A115127 Second (k=2) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).
3, 6, 7, 10, 16, 19, 15, 30, 47, 56, 21, 50, 95, 146, 174, 28, 77, 170, 311, 471, 561, 36, 112, 280, 586, 1043, 1562, 1859, 45, 156, 434, 1015, 2044, 3564, 5291, 6292, 55, 210, 642, 1652, 3682, 7204, 12363, 18226, 21658, 66, 275, 915, 2562, 6230, 13392, 25623
Offset: 2
Examples
[3];[6,7];[10,16,19];[15,30,47,56];... Main diagonal (n-m=1) example: a(3,2)= 7 = 5 + 2 because A115126(3,2)=5 and A115126(2,2)=2. Subdiagonal (n-m>1) example: a(4,2)= 16 = 9 + 7 because A115126(4,2)=9 and a(3,2)=7.
Links
- W. Lang: First 10 rows.
Crossrefs
Row sums give A115128.
Formula
A115132 Partial sums of A064059.
132, 561, 1562, 3564, 7204, 13392, 23388, 38892, 62148, 96063, 144342, 211640, 303732, 427702, 592152, 807432, 1085892, 1442157, 1893426, 2459796, 3164612, 4034844, 5101492, 6400020, 7970820, 9859707, 12118446, 14805312, 17985684
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).
Programs
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Mathematica
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{132,561,1562,3564,7204,13392,23388,38892},40] (* Harvey P. Dale, Jul 22 2024 *)
A180016 Partial sums of number of n-step closed paths on hexagonal lattice A002898.
1, 1, 7, 19, 109, 469, 2509, 12589, 67399, 358039, 1946395, 10622755, 58600531, 324978643, 1813780243, 10169519635, 57273912685, 323755931917, 1836345339961, 10446793409041, 59591722204861, 340755882430381
Offset: 0
Comments
Also, number of closed paths of length <= n on the honeycomb lattice. The analog on the square lattice is A115130.
The subsequence of primes begins 7, 19, 109, 12589, 67399.
Examples
a(0) = 1 because there is a unique null walk on no points. a(1) = 1 because there are no closed paths of length 1 (which connects the origin with one of 6 other points before symmetry is considered). a(2) = 7 because one adds the 6 closed paths of length 2 (which go from origin to one of 6 surrounding points on the lattice, and return in the opposite directions). a(8) = 1 + 0 + 6 + 12 + 90 + 360 + 2040 + 10080 + 54810 = 67399.
Programs
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Mathematica
Table[Sum[Sum[(-2)^(nn-i)*Binomial[i, j]^3*Binomial[nn, i], {i, 0, nn}, {j, 0, i}],{nn,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 24 2012 *)
Formula
a(n) = Sum_{i=0..n} A002898(i).
D-finite with recurrence: n^2*a(n) = (2*n-1)*n*a(n-1) + (n-1)*(23*n-24)*a(n-2) + 12*(n-4) * (n-1)*a(n-3) - 36*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 3*sqrt(3)*6^n/(5*Pi*n). - Vaclav Kotesovec, Oct 24 2012
G.f.: hypergeom([1/3,1/3],[1],-27*x*(2*x+1)^2/((3*x+1)*(6*x-1)^2))/((1-x)*(3*x+1)^(1/3)*(1-6*x)^(2/3)). - Mark van Hoeij, Apr 17 2013
Comments