A025192 -id:A025192 - cp's OEIS frontend

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

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

Paul Curtz, Jan 20 2009

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

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

			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

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A001045, A004054, A046936.

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

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

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 *)

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

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)

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

A171501 Inverse binomial transform of A084640.

Original entry on oeis.org

0, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881

Offset: 0

Paul Curtz, Dec 10 2009

a(n) and differences are
0,     1,   3,  -1,    7,   -9,
1,     2,  -4,   8,  -16,   32,  =(-1)^(n+1) * A171449(n),
1,    -6,  12, -24,   48,  -96,
-7,   18, -36,  72, -144,  288,
25,  -54, 108, -216, 432, -864,
Vertical: 1) 0 followed with A168589(n).
          2) (-1 followed with A008776(n) ) signed. See A025192(n).
Main diagonal: see A167747(1+n). - Paul Curtz, Jun 16 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2).

I:=[0, 1, 3]; [n le 3 select I[n] else -Self(n-1) + 2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 18 2012

CoefficientList[Series[x*(1 + 4*x)/((1 + 2*x)*(1 - x)), {x, 0, 30}], x]
LinearRecurrence[{-1,2},{0,1,3},40] (* Harvey P. Dale, Jan 14 2020 *)

a(n) = A140966(n), n>0.
G.f.: x*(1+4*x) / ( (1+2*x)*(1-x) ). - R. J. Mathar, Jun 14 2011
a(1+n)= (-1)^(1+n) * A001045(1+n) + 2. - Paul Curtz, Jun 16 2011

Mathematica program by Olivier Gérard, Jul 06 2011

A202209 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 13, 5, 0, 0, 34, 19, 1, 0, 0, 89, 65, 8, 0, 0, 0, 233, 210, 42, 1, 0, 0, 0, 610, 654, 183, 11, 0, 0, 0, 0, 1597, 1985, 717, 74, 1, 0, 0, 0, 0, 4181, 5911, 2622, 394, 14, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 14 2011

Riordan array ((1-x)/(1-3x+x^2), x^2/(1-3x+x^2)) .

			Triangle begins :
1
2, 0
5, 1, 0
13, 5, 0, 0
34, 19, 1, 0, 0
89, 65, 8, 0, 0, 0
233, 210, 42, 1, 0, 0, 0

Cf. A000045, A000079, A001519, A001870, A001906, A126124, A202207 (antidiagonal sums)

T(n,k) = 3*T(n-1,k) - T(n-2,k) + T(n-2,k-1).
G.f.: (1-x)/(1-3x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively.
T(n,0) = A122367(n) = A000045(2n+1).

A234713 Triangle, read by rows, based on the Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 1, 3, 2, 2, 6, 7, 3, 3, 13, 20, 14, 4, 5, 25, 51, 51, 25, 5, 8, 48, 118, 154, 111, 41, 6, 13, 89, 260, 416, 393, 217, 63, 7, 21, 163, 548, 1042, 1218, 890, 392, 92, 8, 34, 294, 1121, 2465, 3435, 3127, 1842, 666, 129, 9, 55, 525, 2236, 5586, 9035, 9845

Offset: 0

Philippe Deléham, Dec 29 2013

First column is the Fibonacci sequence.

Sum_{k=0..n} T(n,k)*2^k = -A106732(n).

			Triangle begins:
0
1, 1
1, 3, 2
2, 6, 7, 3
3, 13, 20, 14, 4
5, 25, 51, 51, 25, 5
8, 48, 118, 154, 111, 41, 6
13, 89, 260, 416, 393, 217, 63, 7
21, 163, 548, 1042, 1218, 890, 392, 92, 8

Cf. Diagonals: A001477, A004006.
Cf. Columns: A000045 (Fibonacci), A131913, A261054.
Cf. A025192 (row sums for n>0), A006054 (diagonal sums)

G.f.: (y+1)*x/(1-(2y+1)*x+(y^2-1)*x^2).

T(n,k)=T(n-1,k)+2*T(n-1,k-1)+T(n-2,k)-T(n-2,k-2), T(0,0)=0, T(1,0)=1, T(1,1)=1, T(n,k)=0 if k<0 or if k>n.

A238941 Triangle T(n,k), read by rows given by (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 8, 4, 1, 13, 21, 13, 6, 1, 34, 55, 40, 25, 7, 1, 89, 144, 120, 90, 33, 9, 1, 233, 377, 354, 300, 132, 51, 10, 1, 610, 987, 1031, 954, 483, 234, 62, 12, 1, 1597, 2584, 2972, 2939, 1671, 951, 308, 86, 13, 1, 4181, 6765, 8495, 8850, 5561, 3573, 1345, 480, 100, 15, 1

Offset: 0

Philippe Deléham, Mar 07 2014

Row sums are A025192(n).

