A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.
0, 1, 0, -1, 2, 0, 3, -2, 4, 0, -5, 6, -4, 8, 0, 11, -10, 12, -8, 16, 0, -21, 22, -20, 24, -16, 32, 0, 43, -42, 44, -40, 48, -32, 64, 0, -85, 86, -84, 88, -80, 96, -64, 128, 0, 171, -170, 172, -168, 176, -160, 192, -128, 256, 0, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 0
Offset: 0
Examples
Triangle begins as:
0;
1, 0;
-1, 2, 0;
3, -2, 4, 0;
-5, 6, -4, 8, 0;
11, -10, 12, -8, 16, 0;
-21, 22, -20, 24, -16, 32, 0;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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Magma
[2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023
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Maple
A001045:= n -> (2^n-(-1)^n)/3; A140944:= proc(n,k) if n = 0 then A001045(k); else procname(n-1,k+1)-procname(n-1,k) ; fi; end: seq(seq(A140944(n,k),k=0..n),n=0..10); # R. J. Mathar, Sep 07 2009
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Mathematica
T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2014 *) Table[2^k*(1-(-2)^(n-k))/3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2023 *) -
PARI
T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022
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SageMath
def A140944(n,k): return 2^k*(1 - (-2)^(n-k))/3 flatten([[A140944(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 18 2023
Formula
T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).
T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - Jianing Song, Aug 11 2022
From G. C. Greubel, Feb 18 2023: (Start)
T(n, n-1) = A000079(n).
T(2*n, n) = (-1)^(n+1)*A192382(n+1).
T(2*n, n-1) = (-1)^n*A246036(n-1).
T(2*n, n+1) = A083086(n).
T(3*n, n) = -A115489(n).
Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).
Sum_{k=0..n} 2^k*T(n, k) = A084175(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).
Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)
Extensions
Edited and extended by R. J. Mathar, Sep 07 2009
Comments