cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

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

Views

Author

Paul Curtz, Jul 24 2008

Keywords

Comments

A variant of the triangle A140503, now including the diagonal.
Since the diagonal contains zeros, rows sums are those of A140503.

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

Crossrefs

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.
Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

Programs

  • Magma
    [2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023
  • 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
  • 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
  • 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

A140950 a(n) = A140944(n+1) - 3*A140944(n).

Original entry on oeis.org

1, -3, -1, 5, -6, 3, -11, 10, -12, -5, 21, -22, 20, -24, 11, -43, 42, -44, 40, -48, -21, 85, -86, 84, -88, 80, -96, 43, -171, 170, -172, 168, -176, 160, -192, -85, 341, -342, 340, -344, 336, -352, 320, -384, 171, -683, 682, -684, 680, -688
Offset: 0

Views

Author

Paul Curtz, Jul 25 2008

Keywords

Comments

Jacobsthal numbers appear twice: 1) A001045(n+2) signed, terms 0, 1, 3, 6, 10 (A000217); 2) A001045(n+1) signed, terms 0, 2, 5, 9 (n*(n+3)/2=A000096); between them are -3; 5, -6; -11, 10, -12; which appear (opposite sign) by rows in A140503 (1, -1, 2, 3, -2, 4) square.
Consider the permutation of the nonnegative numbers
0, 2, 5, 9, 14, 20, 27,
1, 3, 6, 10, 15, 21, 28,
4, 7, 11, 16, 22, 29,
8, 12, 17, 23, 30,
13, 18, 24, 31,
19, 25, 32,
26, 33,
34, etc.
The corresponding distribution of a(n) is
1, -1, 3, -5, 11, -21, 43,
-3, 5, -11, 21, -43, 85, -171,
-6, 10, -22, 42, -86, 170,
-12, 20, -44, 84, -172,
-24, 40, -88, 168,
-48, 80, -176,
-96, 160,
-192, etc.
Column sums: -2, -2, -10, -10, -42, -42, -170, ... duplicate of a bisection of -A078008(n+2).
b(n)= 1, -1, 3, -5, 11, 21, ... = (-1)^n*A001045(n+1) = A077925(n). Every row is b(n) or b(n+2) multiplied by 1, -1, -2, -4, -8, -16, ..., essentially -A011782(n).

Crossrefs

Cf. A000096, A000217, A005015, A001045, A007283, A020988, A077925, A078008, A140503, A146523, A151575, A175805, A084247.

Programs

  • Mathematica
    T[0, 0] = 0; T[1, 0] = T[0, 1] = 1; T[0, n_] := T[0, n] = T[0, n - 1] + 2*T[0, n - 2]; T[d_, d_] = 0; T[d_, n_] := T[d, n] = T[d - 1, n + 1] - T[d - 1, n]; A140944 = Table[T[d, n], {d, 0, 10}, {n, 0, d}] // Flatten; a[n_] := A140944[[n + 2]] - 3*A140944[[n + 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 18 2014 *)

Extensions

More terms and a(19)=-48 instead of 42 corrected by Jean-François Alcover, Dec 22 2014

A140951 Based on Jacobsthal numbers. Increasing order of different positive terms of A140950.

Original entry on oeis.org

1, 3, 5, 6, 10, 11, 12, 20, 21, 22, 24, 40, 42, 43, 44, 48, 80, 84, 85, 86, 88, 96, 160, 168, 170, 171, 172, 176, 192, 320, 336, 340, 341, 342, 344, 352, 384, 640
Offset: 0

Views

Author

Paul Curtz, Jul 25 2008

Keywords

Comments

Two possibilities of triangle on line. 1) From 1: 1; 3, 5; 6, 10, 11; 12, 20, 21, 22; 24, 40, 42, 43, 44; . 2) After 1: 3; 5, 6; 10, 11, 12; 20, 21, 22, 24; .

Formula

Also A140642 (1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 16, 20) without A000079(n+1). Note position of A001045(n+2) terms: 0, 1, 2, 5, 8, 13 =A000982. See A140503 square .
Showing 1-3 of 3 results.

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