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Search: a047406 -id:a047406
Displaying 1-7 of 7 results found. page 1
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A042964 Numbers that are congruent to 2 or 3 mod 4. +10
34
2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99, 102, 103, 106, 107, 110, 111, 114, 115, 118, 119, 122, 123, 126, 127 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Also numbers m such that binomial(m+2, m) mod 2 = 0. - Hieronymus Fischer, Oct 20 2007
Also numbers m such that floor(1+(m/2)) mod 2 = 0. - Hieronymus Fischer, Oct 20 2007
Partial sums of the sequence 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, ... which has period 2. - Hieronymus Fischer, Oct 20 2007
In groups of four add and divide by two the odd and even numbers. - George E. Antoniou, Dec 12 2001
From Jeremy Gardiner, Jan 22 2006: (Start)
Comments on the "mystery calculator". There are 6 cards.
Card 0: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, ... (A005408 sequence).
Card 1: 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, ... (this sequence).
Card 2: 4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, ... (A047566).
Card 3: 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 40, 41, 42, ... (A115419).
Card 4: 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50, ... (A115420).
Card 5: 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (A115421).
The trick: You secretly select a number between 1 and 63 from one of the cards. You indicate to me the cards on which that number appears; I tell you the number you selected!
The solution: I add together the first term from each of the indicated cards. The total equals the selected number. The numbers in each sequence all have a "1" in the same position in their binary expansion. Example: You indicate cards 1, 3 and 5. Your selected number is 2 + 8 + 32 = 42.
Numbers having a 1 in position 1 of their binary expansion. One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. (End)
Complement of A042948. - Reinhard Zumkeller, Oct 03 2008
Also the 2nd Witt transform of A040000 [Moree]. - R. J. Mathar, Nov 08 2008
In general, sequences of numbers congruent to {a,a+i} mod k will have a closed form of (k-2*i)*(2*n-1+(-1)^n)/4+i*n+a, from offset 0. - Gary Detlefs, Oct 29 2013
Union of A004767 and A016825; Fixed points of A098180. - Wesley Ivan Hurt, Jan 14 2014, Oct 13 2015
LINKS
David Lovler, Table of n, a(n) for n = 1..10000
Maths Magic, Mystery Calculator.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = A047406(n)/2.
From Michael Somos, Jan 12 2000: (Start)
G.f.: x*(2+x+x^2)/((1-x)*(1-x^2)).
a(n) = a(n-1) + 2 + (-1)^n. (End)
a(n) = 2n if n is odd, otherwise n = 2n - 1. - Amarnath Murthy, Oct 16 2003
a(n) = (3 + (-1)^(n-1))/2 + 2*(n-1) = 2n + 2 - (n mod 2). - Hieronymus Fischer, Oct 20 2007
A133872(a(n)) = 0. - Reinhard Zumkeller, Oct 03 2008
a(n) = 4*n - a(n-1) - 3 (with a(1) = 2). - Vincenzo Librandi, Nov 17 2010
a(n) = 2*n + ((-1)^(n-1) - 1)/2. - Gary Detlefs, Oct 29 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 - log(2)/4. - Amiram Eldar, Dec 05 2021
E.g.f.: 1 + ((4*x - 1)*exp(x) - exp(-x))/2. - David Lovler, Aug 08 2022
MAPLE
A042964:=n->2*n+((-1)^(n-1)-1)/2; seq(A042964(n), n=1..100); # Wesley Ivan Hurt, Jan 07 2014
MATHEMATICA
Flatten[Table[4n + {2, 3}, {n, 0, 31}]] (* Alonso del Arte, Feb 07 2013 *)
Select[Range[200], MemberQ[{2, 3}, Mod[#, 4]]&] (* or *) LinearRecurrence[ {1, 1, -1}, {2, 3, 6}, 90] (* Harvey P. Dale, Nov 28 2018 *)
PROG
(PARI) a(n)=2*n+2-n%2
(Magma) [2*n+((-1)^(n-1)-1)/2 : n in [1..100]]; // Wesley Ivan Hurt, Oct 13 2015
(Magma) [n: n in [1..150] | n mod 4 in [2, 3]]; // Vincenzo Librandi, Oct 13 2015
(PARI) Vec((2+x+x^2)/((1-x)*(1-x^2)) + O(x^100)) \\ Altug Alkan, Oct 13 2015
CROSSREFS
Cf. A000040, A133620, A133621, A133622, A133630, A133635.
Cf. A133872, A133882, A133890, A133900, A133910.
Card trick: A005408, A047566, A115419, A115420, A115421.
