A135231 Row sums of triangle A135230.
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
Offset: 0
Keywords
Examples
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.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..500
Programs
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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
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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 *) -
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
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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
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)
Extensions
Terms a(16) onward added and offset changed by G. C. Greubel, Nov 20 2019