A138894 -id:A138894 - cp's OEIS frontend

A112468 Riordan array (1/(1-x), x/(1+x)).

1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 0, 2, -2, 1, 1, 1, -2, 4, -3, 1, 1, 0, 3, -6, 7, -4, 1, 1, 1, -3, 9, -13, 11, -5, 1, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A040000. Diagonal sums are A112469. Inverse is A112467. Row sums of k-th power are 1, k+1, k+1, k+1, .... Note that C(n,k) = Sum_{j=0..n-k} C(n-j-1, n-k-j).
Equals row reversal of triangle A112555 up to sign, where log(A112555) = A112555 - I. Unsigned row sums equals A052953 (Jacobsthal numbers + 1). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms. - Paul D. Hanna, Jan 20 2006
Sum_{k=0..n} T(n,k)*x^k = A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively (see the square array in A112739). - Philippe Deléham, Feb 22 2014

			Triangle starts
  1;
  1,  1;
  1,  0,  1;
  1,  1, -1,  1;
  1,  0,  2, -2,  1;
  1,  1, -2,  4, -3,  1;
  1,  0,  3, -6,  7, -4,  1;
Matrix log begins:
  0;
  1,  0;
  1,  0,  0;
  1,  1, -1,  0;
  1,  1,  1, -2,  0;
  1,  1,  1,  1, -3,  0; ...
Production matrix begins
  1,  1,
  0, -1,  1,
  0,  0, -1,  1,
  0,  0,  0, -1,  1,
  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0,  0, -1,  1.
- _Paul Barry_, Apr 08 2011

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
H. Belbachir and F. Bencherif, On some properties of bivariate Fibonacci and Lucas Polynomials, JIS 11 (2008) 08.2.6.
Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
H. Prodinger, On the expansion of Fibonacci and Lucas Polynomials, JIS 12 (2009) 09.1.6.

Cf. A174294, A174295, A174296, A174297. - Mats Granvik, Mar 15 2010
Cf. A072547 (central terms), A112555 (reversed rows), A112465, A052953, A112556, A112739, A119258.
See A279006 for another version.

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return T(n-1,k-1) - T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 13 2019

a112468 n k = a112468_tabl !! n !! k
a112468_row n = a112468_tabl !! n
a112468_tabl = iterate (\xs -> zipWith (-) ([2] ++ xs) (xs ++ [0])) [1]
-- Reinhard Zumkeller, Jan 03 2014

function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1,k-1) - T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2019

T := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)*hypergeom( [1,n+1],[k+2],m)/(k+1)!; A112468 := (n,k) -> T(n,n-k,-1);
seq(print(seq(simplify(A112468(n,k)),k=0..n)),n=0..10); # Peter Luschny, Jul 25 2014

T[n_, 0] = 1; T[n_, n_] = 1; T[n_, k_ ]:= T[n, k] = T[n-1, k-1] - T[n-1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Mar 06 2013 *)

{T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x),n,X),k,Y)} \\ Paul D. Hanna, Jan 20 2006

T(n,k) = if(k==0 || k==n, 1, T(n-1, k-1) - T(n-1, k)); \\ G. C. Greubel, Nov 13 2019

@CachedFunction
def T(n, k):
    if (k<0 or n<0): return 0
    elif (k==0 or k==n): return 1
    else: return T(n-1, k-1) - T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 13 2019

Triangle T(n,k) read by rows: T(n,0)=1, T(n,k) = T(n-1,k-1) - T(n-1,k). - Mats Granvik, Mar 15 2010
Number triangle T(n, k)= Sum_{j=0..n-k} C(n-j-1, n-k-j)*(-1)^(n-k-j).
G.f. of matrix power T^m: (1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x). G.f. of matrix log: x*(1-2*x*y+x^2*y)/(1-x*y)^2/(1-x). - Paul D. Hanna, Jan 20 2006
T(n, k) = R(n,n-k,-1) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)*hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

A096053 a(n) = (3*9^n - 1)/2.