			Triangle begins:
1;
1, 1;
2, 3, 1;
5, 8, 4, 1;
13, 21, 13, 6, 1;
34, 55, 40, 25, 7, 1;
89, 144, 120, 90, 33, 9, 1;
233, 377, 354, 300, 132, 51, 10, 1;

Indranil Ghosh, Rows 0..100, flattened

Cf. Columns: A001519, A001906, A238846, A001871.

Cf. Diagonals: A000012, A032766.

nmax=10; Column[CoefficientList[Series[CoefficientList[Series[(1 - 2*x + x*y)/(1 - 3*x + x^2 - x^2*y^2), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 14 2017 *)

G.f. for the column k: x^k*(1-2*x)^A059841(k)/(1-3*x+x^2)^A008619(k).
G.f.: (1-2*x+x*y)/(1-3*x+x^2-x^2*y^2).
T(n,k) = 3*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000007(n), A001519(n), A025192(n), A030195(n+1) for x = -1, 0, 1, 2 respectively.
Sum_{k = 0..n} T(n,k)*3^k = A015525(n) + A015525(n+1).

Data section corrected and extended by Indranil Ghosh, Mar 14 2017

A291931 Primitive elements of A290002.

Original entry on oeis.org

1, 10, 18, 54, 70, 78, 110, 162, 174, 198, 222, 230, 234, 246, 294, 414, 426, 438, 450, 470, 486, 534, 594, 666, 702, 770, 846, 858, 882, 910, 1070, 1158, 1218, 1242, 1314, 1350, 1458, 1610, 1722, 1782, 1794, 1866, 1914, 1926, 1938, 1950, 1998, 2058, 2106, 2250, 2442, 2530, 2538, 2574, 2590, 2646, 2886

Offset: 1

Robert Israel, Sep 06 2017

nonn

Members k of A290002 such that k/2 is not in A290002.

Includes all members of A025192 except 2 and 6.

			a(3) = 18 is in the sequence because psi(phi(18)) = phi(psi(18)) = 12 but psi(phi(9)) = 12 <> 4 = phi(psi(9)).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A025192, A290002.

psi:= proc(n)  n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
A290002:= select(psi @ numtheory:-phi = numtheory:-phi @ psi, {$1..3000}):
sort(convert(A290002 minus map(`*`,A290002,2), list));

f[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]; With[{s = Select[Range[3000], f[EulerPhi@ #] == EulerPhi[f@ #] &]}, Select[s, FreeQ[s, #/2] &]] (* Michael De Vlieger, Sep 06 2017 *)

A380905 Smallest number k such that k^(2*3^n) - 6 is prime.

Original entry on oeis.org

3, 5, 23, 7, 433, 2447, 9377, 82597, 134687

Offset: 0

Jakub Buczak, Feb 07 2025

Terms must have an ending digit of 3, 5 or 7. If k ends in 1 or 9, then k^(2*3^n)-6 ends in a 5, which is not prime.

a(7) is the first composite term. - Michael S. Branicky, Feb 24 2025

			For n=0, k^(2*3^0) - 6 is prime for the first time at a(0) = k = 3.
For n=5, k^(2*3^5) - 6 is prime for the first time at a(5) = k = 2447.

Cf. Subsequence of A382246.
Cf. A008776, A025192.
Cf. A028879 (a(0)), A239414 (a(1)) for the first term.

a(n) = my(p=3,q=2*3^n); while (!ispseudoprime(p^q-6), p+=2); p; \\ Michel Marcus, Feb 08 2025

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(2) if k%10 in {3,5,7} and isprime(k**(2*3**n)-6))

a(7) from Michael S. Branicky, Feb 24 2025

a(8) from Georg Grasegger, Apr 17 2025

A134318 A007318 * A134317.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 6, 4, 8, 8, 14, 16, 8, 16, 16, 30, 44, 40, 16, 32, 32, 62, 104, 128, 96, 32, 64, 64, 126, 228, 336, 352, 224, 64, 128, 128, 254, 480, 792, 1024, 928, 512, 128, 256, 256, 510, 988, 1752, 2608, 2976, 2368, 1152, 256

Offset: 0

Gary W. Adamson, Oct 19 2007

Row sums = A025192: (1, 2, 6, 18, 54, 162, ...).

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  2;
   4,  4,  6,   4;
   8,  8, 14,  16,   8;
  16, 16, 30,  44,  40, 16;
  32, 32, 62, 104, 128, 96, 32;
  ...

