 03 2021: (Start)
T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
Sum_{k=0..n} T(n-k, k) = A024206(n).
Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).
T(2*n-2, n) = A093178(n). (End)
EXAMPLE
The array T(n,k) starts in row n=2 with columns k>=0 as:
1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ... A000034;
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3 ... A010684;
1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4 ... A010685;
1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5 ... A010686;
1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6 ... A010687;
1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7 ... A010688;
1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8 ... A010689;
1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9 ... A010690;
1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
Antidiagonal triangle begins as:
1;
1, 2;
1, 3, 1;
1, 4, 1, 2;
1, 5, 1, 3, 1;
1, 6, 1, 4, 1, 2;
1, 7, 1, 5, 1, 3, 1;
1, 8, 1, 6, 1, 4, 1, 2;
1, 9, 1, 7, 1, 5, 1, 3, 1;
1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;
1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;
1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
MATHEMATICA
T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
Table[T[n-k, k], {n, 2, 17}, {k, 2, n}]//Flatten (* G. C. Greubel, Dec 03 2021 *)
PROG
(Sage) flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # G. C. Greubel, Dec 03 2021
CROSSREFS
Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.
Cf. A001045, A024206, A084639, A093178, A173114.
KEYWORD
nonn,tabl,easy
AUTHOR
Paul Curtz, Feb 14 2010
STATUS
approved
A190906
a(n) = gcd(n! / floor(n/2)!^2, 3^n).
+10
1
1, 1, 1, 3, 3, 3, 1, 1, 1, 9, 9, 9, 3, 3, 3, 9, 9, 9, 1, 1, 1, 3, 3, 3, 1, 1, 1, 27, 27, 27, 9, 9, 9, 27, 27, 27, 3, 3, 3, 9, 9, 9, 3, 3, 3, 27, 27, 27, 9, 9, 9, 27, 27, 27, 1, 1, 1, 3, 3, 3, 1, 1, 1, 9, 9, 9, 3, 3, 3, 9, 9, 9, 1, 1, 1, 3, 3, 3, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
FORMULA
a(n) = gcd(A056040(n), 3^n).
a(n) <= n. - Charles R Greathouse IV, Jun 30 2011
From Johannes W. Meijer, Jun 30 2011: (Start)
a(3*n) = a(3*n+1) = a(3*n+2) = A010684(n)*a(n) for n > 1 with a(0) = a(1) = a(2) = 1.
a(9*n+3) = a(9*n+4) = a(9*n+5) = 3*a(n).
a(9*n) = a(9*n+1) = a(9*n+2) = a(9*n+6) = a(9*n+7) = a(9*n+8) = A010690(n)*a(n). (End)
MAPLE
A190906 := n -> igcd(n!/iquo(n, 2)!^2, 3^n): seq(A190906(n), n=0..80);
MATHEMATICA
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := GCD[sf[n], 3^n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 29 2013 *)
PROG
(PARI) a(n)=gcd(n!/(n\2)!^2, 3^n) \\ Charles R Greathouse IV, Jun 30 2011
(PARI) a(n)=my(s); while(n\=3, s+=n%2); 3^s \\ Charles R Greathouse IV, Jun 30 2011
CROSSREFS
Cf. A060632.
KEYWORD
nonn,easy,look,hear
AUTHOR
Peter Luschny, Jun 30 2011
STATUS
approved
page 1

Search completed in 0.011 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified November 7 17:42 EST 2024. Contains 377536 sequences.
License Agreements, Terms of Use, Privacy Policy