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A276382
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a(1) = 1, and a(n) = a(n-1) + floor(3*n/2) + 1 for n >= 2.
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2
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1, 5, 10, 17, 25, 35, 46, 59, 73, 89, 106, 125, 145, 167, 190, 215, 241, 269, 298, 329, 361, 395, 430, 467, 505, 545, 586, 629, 673, 719, 766, 815, 865, 917, 970, 1025, 1081, 1139, 1198, 1259, 1321, 1385, 1450, 1517, 1585, 1655, 1726, 1799, 1873, 1949, 2026
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OFFSET
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1,2
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COMMENTS
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Given 3 distinct numbers i, j and k whose prime signatures are exactly n 1's, then a(n) is the number of prime signatures for all permutations of i*j*k.
a(n) is the number of partitions of 3n such that there are no more than n-1 3's and no parts > 3.
a(3n+1) represents the number of prime signature sets whose members are excluded as terms in A026477, as a consequence of being products of three smaller terms whose prime signatures are exactly 3n+1 1's.
First differences are floor(3n/2) + 1 (A001651(n+1)); second differences are 1 when n is even and 2 when n is odd; third differences are 1 when n is even and -1 when n is odd.
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LINKS
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FORMULA
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a(n) = ((-1)^n + 12*n + 6*n^2 - 9)/8 for n > 0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 4.
G.f.: x*(1 + 3*x - x^3) / ((1-x)^3*(1+x)). (End)
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EXAMPLE
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a(2)=5; the 5 prime signatures / partitions are: {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1} and {1,1,1,1,1,1}.
G.f. = x + 5*x^2 + 10*x^3 + 17*x^4 + 25*x^5 + 35*x^6 + 46*x^7 + ... - Michael Somos, Sep 08 2023
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[3 n/2] + 1 ; Array[a, 51] (* Michael De Vlieger, Sep 01 2016 *)
a[n_] := Floor[(3*(n+1)^2 - 7)/4]; (* Michael Somos, Sep 08 2023 *)
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PROG
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(PARI) Vec(x*(1+3*x-x^3)/((1-x)^3*(1+x)) + O(x^60)) \\ Colin Barker, Sep 01 2016
(PARI) {a(n) = (3*(n+1)^2 - 7)\4}; /* Michael Somos, Sep 09 2023 */
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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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