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 A135231 Row sums of triangle A135230. 2
 1, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532, 5726623062, 11453246124, 22906492246, 45812984492 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..500 FORMULA a(2*n+1) = A005578(n+1) if n is odd. Conjectures from Chai Wah Wu, Aug 31 2023: (Start) a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 3. G.f.: (-2*x^3 - x^2 + 1)/((x - 1)*(x + 1)*(2*x - 1)). (End) EXAMPLE a(3) = 6 = sum of row 4 terms of triangle A135230; (1 + 2 + 2 + 1). a(5) = 22 = A005578(6). a(6) = 44 = A005578(7) + 1. MAPLE T:= proc(n, k) option remember; if k=n then 1 elif k=0 then (3+(-1)^n)/2 else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2)) fi; end: seq( add(T(n, j), j=0..n), n=0..40); # G. C. Greubel, Nov 20 2019 MATHEMATICA T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3+(-1)^n)/2, Sum[Binomial[n-1 - 2*j, k-1], {j, 0, Floor[(n-1)/2]}]]]; Table[Sum[T[n, j], {j, 0, n}], {n, 0, 40}] (* G. C. Greubel, Nov 20 2019 *) PROG (PARI) T(n, k) = if(k==n, 1, if(k==0, (3+(-1)^n)/2, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) )); \\ G. C. Greubel, Nov 20 2019 (Magma) function T(n, k) if k eq n then return 1; elif k eq 0 then return (3+(-1)^n)/2; else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]); end if; return T; end function; [(&+[T(n, j): j in [0..n]]): n in [0..40]]; // G. C. Greubel, Nov 20 2019 (Sage) @CachedFunction def T(n, k): if (k==n): return 1 elif (k==0): return (3+(-1)^n)/2 else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2))) [sum(T(n, j) for j in (0..n)) for n in (0..40)] # G. C. Greubel, Nov 20 2019 CROSSREFS Cf. A005578, A135230. Sequence in context: A341582 A370648 A326114 * A326489 A217356 A030793 Adjacent sequences: A135228 A135229 A135230 * A135232 A135233 A135234 KEYWORD nonn AUTHOR Gary W. Adamson, Nov 23 2007 EXTENSIONS Terms a(16) onward added and offset changed by G. C. Greubel, Nov 20 2019 STATUS approved

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Last modified August 6 02:44 EDT 2024. Contains 374957 sequences. (Running on oeis4.)