Cf. A134317, A025192.

Binomial transform of A134317, as infinite lower triangular matrices.

A140320 a(n) = A137576((3^n-1)/2).

Original entry on oeis.org

1, 3, 13, 55, 217, 811, 2917, 10207, 34993, 118099, 393661, 1299079, 4251529, 13817467, 44641045, 143489071, 459165025, 1463588515, 4649045869, 14721978583, 46490458681, 146444944843, 460255540933, 1443528742015, 4518872583697, 14121476824051, 44059007691037, 137260754729767

Offset: 0

Vladimir Shevelev, May 26 2008

nonn

Conjecture. a(n) = 2n*3^(n-1)+1.
If conjecture is true then limsup(A137576(n)/n)=infinity while liminf(A137576(n)/n)=2 with a realization on primes.
a(n) is also the number of edges in the graph generated from the n-dimensional hypercube (plus 1) in the following manner: connect all (d + 1)-dimensional faces to the d faces that are incident. Each d-dimensional face should be incident on (n - d) (d + 1)-dimensional faces. [Roy Liu (royliu(AT)cs.ucsd.edu), Jul 26 2010]

Wikipedia, Hypercube

Cf. A037576, A002326, A006694, A025192.

a137576(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1;
a(n) = a137576((3^n-1)/2); \\ Michel Marcus, Dec 18 2018

Sum_{m = 0}^{n} 2^(n - m) * binomial(n,m) is the number of m-dimensional faces in the n-dimensional hypercube. Consequently, Sum_{m = 0..n} (n - m) * 2^(n - m) * binomial(n,m) gives the number of incidence edges, which yields said sequence minus 1. The recurrence relation is: a(n) = 3 * a(n - 1) + 2 * 3^(n - 1) - 2. [Roy Liu (royliu(AT)cs.ucsd.edu), Jul 26 2010]

Empirical G.f.: (1-4*x+7*x^2)/(1-7*x+15*x^2-9*x^3). [Colin Barker, Jan 09 2012]

More terms from Michel Marcus, Dec 18 2018

A153310 Coefficient triangle sequence of a polynomial recursion: p(x,n)=(x + 1)(p(x, n - 1) + 3^(n - 1)x); Row sums are 2*3^n.

Original entry on oeis.org

2, 3, 3, 2, 14, 2, 2, 25, 25, 2, 2, 54, 77, 27, 2, 2, 137, 212, 104, 29, 2, 2, 382, 592, 316, 133, 31, 2, 2, 1113, 1703, 908, 449, 164, 33, 2, 2, 3302, 5003, 2611, 1357, 613, 197, 35, 2, 2, 9865, 14866, 7614, 3968, 1970, 810, 232, 37, 2, 2, 29550, 44414, 22480, 11582

Offset: 0

Roger L. Bagula, Dec 23 2008

Row sums:

{2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098,...}.

			{2},
{3, 3},
{2, 14, 2},
{2, 25, 25, 2},
{2, 54, 77, 27, 2},
{2, 137, 212, 104, 29, 2},
{2, 382, 592, 316, 133, 31, 2},
{2, 1113, 1703, 908, 449, 164, 33, 2},
{2, 3302, 5003, 2611, 1357, 613, 197, 35, 2},
{2, 9865, 14866, 7614, 3968, 1970, 810, 232, 37, 2},
{2, 29550, 44414, 22480, 11582, 5938, 2780, 1042, 269, 39, 2}

A025192

Clear[p, n, m, x];
p[x, 0] = 2; p[x, 1] = 3*x + 3; p[x, 2] = 2*x^2 + 14*x + 2;
p[x_, n_] := p[x, n] = (x + 1)*(p[x, n - 1] + 3^(n - 1)*x);
Table[ExpandAll[p[x, n]], {n, 0, 10}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

p(x,n)=(x + 1)*(p(x, n - 1) + 3^(n - 1)*x).

cp's OEIS Frontend

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

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A171501 Inverse binomial transform of A084640.

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A202209 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A234713 Triangle, read by rows, based on the Fibonacci numbers.

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A238941 Triangle T(n,k), read by rows given by (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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Mathematica

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A291931 Primitive elements of A290002.

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Maple

Mathematica

A380905 Smallest number k such that k^(2*3^n) - 6 is prime.

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A134318 A007318 * A134317.

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A153310 Coefficient triangle sequence of a polynomial recursion: p(x,n)=(x + 1)(p(x, n - 1) + 3^(n - 1)x); Row sums are 2*3^n.