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 A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n. 1
 1, 3, 2, 4, 3, 1, 7, 6, 4, 3, 8, 7, 5, 4, 1, 10, 9, 7, 6, 3, 2, 11, 10, 8, 7, 4, 3, 1, 15, 14, 12, 11, 8, 7, 5, 4, 16, 15, 13, 12, 9, 8, 6, 5, 1, 18, 17, 15, 14, 11, 10, 8, 7, 3, 2, 19, 18, 16, 15, 12, 11, 9, 8, 4, 3, 1, 22, 21, 19, 18, 15, 14, 12, 11, 7, 6, 4, 3 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums give A143157. Left border gives A005187. Right border gives A001511. LINKS FORMULA Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n. From Kevin Ryde, Oct 07 2021: (Start) T(n,k) = A005187(n) - A005187(k-1). G.f.: (V(x) - V(x*y)) * y/((1-x)*(1-y)) where V(x) is the g.f. of A001511. (End) EXAMPLE First few rows of the triangle =         k=1 k=2 k=3 k=4 k=5 k=6 k=7   n=1:   1;   n=2:   3,  2;   n=3:   4,  3,  1;   n=4:   7,  6,  4,  3;   n=5:   8,  7,  5,  4,  1;   n=6:  10,  9,  7,  6,  3,  2;   n=7:  11, 10,  8,  7,  4,  3,  1;   ... Row 6 = (10, 9, 7, 6, 3, 2) = partial sums of the first 6 terms of the ruler sequence, starting from the right: (1, 2, 1, 3, 1, 2,...). PROG (PARI) T(n, k) = k--; 2*(n-k) - hammingweight(n) + hammingweight(k); \\ Kevin Ryde, Oct 07 2021 CROSSREFS Cf. A001511, A000012, A091512, A005187. Sequence in context: A334662 A016559 A104566 * A227471 A101403 A025509 Adjacent sequences:  A143153 A143154 A143155 * A143157 A143158 A143159 KEYWORD nonn,easy,tabl AUTHOR Gary W. Adamson, Jul 27 2008 STATUS approved

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Last modified January 20 18:38 EST 2022. Contains 350472 sequences. (Running on oeis4.)