A168380 Row sums of A168281.

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

Offset: 1

Author

Paul Curtz, Nov 24 2009

Keywords

• nonn
• easy

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

Examples

			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)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 4.
• Lara Pudwell, Packing patterns in restricted permutations, 2019.
• 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.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Programs

Magma

[(n+1)*(3+2*n^2+4*n-3*(-1)^n)/12: n in [1..50] ]; // Vincenzo Librandi, Aug 06 2011

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 *)

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

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