A084175 -id:A084175 - cp's OEIS frontend

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

Paul Curtz, Jul 24 2008

A variant of the triangle A140503, now including the diagonal.

Since the diagonal contains zeros, rows sums are those of A140503.

			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;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.

Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

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

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

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 *)

T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022

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

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)

Edited and extended by R. J. Mathar, Sep 07 2009

A084177 Binomial transform of Jacobsthal oblongs.

Original entry on oeis.org

0, 1, 5, 27, 137, 691, 3465, 17347, 86777, 433971, 2170025, 10850467, 54253017, 271266451, 1356334985, 6781680387, 33908412857, 169542086131, 847710474345, 4238552459107, 21192762470297, 105963812701011, 529819064204105

Offset: 0

Paul Barry, May 18 2003

Binomial transform of A084175.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-3,-10).

Cf. A001045.

[(2*5^n-(-1)^n-2^n)/9: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

a(n) = (2*5^n - (-1)^n - 2^n)/9.
G.f.: x*(1-x)/((1+x)*(1-2*x)*(1-5*x)).
a(n) = 6*a(n-1) - 3*a(n-2) - 10*a(n-3).
E.g.f.: (2*exp(5*x) - exp(2*x) - exp(-x))/9.

A128210 Number triangle T(n,k) = (-1)^(n-k)[k<=n]Product_{i=k+1..n} Sum_{j=0..i-1} A078008(j-1).

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -3, 3, -3, 1, 15, -15, 15, -5, 1, -165, 165, -165, 55, -11, 1, 3465, -3465, 3465, -1155, 231, -21, 1, -148995, 148995, -148995, 49665, -9933, 903, -43, 1, 12664575, -12664575, 12664575

Offset: 0

Paul Barry, Feb 19 2007

Inverse is A128208. Subdiagonals include -A001045(n+1) and A084175(n+1).

			Triangle begins:
        1;
       -1,      1;
        1,     -1,       1;
       -3,      3,      -3,     1;
       15,    -15,      15,    -5,     1;
     -165,    165,    -165,    55,   -11,   1;
     3465,  -3465,    3465, -1155,   231, -21,   1;
  -148995, 148995, -148995, 49665, -9933, 903, -43, 1;
  ...

A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

Original entry on oeis.org

1, 10, 220, 3080, 52976, 818720, 13333440, 211474560, 3398520576, 54257082880, 869067996160, 13897453373440, 222420341682176, 3558236809994240, 56935698394234880, 910939899548958720, 14575288593717067776, 233202615903456460800

Offset: 0

Ralf Steiner, May 16 2020

Matrix {{2, 0}, {1, -1}} is [g_{-2}] given by Firstov in eq. (24).
These primitive Pythagorean triples are also given by Lee Price as (M_2)^n (3,4,5)^T (T for transposed), with M_2 = {{2, 1, 1}, {2, -2, 2}, {2, -1, 3}}.
For a primitive Pythagorean triangle (x, y, z) = (u^2-v^2, 2*u*v, u^2+v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction. Here:
x(n) = A084175(n+2).
y(n) = 4*(A084175(n+1) - A084175(n)) = A054881(n+2).
     = 2*A192382(n+1) = 4*A003683(n+1).
z(n) = A084175(n+2) + 2*A084175(n+1) - 4*A084175(n).
     = A108924(n+2)/2 = A084175(n+2) + 2*A139818(n+1).
     = A000302(n+1) + A139818(n+1).
u(n) = A000079(n+1) = 2^(n+1).
v(n) = A001045(n+1) = (2^(n+1) + (-1)^n)/3.
For the area A(n): Limit_{n -> oo} (3^3/(2^(4*n+7)))*A(n) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

			a(0) = 3*4/12 = 1 for the triangle (3, 4, 5).

