A165148 -id:A165148 - cp's OEIS frontend

A165149 a(n) = (3*9^n - 5^n)/2.

1, 11, 109, 1031, 9529, 87011, 789349, 7135391, 64374769, 580154171, 5225293789, 47047175351, 423522234409, 3812188390931, 34312136924629, 308821439352911, 2779453989332449, 25015391079773291, 225140045596865869, 2026268039766324071, 18236450504869572889, 164128245278689437251

Offset: 0

Klaus Brockhaus, Sep 15 2009

Binomial transform of A165148. Inverse binomial transform of A165150.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-45).

Cf. A165148, A165150.

Magma
```
[ (3*9^n-5^n)/2: n in [0..18] ];
```

A165149:=n->(3*9^n-5^n)/2; seq(A165149(n), n=0..30); # Wesley Ivan Hurt, Dec 10 2013

Table[(3*9^n - 5^n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Dec 10 2013 *)

a(n) = 14*a(n-1) - 45*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
G.f.: (1 - 3*x)/((1 - 5*x)*(1 - 9*x)).
E.g.f.: exp(5*x)*(3*exp(4*x) - 1)/2. - Stefano Spezia, Mar 21 2023

A165147 a(n) = (3*7^n-3^n)/2.

Original entry on oeis.org

1, 9, 69, 501, 3561, 25089, 176109, 1234221, 8643921, 60520569, 423683349, 2965901541, 20761665081, 145332718449, 1017332217789, 7121335090461, 49849374331041, 348945706410729, 2442620203155429, 17098342196928981

Offset: 0

Klaus Brockhaus, Sep 15 2009

Partial sums are in A016138.

Binomial transform of A016129. Inverse binomial transform of A165148.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Cf. A016138, A016129, A165148.

Magma
```
[ (3*7^n-3^n)/2: n in [0..19] ];
```

LinearRecurrence[{10, -21}, {1, 9}, 25] (* Paolo Xausa, Apr 22 2024 *)
Table[(3*7^n-3^n)/2,{n,0,20}] (* Harvey P. Dale, Aug 05 2025 *)

a(n) = 10*a(n-1)-21*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

G.f.: (1-x)/((1-3*x)*(1-7*x)).

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

Original entry on oeis.org

1, 10, 6, 33, 24, 24, 88, 48, 72, 48, 145, 120, 120, 120, 120, 330, 144, 240, 198, 240, 144, 385, 336, 336, 336, 336, 336, 336, 736, 384, 576, 384, 672, 384, 576, 384, 945, 648, 648, 864, 648, 648, 864, 648, 648, 1450, 720, 1200, 720, 1200, 870, 1200, 720, 1200, 720

