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A168380 Row sums of A168281. 19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 4.
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.
Philip Stewart, Charles Janet: unrecognized genius of the Periodic System. Foundations of Chemistry (2010), p. 9.
FORMULA
a(n) = 2*A005993(n-1).
a(n) = (n+1)*(3 + 2*n^2 + 4*n - 3*(-1)^n)/12.
a(n+1) - a(n) = A093907(n) = A137583(n+1).
a(2n+1) = A035597(n+1), a(2n) = A002492(n).
a(n) = A099956(n-1), 2 <= n <= 7.
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 ).
a(n) = A000292(n) + A027656(n-1). - Paul Curtz, Oct 26 2012
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
EXAMPLE
From Lara Pudwell, Jun 09 2019: (Start)
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)
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A090922 A056228 A166869 * A335730 A308286 A297184
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Nov 24 2009
STATUS
approved

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Last modified July 25 12:21 EDT 2024. Contains 374588 sequences. (Running on oeis4.)