Original entry on oeis.org

1, 13, 121, 1093, 9841, 88573, 797161, 7174453, 64570081, 581130733, 5230176601, 47071589413, 423644304721, 3812798742493, 34315188682441, 308836698141973, 2779530283277761, 25015772549499853, 225141952945498681

Offset: 0

Benoit Cloitre, Jun 18 2004

Generalized NSW numbers. - Paul Barry, May 27 2005
Counts total area under elevated Schroeder paths of length 2n+2, where area under a horizontal step is weighted 3. Case r=4 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315. Fifth binomial transform of (1+8x)/(1-16x^2), A107906. - Paul Barry, May 27 2005
Primes in this sequence include: a(2) = 13, a(4) = 1093, a(7) = 797161. Semiprimes in this sequence include: a(3) = 121 = 11^2, a(5) = 9841 = 13 * 757, a(6) = 88573 = 23 * 3851, a(9) = 64570081 = 1871 * 34511, a(10) = 581130733 = 1597 * 363889, a(12) = 47071589413 = 47 * 1001523179, a(19) = 225141952945498681 = 13097927 * 17189128703.
Sum of divisors of 9^n. - Altug Alkan, Nov 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A083420, A096045, A096046, A096047, A096054.

Cf. A107903, A138894 ((5*9^n-1)/4).

[(3*9^n-1)/2: n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

Table[(3*9^n - 1)/2, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

a(n)=(3*9^n-1)/2 \\ Charles R Greathouse IV, Sep 28 2015

vector(30, n, n--; sigma(9^n)) \\ Altug Alkan, Nov 10 2015

From Paul Barry, May 27 2005: (Start)
G.f.: (1+3*x)/(1-10*x+9*x^2);
a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*4^k;
a(n) = ((1+sqrt(4))*(5+2*sqrt(4))^n+(1-sqrt(4))*(5-2*sqrt(4))^n)/2. (End)
a(n-1) = (-9^n/3)*B(2n,1/3)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k)=B(k,0) is the k-th Bernoulli number.
a(n) = 10*a(n-1) - 9*a(n-2).
a(n) = 9*a(n-1) + 4. - Vincenzo Librandi, Nov 01 2011
a(n) = A000203(A001019(n)). - Altug Alkan, Nov 10 2015
a(n) = A320030(3^n-1). - Nathan M Epstein, Jan 02 2019

Edited by N. J. A. Sloane, at the suggestion of Andrew S. Plewe, Jun 15 2007

A112739 Array counting nodes in rooted trees of height n in which the root and internal nodes have valency k (and the leaf nodes have valency one).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 5, 2, 1, 1, 5, 10, 7, 2, 1, 1, 6, 17, 22, 9, 2, 1, 1, 7, 26, 53, 46, 11, 2, 1, 1, 8, 37, 106, 161, 94, 13, 2, 1, 1, 9, 50, 187, 426, 485, 190, 15, 2, 1, 1, 10, 65, 302, 937, 1706, 1457, 382, 17, 2, 1, 1, 11, 82, 457, 1814, 4687, 6826, 4373, 766, 19

Offset: 0

Paul Barry, Sep 16 2005

Rows of the square array have g.f. (1+x)/((1-x)(1-kx)). They are the partial sums of the coordination sequences for the infinite tree of valency k. Row sums are A112740.

Rows of the square array are successively: A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023. - Philippe Deléham, Feb 22 2014

			As a square array, rows begin
1,1,1,1,1,1,... (A000012)
1,2,2,2,2,2,... (A040000)
1,3,5,7,9,11,... (A005408)
1,4,10,22,46,94,... (A033484)
1,5,17,53,161,485,... (A048473)
1,6,26,106,426,1706,... (A020989)
1,7,37,187,937,4687,... (A057651)
1,8,50,302,1814,10886,... (A061801)
As a number triangle, rows start
1;
1,1;
1,2,1;
1,3,2,1;
1,4,5,2,1;
1,5,10,7,2,1;

L. He, X. Liu and G. Strang, (2003) Trees with Cantor Eigenvalue Distribution. Studies in Applied Mathematics 110 (2), 123-138.
L. He, X. Liu and G. Strang, Laplacian eigenvalues of growing trees, Proc. Conf. on Math. Theory of Networks and Systems, Perpignan (2000).

Cf. A112468, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023.