Offset: 1

Erdos Pal, Oct 03 2010

The n-th row is {T(n,0),T(n,1),...,T(n,n-1)}.
Let m denote the prime power p^e.
T(m,0) = A020478(m) = (p^(e+1) + p^e-1)*p^(2*e-1).
T(m,1) = A000056(m) = (p^2-1)*p^(3*e-2).
T(prime(n),1) = A127917(n).
Sum_{k=1..n-1} T(n,k) = A005353(n).
T(n,1) = n*A007434(n) for n>=1 because A000056(n) = n*Jordan_Function_J_2(n).
T(2^n,1) = A083233(n) = A164640(2n) for n>=1. Proof: a(n):=T(2^n,1); a(1)=6, a(n)=8*a(n-1); A083233(1)=6 and A083233(n) is a geometric series with ratio 8 (because of its g.f.), too; A164640 = {b(1)=1, b(2)=6, b(n)=8*b(n-2)}.
T(2^n,0) = A165148(n) for n>=0, because  2*T(2^n,0) = (3*2^n-1)*4^n.
T(2^e,2) = A003951(e) for 2 <= e. Proof: T(2^e,2) = 9*8^(e-1) is a series with ratio 8 and initial term 72, as A003951(2...inf) is.
Working with consecutive powers of a prime p, we need a definition (0 <= i < e):
N(p^e,i):=#{k: 0 < k < p^e, gcd(k,p^e) = p^i} = (p-1)*p^(e-1-i). We say that these k's belong to i (respect to p^e). Note that N(p^e,0) = EulerPhi(p^e), and if 0 < k < p^e then gcd(k,p^e) = gcd(k,p^(e+1)). Let T(p^e,[i]) denote the common value of T(p^e,k)'s, where k's belong to i (q.v.PROGRAM); for example, T(p^e,[0]) = T(p^e,1). The number of the 2 X 2 matrices over Z(p^e), T(p^e,0) + Sum_{i=0..e-1} T(p^e,[i])*N(p^e,i) = p^(4e) will be useful.
On the hexagon property: Let prime p be given and let T(p^e,[0]), T(p^e,[1]), T(p^e,[2]), ..., T(p^e,[e-2]), T(p^e,[e-1]) form the e-th row of a Pascal-like triangle, e>=1. Let denote X(r,s) an element of the triangle and its value T(p^r,[s]). Let positive integers a and b given, so that the entries A(m-a,n-b), B(m-a,n), C(m,n+a), D(m+b,n+a), E(m+b,n), F(m,n-b) of the triangle form a hexagon spaced around T(p^m,[n]); if a=b=1 then they surround it. If A*C*E = B*D*F, then we say that the triangle T(.,.) has the "hexagon property". (In the case of binomial coefficients X(r,s) = COMB(r,s), the "hexagon property" holds (see [Gupta]) and moreover gcd(A,C,E) = gcd(B,D,F) (see [Hitotumatu & Sato]).)
Corollary 2.2 in [Brent & McKay] says that, for the d X d matrices over Z(p^e), (mutatis mutandis) T_d(p^e,0) = K*(1-P(d+e-1)/P(e-1)) and T_d(p^e,[i]) = K*(q^e)*((1-q^d)/(1-q))*P(d+i-1)/P(i), where q=1/p, K=(p^e)^(d^2), P(t) = Product_{j=1..t} (1-q^j), P(0):=1. (For the case d=2, we have T(p^e,[i]) = (p+1)*(p^(i+1)-1)*p^(3*e-i-2).) Due to [Brent & McKay], it can be simply proved that for d X d matrices the "hexagon property" is true. The formulation implies an obvious generalization: For the entries A(r,u), B(r,v), C(s,w), D(t,w), E(t,v), F(s,u) of the T_d(.,.)-triangle, a hexagon-like property A*C*E = B*D*F holds. This is false in general for the COMB(.,.)-triangle.
Another (rotated-hexagon-like) property: for the entries A(m-b1,n), B(m-a1,n+c2), C(m+a2,n+c2), D(m+b2,n), E(m+a2,n-c1), F(m-a1,n-c1) of the T_d(.,.)-triangle, the property A*C*E = B*D*F holds, if and only if 2*(a1 + a2) = b1 + b2. This is also in general false for COMB(.,.)-triangle.

			From _Andrew Howroyd_, Jul 16 2018: (Start)
Triangle begins:
    1;
   10,   6;
   33,  24,  24;
   88,  48,  72,  48;
  145, 120, 120, 120, 120;
  330, 144, 240, 198, 240, 144;
  385, 336, 336, 336, 336, 336, 336;
  736, 384, 576, 384, 672, 384, 576, 384;
  945, 648, 648, 864, 648, 648, 864, 648, 648;
  ... (End)

Erdos Pal, Rows n=1..100 of triangle, flattened
Richard P. Brent and Brendan D. McKay, Determinants and ranks of random matrices over Z_m, Discrete Mathematics 66 (1987) pp. 35-49.
A. K. Gupta, Generalized hidden hexagon squares, The Fibonacci Quarterly, Vol 12, Number 1, Feb.1974, pp. 45-46.
S. Hitotumatu, D. Sato, Star of David theorem (I), The Fibonacci Quarterly, Vol 13, Number 1, Feb.1975, p. 70.

Column k=0 is A020478.
Column k=1 is A000056.
Row sums are A005353.
Cf. A000252, A127917.

  (* computing T(p^e,k) ; p=prime, 1<=e, 0<=k

S(p,e)={my(u=vector(p^e)); my(t=(p-1)*p^(e-1)); u[1] = p^e + e*t; for(j=1, p^e-1, u[j+1] = t*(1+valuation(j, p))); vector(#u, k, sum(j=0, #u-1, u[j + 1]*u[(j+k-1) % #u + 1]))}
T(n)={my(f=factor(n), v=vector(n,i,1)); for(i=1, #f~, my(r=S(f[i,1], f[i,2])); for(j=0, #v-1, v[j + 1] *= r[j % #r + 1])); v}
for(n=1, 10, print(T(n))); \\ Andrew Howroyd, Jul 16 2018

T(a*b,k) = T(a,(k mod a))*T(b,(k mod b)) if gcd(a,b) = 1.

Sum_{k=1..n-1, gcd(k,n)=1} T(n,k) = A000252(n). - Andrew Howroyd, Jul 16 2018

Terms a(24)-a(55) from b-file by Andrew Howroyd, Jul 16 2018

cp's OEIS Frontend

A165149 a(n) = (3*9^n - 5^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A165147 a(n) = (3*7^n-3^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Other

PARI

Formula

Extensions