G. C. Greubel, Table of n, a(n) for n = 0..825
V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
H. Lee Price, The Pythagorean Tree: A New Species, arXiv:0809.4324 [math.HO], 2008-2011
R. Steiner, Spezielle Folge primitiver pythagoräischer Dreiecke, researchgate.net, 2020
Index entries for linear recurrences with constant coefficients, signature (10,120,-320,-1024).

Cf. A000079, A000302, A001045, A003683, A054881, A084175, A108924, A139818, A192382.

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81: n in [0..40]]; // G. C. Greubel, Feb 18 2023

Table[(2^(2*n+1)*(2^(2*n+5) -3) + (-2)^n*(3*2^(2*n+3) -1))/3^4, {n,0,40}]

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81 for n in range(41)] # G. C. Greubel, Feb 18 2023

a(n) = ( 2^(4*n+6) - 3*2^(2*n+1) - 3*(-2)^(3*n+3) - (-2)^n )/3^4.
G.f.: 1 / ((1 + 2*x)*(1 - 4*x)*(1 + 8*x)*(1 - 16*x)). - Colin Barker, Jun 11 2020
E.g.f.: (1/81)*(24*exp(-8*x) - exp(-2*x) - 6*exp(4*x) + 64*exp(16*x)). - G. C. Greubel, Feb 18 2023

A374098 a(n) = A112387(n)^2.

Original entry on oeis.org

1, 1, 4, 1, 16, 9, 64, 25, 256, 121, 1024, 441, 4096, 1849, 16384, 7225, 65536, 29241, 262144, 116281, 1048576, 466489, 4194304, 1863225, 16777216, 7458361, 67108864, 29822521, 268435456, 119311929, 1073741824, 477204025, 4294967296, 1908903481, 17179869184

Offset: 0

Paul Curtz, Jun 28 2024

Index entries for linear recurrences with constant coefficients, signature (0,3,0,6,0,-8).

Cf. A000302, A003683, A084175, A112387, A139818.

LinearRecurrence[{0, 3, 0, 6, 0, -8}, {1, 1, 4, 1, 16, 9}, 35] (* Amiram Eldar, Jul 01 2024 *)

a(2*n) = A000302(n); a(2*n+1) = A139818(n+1).
(a(2*n) + a(2*n-1))^2 = A084175(n+1)^2 + 16*A003683(n)^2, for n >= 1. - Thomas Scheuerle, Jun 28 2024
G.f. ( 1+x+x^2-2*x^3-2*x^4 ) / ( (x-1)*(2*x+1)*(2*x-1)*(1+x)*(2*x^2+1) ). - R. J. Mathar, Aug 02 2024

A378676 a(n) = J(n) * J(n+2) where J(n) = Jacobsthal(n) = A001045(n).

Original entry on oeis.org

0, 3, 5, 33, 105, 473, 1785, 7353, 28985, 116793, 465465, 1865273, 7454265, 29830713, 119295545, 477236793, 1908837945, 7635570233, 30541844025, 122168249913, 488671252025, 1954688503353, 7818747022905, 31275002072633, 125099980328505, 500399977238073, 2001599797104185, 8006399412112953, 32025597201059385

Offset: 0

Werner Schulte, Dec 03 2024

Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Cf. A001045, A084175.

a[n_] := (4^(n+1) - 5*(-2)^n + 1)/9; Array[a, 30, 0] (* Amiram Eldar, Dec 06 2024 *)

PARI
```
a(n)=(4^(n+1)-5*(-2)^n+1)/9
```

a(n) = (2^n - (-1)^n) * (2^(n+2) - (-1)^n) / 9 = (4 * 4^n - 5 * (-2)^n + 1) / 9.
G.f.: x * (3 - 4*x) / ((1-x) * (1+2*x) * (1-4*x)).
a(n) = 3 * a(n-1) + 6 * a(n-2) - 8 * a(n-3) for n > 2 with initial values a(0) = 0, a(1) = 3, and a(2) = 5.
Sum_{k=1..n-1} 2^(k-1) / a(k) = 1 - 2^(n-1) / A084175(n) for n > 0.
Sum_{k>0} 2^(k-1) / a(k) = 1.
E.g.f.: exp(x)*(1 - cosh(3*x) + 9*sinh(3*x))/9. - Stefano Spezia, Dec 06 2024

A378931 Triangle read by rows, based on products of Jacobsthal numbers (A001045).

