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A289199
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a(n) is the number of odd integers divisible by 13 in the open interval (12*(n-1)^2, 12*n^2).
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2
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0, 0, 2, 2, 3, 5, 5, 6, 7, 7, 9, 10, 10, 12, 12, 14, 14, 15, 17, 17, 18, 19, 19, 21, 22, 22, 24, 24, 26, 26, 27, 29, 29, 30, 31, 31, 33, 34, 34, 36, 36, 38, 38, 39, 41, 41, 42, 43, 43, 45, 46, 46, 48, 48, 50, 50, 51, 53, 53, 54, 55, 55, 57, 58, 58, 60, 60, 62, 62, 63, 65
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OFFSET
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0,3
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COMMENTS
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This sequence has the form (0+12k, 0+12k, 2+12k, 2+12k, 3+12k, 5+12k, 5+12k, 6+12k, 7+12k, 7+12k, 9+12k, 10+12k, 10+12k) for k >= 0.
Theorems: A) Generally for an interval (2*m*(n-1)^2,2*m*n^2) and a divisor d with 2*m < d there is a unique d-length form (e_i+2*m*k)_{i=0..d-1, k>=0} with e_i in [0,2*m]; here m = 6, d = 13.
B) Sum_{i=0..d-1}e_i = m*(d-2); here 66 = 6*(13-2).
Proof:
A) In d consecutive intervals
(2*m*(n-1)^2,2*m*(n+2)^2) there are m*d*(2*k+d) consecutive odd numbers and therefore m*(2*k+d) multiples of d where k=floor((n-1)/d).
B) With initial value a(0)=0 we have a(d)=2*m and thus Sum_{i=0..d-1} e_i = Sum_{i=1..d}a(i)-a(d) = m(2*0+d)-2*m = m*(d-2). Q.E.D.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,1,-1).
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FORMULA
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a(n + 13*k) = a(n) + 12*k.
G.f.: x^2*(1 + x)*(1 - x + x^2)*(2 + x^2 + x^6 + 2*x^8) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)).
a(n) = a(n-1) + a(n-13) - a(n-14) for n>13.
(End)
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MATHEMATICA
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Table[Count[Mod[Table[2(6(n-1)^2 +k)-1, {k, 12 n-6}], 13], 0], {n, 0, 70}]
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PROG
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(PARI) concat(vector(2), Vec(x^2*(1 + x)*(1 - x + x^2)*(2 + x^2 + x^6 + 2*x^8) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)) + O(x^100))) \\ Colin Barker, Jul 03 2017
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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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