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2, 4, 12, 20, 38, 56, 88, 120, 170, 220, 292, 364, 462, 560, 688, 816, 978, 1140, 1340, 1540, 1782, 2024, 2312, 2600, 2938, 3276, 3668, 4060, 4510, 4960, 5472, 5984, 6562, 7140, 7788, 8436, 9158, 9880, 10680, 11480, 12362, 13244, 14212, 15180, 16238, 17296, 18448, 19600, 20850, 22100
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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The atomic numbers of the augmented alkaline earth group in Charles Janet's spiral periodic table are 0 and the first eight terms of this sequence (see Stewart reference). - Alonso del Arte, May 13 2011
Maximum number of 123 patterns in an alternating permutation of length n+3. - Lara Pudwell, Jun 09 2019
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LINKS
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FORMULA
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a(n) = (n+1)*(3 + 2*n^2 + 4*n - 3*(-1)^n)/12.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: 2*x*(1 + x^2) / ( (1+x)^2*(x-1)^4 ).
E.g.f.: (1/12)*(3*(x - 1) + (3 + 15*x + 12*x^2 + 2*x^3)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 19 2016
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EXAMPLE
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a(1)=2. The alternating permutation of length 1+3=4 with the maximum number of copies of 123 is 1324. The two copies are 124 and 134.
a(2)=4. The alternating permutation of length 2+3=5 with the maximum number of copies of 123 is 13254. The four copies are 124, 125, 134, and 135.
a(3)=12. The alternating permutation of length 3+3=6 with the maximum number of copies of 123 is 132546. The twelve copies are 124, 125, 126, 134, 135, 136, 146, 156, 246, 256, 346, and 356. (End)
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MATHEMATICA
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LinearRecurrence[{2, 1, -4, 1, 2, -1}, {2, 4, 12, 20, 38, 56}, 50] (* G. C. Greubel, Jul 19 2016 *)
Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12, {n, 50}] (* Michael De Vlieger, Jul 20 2016 *)
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PROG
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(Magma) [(n+1)*(3+2*n^2+4*n-3*(-1)^n)/12: n in [1..50] ]; // Vincenzo Librandi, Aug 06 2011
(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; -1, 2, 1, -4, 1, 2]^(n-1)*[2; 4; 12; 20; 38; 56])[1, 1] \\ Charles R Greathouse IV, Jul 21 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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