Original entry on oeis.org

1, -1, 3, -2, -9, 15, -4, -18, -25, 55, -8, -36, -50, -121, 231, -16, -72, -100, -242, -441, 903, -32, -144, -200, -484, -882, -1849, 3655, -64, -288, -400, -968, -1764, -3698, -7225, 14535, -128, -576, -800, -1936, -3528, -7396, -14450, -29241, 58311, -256, -1152, -1600, -3872, -7056, -14792, -28900, -58482, -116281, 232903

Offset: 1

Werner Schulte, Dec 11 2024

Let M = T^(-1) be matrix inverse of T seen as a lower triangular matrix. M is a harmonic triangle with M(n, k) = 1 / A084175(n) if k = n, and 1 / A378676(k) if 1 <= k < n. Triangle M(n, k) for 1 <= k <= n starts:
  1/1
  1/3  1/3
  1/3  1/5  1/15
  1/3  1/5  1/33   1/55
  1/3  1/5  1/33  1/105  1/231
  1/3  1/5  1/33  1/105  1/473  1/903
  etc.
  Sum_{k=1..n} M(n, k) * 2^(k-1) = 1.
  Sum_{k=1..n} M(n, k) * (-2)^(k-1) = (-1)^(n-1) / A001045(n+1).
  Sum_{k=1..n} 2^(k-1) / M(n, k) = (8^n - 1) / 7 = A023001(n).

			Triangle T(n, k) for 1 <= k <= n starts:
n\k :     1     2     3      4      5      6       7       8      9
===================================================================
  1 :     1
  2 :    -1     3
  3 :    -2    -9    15
  4 :    -4   -18   -25     55
  5 :    -8   -36   -50   -121    231
  6 :   -16   -72  -100   -242   -441    903
  7 :   -32  -144  -200   -484   -882  -1849    3655
  8 :   -64  -288  -400   -968  -1764  -3698   -7225   14535
  9 :  -128  -576  -800  -1936  -3528  -7396  -14450  -29241  58311
  etc.

Cf. A001045, A023001, A378676.

A084175 (main diagonal), A139818 (1st subdiagonal), A000079 (column 1 and row sums).

T[n_,k_]:=If[k==n, (2*4^n-(-2)^n-1)/9, -2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9]; Table[T[n,k],{n,10},{k,n}]//Flatten (* Stefano Spezia, Dec 11 2024 *)

T(n,k)=if(k==n,(2*4^n-(-2)^n-1)/9,-2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9)

T(n, n) = (2 * 4^n - (-2)^n - 1) / 9 = A084175(n), and T(n, k) = -2^(n-1-k) * (2^(k+1) + (-1)^k)^2 / 9 for 1 <= k < n.

G.f.: x*t * (1 - 3*t - 6*x*t^2 + 8*x^2*t^3) / ((1 - 2*t) * (1 - x*t) * (1 + 2*x*t) * (1 - 4*x*t)).

cp's OEIS Frontend

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

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A084177 Binomial transform of Jacobsthal oblongs.

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A128210 Number triangle T(n,k) = (-1)^(n-k)*[k<=n]*Product_{i=k+1..n} Sum_{j=0..i-1} A078008(j-1).

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A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

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A374098 a(n) = A112387(n)^2.

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A378676 a(n) = J(n) * J(n+2) where J(n) = Jacobsthal(n) = A001045(n).

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A378931 Triangle read by rows, based on products of Jacobsthal numbers (A001045).

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A128210 Number triangle T(n,k) = (-1)^(n-k)[k<=n]Product_{i=k+1..n} Sum_{j=0..i-1} A078008(j-1).