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 A140993 Triangle G(n, k) read by rows, for 1 <= k <= n, where G(n, n) = G(n+1, 1) = 1, G(n+2, 2) = 2, G(n+3, m) = G(n+1, m-1) + G(n+1, m-2) + G(n+2, m-1) for n >= 1 and m = 3..(n+2). 24
 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 7, 1, 1, 2, 5, 11, 12, 1, 1, 2, 5, 12, 23, 20, 1, 1, 2, 5, 12, 28, 46, 33, 1, 1, 2, 5, 12, 29, 63, 89, 54, 1, 1, 2, 5, 12, 29, 69, 137, 168, 88, 1, 1, 2, 5, 12, 29, 70, 161, 289, 311, 143, 1, 1, 2, 5, 12, 29, 70, 168, 367, 594, 567, 232, 1, 1, 2, 5, 12, 29, 70, 169, 399, 817, 1194, 1021, 376, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS From Petros Hadjicostas, Jun 10 2019: (Start) Let b(m) = lim_{n -> infinity} G(n, m) for each m >= 1. Then b(1) = 1, b(2) = 2, and b(m) = 2*b(m-1) + b(m-2) for m >= 3. (The existence of the limit can be proved by induction on m.) This means b(m) = A000129(m) for m >= 1 (known as the Pell numbers). If we want to get the second main diagonal, we let c(n) = G(n+1, n) for n >= 1. Then c(n+2) = G(n+3, n+2) = G(n+1, n+1) + G(n+1, n) + G(n+2, n+1) = 1 + c(n) + c(n+1) with c(1) = G(2, 1) = 1 and c(2) = G(3, 2) = 2, which implies that c(n) = A000071(n+2) = Fibonacci(n+2) - 1 for n >= 1. This array is the mirror image of A140998 (except for a shifting of the indices by 1). Thus, G(n, k) = A140998(n - 1, n - k) for 1 <= k <= n. This array has index of obliqueness e = 1, while array A140998 has index of obliqueness e = 0. Both arrays have the same index of asymmetry (s = 1). (End) LINKS Juri-Stepan Gerasimov, Stepan's triangles and Pascal's triangle are connected by the recurrence relation ... FORMULA From Petros Hadjicostas, Jun 10 2019: (Start) G(n, k) = A140998(n - 1, n - k) for 1 <= k <= n. Bivariate g.f.: Sum_{n >= 1, k >= 1} G(n, k)*x^n*y^k = x*y*(1 - x*y -x^2*y^2 + x^3*y^2)/((1 - x) * (1 - x*y) * (1 - x*y - x^2*y - x^2*y^2)). (Here, we let G(n, k) = 0 when 1 <= n < k.) To get the row sums, we let y = 1 in the above bivariate g.f. and simplify. We get x/(1 - 2*x), which is the g.f. of sequence (A000079(n-1): n >= 1) = (2^(n-1): n >= 1). (End) EXAMPLE Triangle (with rows for n >= 1 and columns for k >= 1) begins as follows:   1   1 1   1 2 1   1 2 4  1   1 2 5  7  1   1 2 5 11 12  1   1 2 5 12 23 20   1   1 2 5 12 28 46  33   1   1 2 5 12 29 63  89  54   1   1 2 5 12 29 69 137 168  88    1   1 2 5 12 29 70 161 289 311  143    1   1 2 5 12 29 70 168 367 594  567  232    1   1 2 5 12 29 70 169 399 817 1194 1021  376   1   1 2 5 12 29 70 169 407 934 1778 2355 1820 609 1   ... MAPLE A140993 := proc(n, k) if k = n then 1; elif k = 1 then 1; elif k = 2 then 2; else procname(n-2, k-1)+procname(n-2, k-2)+procname(n-1, k-1) ; end if; end proc: seq(seq(A140993(n, k), k=1..n), n=1..15) ; # R. J. Mathar, Apr 28 2010 MATHEMATICA t[n_, k_] := If[k == n, 1, If[k == 1, 1, If[k == 2, 2, t[n - 2, k - 1] + t[n - 2, k - 2] + t[n - 1, k - 1]]]]; Flatten[Table[ t[n, k], {n, 13}, {k, n}]] (* Robert G. Wilson v, Dec 22 2011 *) CROSSREFS Cf. A000071, A000079, A000129, A007318, A140994, A140995, A140996, A140997, A140998, A141020, A141021. Sequence in context: A140995 A140994 A245163 * A027935 A137940 A274859 Adjacent sequences:  A140990 A140991 A140992 * A140994 A140995 A140996 KEYWORD nonn,tabl AUTHOR Juri-Stepan Gerasimov, Jul 08 2008 EXTENSIONS Entries checked by R. J. Mathar, Apr 28 2010 Name and offset edited by Petros Hadjicostas, Jun 10 2019 STATUS approved

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Last modified May 29 07:50 EDT 2020. Contains 334697 sequences. (Running on oeis4.)