As a square array read by antidiagonals, T(n, k)=sum{j=0..k, (2-0^j)*(n-1)^(k-j)}; T(n, k)=(n(n-1)^k-2)/(n-2), n<>2, T(2, n)=2n+1; T(n, k)=sum{j=0..k, (n(n-1)^j-0^j)/(n-1)}, j<>1. As a triangle read by rows, T(n, k)=if(k<=n, sum{j=0..k, (2-0^j)*(n-k-1)^(k-j)}, 0).

A191681 a(n) = (9^n - 1)/2.

Original entry on oeis.org

0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044

Offset: 0

Adi Dani, Jun 11 2011

Number of compositions of odd numbers into n parts < 9.
These are also the junctions of the Collatz trajectories of 2^(2k-1)-1 and 2^2k-1. - David Rabahy, Nov 01 2017
a(n) gives the number of turns in the n-th iteration of the Peano curve given by plotting (A163528, A163529) or by (Siromoney 1982). - Jason V. Morgan, Oct 08 2021

			a(2)=40: there are 40 compositions of odd numbers into 2 parts < 9:
1:  (0,1),(1,0);
3:  (0,3),(3,0),(1,2),(2,1);
5:  (0,5),(5,0),(1,4),(4,1),(2,3),(3,2);
7:  (0,7),(7,0),(1,6),(6,1),(2,5),(5,2),(3,4),(4,3);
9:  (1,8),(8,1),(2,7),(7,2),(3,6),(6,3),(4,5),(5,4);
11: (3,8),(8,3),(4,7),(7,4),(5,6),(6,5);
13: (5,8),(8,5),(6,7),(7,6);
15: (7,8),(8,7).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Adi Dani, Restricted compositions of natural numbers
Rani Siromoney and K. G. Subramanian, Space-filling curves and infinite graphs, International Workshop on Graph Grammars and Their Application to Computer Science, 1982.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A002452, A096053, A125857, A138894, A198964.

[(9^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 16 2011

Table[(9^n - 1)/2, {n, 0, 19}]
LinearRecurrence[{10,-9},{0,4},30] (* Harvey P. Dale, Jun 19 2011 *)

a(n)=9^n\2 \\ Charles R Greathouse IV, Oct 16 2015

a(0)=0, a(1)=4, a(n) = 10*a(n-1) - 9*a(n-2). - Harvey P. Dale, Jun 19 2011
G.f.: 4*x / ((x-1)*(9*x-1)). - Colin Barker, May 16 2013
a(n) = 2 * A125857(n+1) = 4 * A002452(n). - Bernard Schott, Oct 29 2021

Example corrected by L. Edson Jeffery, Feb 13 2015

A238275 a(n) = (4*7^n - 1)/3.

Original entry on oeis.org

1, 9, 65, 457, 3201, 22409, 156865, 1098057, 7686401, 53804809, 376633665, 2636435657, 18455049601, 129185347209, 904297430465, 6330082013257, 44310574092801, 310174018649609, 2171218130547265, 15198526913830857, 106389688396816001, 744727818777712009

Offset: 0

Philippe Deléham, Feb 21 2014

Sum of n-th row of triangle of powers of 7: 1; 1 7 1; 1 7 49 7 1; 1 7 49 343 49 7 1; ...

Number of cubes in the crystal structure cubic carbon CCC(n+1), defined in the Baig et al. and in the Gao et al. references. - Emeric Deutsch, May 28 2018

			a(0) = 1;
a(1) = 1 + 7 + 1 = 9;
a(2) = 1 + 7 + 49 + 7 + 1 = 65;
a(3) = 1 + 7 + 49 + 343 + 49 + 7 + 1 = 457; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, Molecular description of carbon graphite and crystal cubic carbon structures, Canadian J. Chem., 95, 674-686, 2017.
W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, Topological characterization of carbon graphite and crystal cubic carbon structures, Molecules, 22, 1496, 1-12, 2017.
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, this sequence, A238276, A138894, A090843, A199023.

Cf. A112468, A112739.

