Search: a047406 -id:a047406
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A042964
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Numbers that are congruent to 2 or 3 mod 4.
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+10
34
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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
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OFFSET
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1,1
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COMMENTS
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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
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.
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
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LINKS
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FORMULA
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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
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
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MAPLE
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MATHEMATICA
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Select[Range[200], MemberQ[{2, 3}, Mod[#, 4]]&] (* or *) LinearRecurrence[ {1, 1, -1}, {2, 3, 6}, 90] (* Harvey P. Dale, Nov 28 2018 *)
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PROG
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(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
(PARI) Vec((2+x+x^2)/((1-x)*(1-x^2)) + O(x^100)) \\ Altug Alkan, Oct 13 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A047431
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Numbers that are congruent to {1, 4, 5, 6} mod 8.
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+10
5
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 24 2021
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MAPLE
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MATHEMATICA
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LinearRecurrence[{2, -2, 2, -1}, {1, 4, 5, 6}, 70] (* Harvey P. Dale, Dec 04 2018 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A047425
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Numbers that are congruent to {3, 4, 5, 6} mod 8.
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+10
3
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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) = 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).
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
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MAPLE
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MATHEMATICA
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Flatten[# + {3, 4, 5, 6} &/@(8*Range[0, 15])] (* Harvey P. Dale, Jun 26 2011 *)
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PROG
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(Magma) [n : n in [0..150] | n mod 8 in [3, 4, 5, 6]]; // Wesley Ivan Hurt, May 31 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A047430
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Numbers that are congruent to {0, 4, 5, 6} mod 8.
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+10
1
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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).
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
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(Magma) [n : n in [0..150] | n mod 8 in [0, 4, 5, 6]]; // Wesley Ivan Hurt, May 25 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A047433
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Numbers that are congruent to {2, 4, 5, 6} mod 8.
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+10
1
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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
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OFFSET
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1,1
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LINKS
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FORMULA
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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
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).
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
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MAPLE
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MATHEMATICA
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Select[Range[120], MemberQ[{2, 4, 5, 6}, Mod[#, 8]]&] (* Harvey P. Dale, Mar 04 2011 *)
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PROG
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(Magma) [n : n in [0..150] | n mod 8 in [2, 4, 5, 6]]; // Wesley Ivan Hurt, May 26 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A160108
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Lodumo_8 of Fibonacci numbers.
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+10
1
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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
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OFFSET
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0,3
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COMMENTS
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Some integers (see A047406) are missing.
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LINKS
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FORMULA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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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
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OFFSET
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0,1
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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)
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MATHEMATICA
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Table[2 n^2 + 12, {n, 0, 50}]
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PROG
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(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]];
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CROSSREFS
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Cf. similar sequences listed in A255843.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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