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 A274706 Irregular triangle read by rows. T(n,k) (n >= 0) is a statistic on orbital systems over n sectors: the number of orbitals which have an integral whose absolute value is k. 10
 1, 1, 0, 2, 0, 4, 2, 2, 0, 2, 0, 2, 6, 4, 6, 4, 4, 4, 2, 0, 6, 0, 6, 0, 4, 0, 2, 0, 2, 6, 24, 16, 20, 14, 16, 12, 8, 6, 8, 4, 4, 2, 8, 0, 14, 0, 14, 0, 10, 0, 10, 0, 6, 0, 4, 0, 2, 0, 2, 36, 52, 68, 48, 64, 48, 48, 40, 44, 32, 36, 24, 22, 16, 16, 8, 10, 8, 4, 4, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS For the combinatorial definitions see A232500. The absolute integral of an orbital w over n sectors is abs(Sum_{1<=k<=n} Sum_{1<=i<=k} w(i)))) where w(i) are the jumps of the orbital represented by -1, 0, 1. An orbital is balanced if its integral is 0 (A241810). LINKS Peter Luschny, Orbitals EXAMPLE The length of row n is 1+floor(n^2//4). The triangle begins: [n] [k=0,1,2,...] [row sum] [0] [1] 1 [1] [1] 1 [2] [0, 2] 2 [3] [0, 4, 2] 6 [4] [2, 0, 2, 0, 2] 6 [5] [6, 4, 6, 4, 4, 4, 2] 30 [6] [0, 6, 0, 6, 0, 4, 0, 2, 0, 2] 20 [7] [6, 24, 16, 20, 14, 16, 12, 8, 6, 8, 4, 4, 2] 140 [8] [8, 0, 14, 0, 14, 0, 10, 0, 10, 0, 6, 0, 4, 0, 2, 0, 2] 70 T(5, 4) = 4 because the integral of four orbitals have the absolute value 4: Integral([-1, -1, 1, 1, 0]) = -4, Integral([0, -1, -1, 1, 1]) = -4, Integral([0, 1, 1, -1, -1]) = 4, Integral([1, 1, -1, -1, 0]) = 4. PROG # Brute force counting, function unit_orbitals defined in A274709. def orbital_integral(n):     if n == 0: return [1]     S = [0]*(1+floor(n^2//4))     for u in unit_orbitals(n):         L = list(accumulate(accumulate(u)))         S[abs(L[-1])] += 1     return S for n in (0..8): print orbital_integral(n) CROSSREFS Cf. A056040 (row sum), A232500, A241810 (col. 0), A242087. Other orbital statistics: A241477 (first zero crossing), A274708 (number of peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent). Sequence in context: A339941 A211318 A324239 * A037035 A159984 A240697 Adjacent sequences:  A274703 A274704 A274705 * A274707 A274708 A274709 KEYWORD nonn,tabf AUTHOR Peter Luschny, Jul 10 2016 STATUS approved

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Last modified April 17 02:26 EDT 2021. Contains 343059 sequences. (Running on oeis4.)