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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A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 5, 8, 12, 18, 27, 11, 16, 24, 36, 54, 81, 21, 32, 48, 72, 108, 162, 243, 43, 64, 96, 144, 216, 324, 486, 729, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 171, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 341, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683
Offset: 0

Views

Author

Paul Curtz, Jan 20 2009

Keywords

Comments

Deleting column k=0 and reading by antidiagonals yields A036561.
Deleting column k=0 and reading the antidiagonals downwards yields A175840.

Examples

			The array starts in row n=0 with columns k>=0 as:
   0   1    3    9    27    81    243    729    2187  ... A140429;
   1   2    6   18    54   162    486   1458    4374  ... A025192;
   1   4   12   36   108   324    972   2916    8748  ... A003946;
   3   8   24   72   216   648   1944   5832   17496  ... A080923;
   5  16   48  144   432  1296   3888  11664   34992  ... A257970;
  11  32   96  288   864  2592   7776  23328   69984  ...
  21  64  192  576  1728  5184  15552  46656  139968  ...
Antidiagonal triangle begins as:
   0;
   1,   1;
   1,   2,   3;
   3,   4,   6,   9;
   5,   8,  12,  18,  27;
  11,  16,  24,  36,  54,  81;
  21,  32,  48,  72, 108, 162, 243;
  43,  64,  96, 144, 216, 324, 486, 729;
  85, 128, 192, 288, 432, 648, 972, 1458, 2187; - _G. C. Greubel_, Mar 25 2021

Links

Crossrefs

Cf. A001045, A004054, A046936.

Programs

  • Magma
    t:= func< n,k | k eq 0 select (2^(n-k) -(-1)^(n-k))/3 else 2^(n-k)*3^(k-1) >;
[t(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 25 2021
  • Maple
    T:=proc(n,k)if(k>0)then return 2^n*3^(k-1):else return (2^n - (-1)^n)/3:fi:end:
for d from 0 to 8 do for m from 0 to d do print(T(d-m,m)):od:od: # Nathaniel Johnston, Apr 13 2011
  • Mathematica
    t[n_, k_]:= If[k==0, (2^(n-k) -(-1)^(n-k))/3, 2^(n-k)*3^(k-1)];
Table[t[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 25 2021 *)
  • Sage
    def A155118(n,k): return (2^(n-k) -(-1)^(n-k))/3 if k==0 else 2^(n-k)*3^(k-1)
flatten([[A155118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 25 2021

Formula

For the square array:
T(n,k) = 2^n*3^(k-1), k>0.
T(n,k) = T(n-1,k+1) - T(n-1,k), n>0.
Rows:
T(0,k) = A140429(k) = A000244(k-1).
T(1,k) = A025192(k).
T(2,k) = A003946(k).
T(3,k) = A080923(k+1).
T(4,k) = A257970(k+3).
Columns:
T(n,0) = A001045(n) (Jacobsthal numbers J_{n}).
T(n,1) = A000079(n).
T(n,2) = A007283(n).
T(n,3) = A005010(n).
T(n,4) = A175806(n).
T(0,k) - T(k+1,0) = 4*A094705(k-2).
From G. C. Greubel, Mar 25 2021: (Start)
For the antidiagonal triangle:
t(n, k) = T(n-k, k).
t(n, k) = (2^(n-k) - (-1)^(n-k))/3 (J_{n-k}) if k = 0 else 2^(n-k)*3^(k-1).
Sum_{k=0..n} t(n, k) = 3^n - J_{n+1}, where J_{n} = A001045(n).
Sum_{k=0..n} t(n, k) = A004054(n-1) for n >= 1. (End)

Extensions

a(22) - a(57) from Nathaniel Johnston, Apr 13 2011

A322116 Main diagonal of triangle A321600; a(n) = A321600(n,n-1) for n >= 1.

Original entry on oeis.org

2, 6, 26, 78, 242, 726, 2186, 6558, 19682, 59046, 177146, 531438, 1594322, 4782966, 14348906, 43046718, 129140162, 387420486, 1162261466, 3486784398, 10460353202, 31381059606, 94143178826, 282429536478, 847288609442, 2541865828326, 7625597484986, 22876792454958, 68630377364882, 205891132094646, 617673396283946, 1853020188851838, 5559060566555522, 16677181699666566
Offset: 1

Views

Author

Paul D. Hanna, Nov 26 2018

Keywords

Comments

Triangle A321600 describes log( (1-y)*Sum_{n=-oo...+oo} (x^n + y)^n )/(1-y).

Examples

			G.f.: A(x) = 2*x + 6*x^2 + 26*x^3 + 78*x^4 + 242*x^5 + 726*x^6 + 2186*x^7 + 6558*x^8 + 19682*x^9 + 59046*x^10 + ...
L.g.f.: L(x)  =  log( (1-x)*(1-x^2)/(1-3*x) )  =  2*x + 6*x^2/2 + 26*x^3/3 + 78*x^4/4 + 242*x^5/5 + 726*x^6/6 + 2186*x^7/7 + 6558*x^8/8 + 19682*x^9/9 + 59046*x^10/10 + 177146*x^11/11 + ... + A321600(n,n-1)*x^n/n + ...
such that
exp(L(x)) = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 144*x^5 + 432*x^6 + 1296*x^7 + 3888*x^8 + 11664*x^9 + 34992*x^10 + 104976*x^11 + ... + A257970(n)*x^n + ...
exp(L(x)/2) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 44*x^5 + 122*x^6 + 342*x^7 + 966*x^8 + 2746*x^9 + 7846*x^10 + 22514*x^11 + 64836*x^12 + ... + A105696(n)*x^n + ...

Links

Crossrefs

Cf. A321600, A257970, A105696.

Programs

  • PARI
    {a(n) = n*polcoeff( log((1 - x)*(1 - x^2)/(1 - 3*x +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))

Formula

L.g.f.: log( (1 - x)*(1 - x^2)/(1 - 3*x) ).
G.f.: 2*x*(1 + 3*x^2)/((1 - x^2)*(1 - 3*x)).

A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

Original entry on oeis.org

1, 3, 4, 9, 12, 16, 27, 36, 48, 64, 81, 108, 144, 192, 256, 243, 324, 432, 576, 768, 1024, 729, 972, 1296, 1728, 2304, 3072, 4096, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 6561, 8748, 11664, 15552, 20736, 27648, 36864, 49152, 65536, 19683, 26244, 34992, 46656, 62208, 82944, 110592, 147456, 196608, 262144
Offset: 0

Views

Author

Wolfdieter Lang, Aug 01 2022

Keywords

Comments

This is Boethius's triangle, with rows read as columns. See the link and reference.

Examples

			The triangle T begins:
n\k     0     1      2      3      4      5      6      7      8      9  ...
0:      1
1:      3     4
2:      9    12     16
3:     27    36     48     64
4:     81   108    144    192    256
5:    243   324    432    576    768   1024
6:    729   972   1296   1728   2304   3072   4096
7:   2187  2916   3888   5184   6912   9216  12288  16384
8:   6561  8748  11664  15552  20736  27648  36864  49152  65536
9:  19683 26244  34992  46656  62208  82944 110592 147456 196608 262144
...

References

  • Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.

Links

Crossrefs

Columns: A000244, A003946, A257970, ...
Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...
Row sums: A005061(n+1).

Programs

  • Mathematica
    T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 05 2022 *)

Formula

T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.
G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).
Showing 1-3 of 3 results.

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