Cf. A004767, A016825 A040000, A042948, A047406, A098180.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar
Corrected by Jaroslav Krizek, Dec 18 2009
STATUS
approved
A047431 Numbers that are congruent to {1, 4, 5, 6} mod 8. +10
5
1, 4, 5, 6, 9, 12, 13, 14, 17, 20, 21, 22, 25, 28, 29, 30, 33, 36, 37, 38, 41, 44, 45, 46, 49, 52, 53, 54, 57, 60, 61, 62, 65, 68, 69, 70, 73, 76, 77, 78, 81, 84, 85, 86, 89, 92, 93, 94, 97, 100, 101, 102, 105, 108, 109, 110, 113, 116, 117, 118, 121, 124 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..62.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
FORMULA
G.f.: x*(1+2*x-x^2+2*x^3)/((x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = (-2-(-i)^n-i^n+4n)/2 where i=sqrt(-1). - Colin Barker, Jun 06 2012
From Wesley Ivan Hurt, May 30 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(2k) = A047406(k), a(2k-1) = A016813(k-1) k>0. (End)
E.g.f.: 2 - cos(x) - (1 - 2*x)*exp(x). - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 24 2021
MAPLE
A047431:=n->(4*n-2-(-I)^n-I^n)/2: seq(A047431(n), n=1..100); # Wesley Ivan Hurt, May 30 2016
MATHEMATICA
Table[(4n-2-(-I)^n-I^n)/2, {n, 80}] (* Wesley Ivan Hurt, May 30 2016 *)
LinearRecurrence[{2, -2, 2, -1}, {1, 4, 5, 6}, 70] (* Harvey P. Dale, Dec 04 2018 *)
PROG
(Sage) [lucas_number1(n, 0, 1)+2*n+1 for n in range(0, 56)] # Zerinvary Lajos, Jul 06 2008
(Magma) [n : n in [0..150] | n mod 8 in [1, 4, 5, 6]]; // Wesley Ivan Hurt, May 30 2016
CROSSREFS
Cf. A016813, A047404, A047406, A047546, A056594.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047425 Numbers that are congruent to {3, 4, 5, 6} mod 8. +10
3
3, 4, 5, 6, 11, 12, 13, 14, 19, 20, 21, 22, 27, 28, 29, 30, 35, 36, 37, 38, 43, 44, 45, 46, 51, 52, 53, 54, 59, 60, 61, 62, 67, 68, 69, 70, 75, 76, 77, 78, 83, 84, 85, 86, 91, 92, 93, 94, 99, 100, 101, 102, 107, 108, 109, 110, 115, 116, 117, 118, 123, 124 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Complement of numbers congruent to {0, 1, 2, 7} mod 8. - Jaroslav Krizek, Dec 19 2009
In general, sequences congruent to {a, a + i, a + 2i, ..., a + pi} mod k and a + p*i < k have a general form of (k - i*p)*floor(n/p) + i*n + a, from offset 0. - Gary Detlefs, Oct 20 2013
LINKS
Table of n, a(n) for n=1..62.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
G.f.: x*(3+x+x^2+x^3+2*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = 8*floor((n-1)/4) + ((n-1) mod 4) + 3.
a(n) = OR(n-1, 1) + OR(n-1, 2). - Gary Detlefs, Oct 20 2013
From Wesley Ivan Hurt, May 31 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (4*n-1-i^(2*n)-(1-i)*i^(-n)-(1+i)*i^n)/2 where i=sqrt(-1).
a(2k) = A047406(k), a(2k-1) = A047621(k). (End)
E.g.f.: 2 + sin(x) - cos(x) + 2*x*sinh(x) + (2*x - 1)*cosh(x). - Ilya Gutkovskiy, May 31 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 + (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2-sqrt(2))/4. - Amiram Eldar, Dec 26 2021
MAPLE
A047425:=n->8*floor((n-1)/4)+((n-1) mod 4)+3: seq(A047425(n), n=1..100); # Wesley Ivan Hurt, May 31 2016
MATHEMATICA
Flatten[# + {3, 4, 5, 6} &/@(8*Range[0, 15])] (* Harvey P. Dale, Jun 26 2011 *)
PROG
(Magma) [n : n in [0..150] | n mod 8 in [3, 4, 5, 6]]; // Wesley Ivan Hurt, May 31 2016
CROSSREFS
Cf. A047406, A047621.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047430 Numbers that are congruent to {0, 4, 5, 6} mod 8. +10
1
0, 4, 5, 6, 8, 12, 13, 14, 16, 20, 21, 22, 24, 28, 29, 30, 32, 36, 37, 38, 40, 44, 45, 46, 48, 52, 53, 54, 56, 60, 61, 62, 64, 68, 69, 70, 72, 76, 77, 78, 80, 84, 85, 86, 88, 92, 93, 94, 96, 100, 101, 102, 104, 108, 109, 110, 112, 116, 117, 118, 120, 124 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..62.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
G.f.: x^2*(4+x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
From Wesley Ivan Hurt, May 25 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-5+i^(2*n)-(2+i)*i^(-n)-(2-i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047406(k), a(2k-1) = A047615(k). (End)
E.g.f.: (4 - sin(x) - 2*cos(x) + (4*x - 3)*sinh(x) + (4*x - 2)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016
Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*log(2+sqrt(2))/8 - (2-sqrt(2))*(Pi-log(2))/16. - Amiram Eldar, Dec 23 2021
MAPLE
A047430:=n->(8*n-5+I^(2*n)-(2+I)*I^(-n)-(2-I)*I^n)/4: seq(A047430(n), n=1..100); # Wesley Ivan Hurt, May 25 2016
MATHEMATICA
Table[(8n-5+I^(2n)-(2+I)*I^(-n)-(2-I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 25 2016 *)
Select[Range[0, 124], MemberQ[{0, 4, 5, 6}, Mod[#, 8]] &] (* Michael De Vlieger, May 25 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 5, 6, 8}, 100] (* Harvey P. Dale, Aug 05 2023 *)
PROG
(Magma) [n : n in [0..150] | n mod 8 in [0, 4, 5, 6]]; // Wesley Ivan Hurt, May 25 2016
CROSSREFS
Cf. A047406, A047615.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047433 Numbers that are congruent to {2, 4, 5, 6} mod 8. +10
1
2, 4, 5, 6, 10, 12, 13, 14, 18, 20, 21, 22, 26, 28, 29, 30, 34, 36, 37, 38, 42, 44, 45, 46, 50, 52, 53, 54, 58, 60, 61, 62, 66, 68, 69, 70, 74, 76, 77, 78, 82, 84, 85, 86, 90, 92, 93, 94, 98, 100, 101, 102, 106, 108, 109, 110, 114, 116, 117, 118, 122, 124 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..62.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
G.f.: x*(2+2*x+x^2+x^3+2*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
From Wesley Ivan Hurt, May 26 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-3-i^(2*n)-(2-i)*i^(-n)-(2+i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047406(k), a(2k-1) = A047617(k). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (3-sqrt(2))*Pi/16 + log(2)/4 + sqrt(2)*log(sqrt(2)-1)/8. - Amiram Eldar, Dec 25 2021
MAPLE
A047433:=n->(8*n-3-I^(2*n)-(2-I)*I^(-n)-(2+I)*I^n)/4: seq(A047433(n), n=1..100); # Wesley Ivan Hurt, May 26 2016
MATHEMATICA
Select[Range[120], MemberQ[{2, 4, 5, 6}, Mod[#, 8]]&] (* Harvey P. Dale, Mar 04 2011 *)
PROG
(Magma) [n : n in [0..150] | n mod 8 in [2, 4, 5, 6]]; // Wesley Ivan Hurt, May 26 2016
CROSSREFS
Cf. A047406, A047617.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A160108 Lodumo_8 of Fibonacci numbers. +10
1
0, 1, 9, 2, 3, 5, 8, 13, 21, 10, 7, 17, 16, 25, 33, 18, 11, 29, 24, 37, 45, 26, 15, 41, 32, 49, 57, 34, 19, 53, 40, 61, 69, 42, 23, 65, 48, 73, 81, 50, 27, 77, 56, 85, 93, 58, 31, 89, 64, 97, 105, 66, 35, 101, 72, 109, 117, 74, 39, 113, 80, 121, 129, 82, 43, 125, 88, 133, 141 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Some integers (see A047406) are missing.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
FORMULA
Lodumo_8 transform of A000045 (for definition see Transforms).
Conjecture: a(n) = 2*a(n-6)-a(n-12). - Colin Barker, Oct 04 2014
Empirical g.f.: x*(7*x^10 +x^9 +6*x^8 +3*x^7 +11*x^6 +8*x^5 +5*x^4 +3*x^3 +2*x^2 +9*x +1) / ((x -1)^2*(x +1)^2*(x^2 -x +1)^2*(x^2 +x +1)^2). - Colin Barker, Oct 04 2014
CROSSREFS
Cf. A000045, A079344.
KEYWORD
nonn
AUTHOR
Philippe Deléham, May 02 2009
EXTENSIONS
Replaced 35 by 33 - R. J. Mathar, May 03 2009
STATUS
approved
A255842 a(n) = 2*n^2 + 12. +10
1
12, 14, 20, 30, 44, 62, 84, 110, 140, 174, 212, 254, 300, 350, 404, 462, 524, 590, 660, 734, 812, 894, 980, 1070, 1164, 1262, 1364, 1470, 1580, 1694, 1812, 1934, 2060, 2190, 2324, 2462, 2604, 2750, 2900, 3054, 3212, 3374, 3540, 3710, 3884, 4062, 4244, 4430 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.
Equivalently, numbers m such that 2*m-24 is a square.
For n = 0..10, a(n)-1 is prime (see A092968).
LINKS
Table of n, a(n) for n=0..47.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A114949(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)
MATHEMATICA
Table[2 n^2 + 12, {n, 0, 50}]
PROG
(PARI) vector(50, n, n--; 2*n^2+12)
(Sage) [2*n^2+12 for n in (0..50)]
(Magma) [2*n^2+12: n in [0..50]];
CROSSREFS
Cf. A016825 (first differences), A092968, A114949.
Subsequence of A047238 and A047406.
Cf. similar sequences listed in A255843.
KEYWORD
nonn,easy
AUTHOR
Avi Friedlich, Mar 08 2015
EXTENSIONS
Edited by Bruno Berselli, Mar 11 2015
STATUS
approved
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Last modified August 29 21:13 EDT 2024. Contains 375518 sequences. (Running on oeis4.)