[(4*7^n - 1)/3: n in [0..30]]; // Vincenzo Librandi, Feb 23 2014

Table[(4 7^n - 1)/3, {n, 0, 40}] (* Vincenzo Librandi, Feb 23 2014 *)

G.f.: (1+x)/((1-x)*(1-7*x)).
a(n) = 7*a(n-1) + 2, a(0) = 1.
a(n) = 8*a(n-1) - 7*a(n-2), a(0) = 1, a(1) = 9.
a(n) = Sum_{k=0..n} A112468(n,k)*8^k.
E.g.f.: exp(x)*(4*exp(6*x) - 1)/3. - Stefano Spezia, Feb 12 2025

A238276 a(n) = (9*8^n - 2)/7.

Original entry on oeis.org

1, 10, 82, 658, 5266, 42130, 337042, 2696338, 21570706, 172565650, 1380525202, 11044201618, 88353612946, 706828903570, 5654631228562, 45237049828498, 361896398627986, 2895171189023890, 23161369512191122, 185290956097528978, 1482327648780231826

Offset: 0

Philippe Deléham, Feb 21 2014

Sum of n-th row of triangle of powers of 8: 1; 1 8 1; 1 8 64 8 1; 1 8 64 512 64 8 1; ...

			a(0) = 1;
a(1) = 1 + 8 + 1 = 10;
a(2) = 1 + 8 + 64 + 8 + 1 = 82;
a(3) = 1 + 8 + 64 + 512 + 64 + 8 + 1 = 658; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9, -8).

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, this sequence, A138894, A090843, A199023.

Cf. A112468, A112739.

[(9*8^n - 2)/7: n in [0..30]]; // Vincenzo Librandi, Feb 23 2014

Table[(9 8^n - 2)/7, {n, 0, 50}] (* Vincenzo Librandi, Feb 23 2014 *)

G.f.: (1+x)/((1-x)*(1-8*x)).
a(n) = 8*a(n-1) + 2, a(0) = 1.
a(n) = 9*a(n-1) - 8*a(n-2), a(0) = 1, a(1) = 10.
a(n) = Sum_{k=0..n} A112468(n,k)*9^k.

Corrected by Vincenzo Librandi, Feb 23 2014

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A255043 a(n) = (5*9^n - 1)/2.

Original entry on oeis.org

2, 22, 202, 1822, 16402, 147622, 1328602, 11957422, 107616802, 968551222, 8716961002, 78452649022, 706073841202, 6354664570822, 57191981137402, 514727830236622, 4632550472129602, 41692954249166422, 375236588242497802, 3377129294182480222

Offset: 0

L. Edson Jeffery, Feb 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A096053, A138894, A191681, A198964, A198969.

Row 2 of A255044.

[(5*9^n -1)/2: n in [0..20]]; // G. C. Greubel, Feb 07 2021

Table[(5*9^n - 1)/2, {n, 0, 19}]
LinearRecurrence[{10,-9},{2,22},20] (* Harvey P. Dale, Jun 15 2018 *)

[(5*9^n -1)/2 for n in (0..20)] # G. C. Greubel, Feb 07 2021

G.f.: 2*(1+x)/((1-x)*(1-9*x)).
Recurrence: a(n) = 10*a(n-1) - 9*a(n-2), n>=2, a(0) = 2, a(1) = 22.
a(n) = 2*A138894(n).
E.g.f.: (5*exp(9*x) - exp(x))/2. - G. C. Greubel, Feb 07 2021

A138893 A generalized Chamberland function.

Original entry on oeis.org

0, 11, 36, 29, 8, 47, 100, 65, 16, 83, 164, 101, 24, 119, 228, 137, 32, 155, 292, 173, 40, 191, 356, 209, 48, 227, 420, 245, 56, 263, 484, 281, 64, 299, 548, 317, 72, 335, 612, 353, 80, 371, 676, 389, 88, 407, 740, 425, 96, 443, 804, 461, 104, 479, 868, 497

Offset: 0

Paul Barry, Apr 02 2008

The orbit of a(n) beginning at 1 is A138894.

M. Chamberland, A Continuous Extension of the 3x+1 Problem to the Real Line, Dynamics of Continuous, Discrete and Impulsive Dynamical Systems 2(1996), 495-509.

Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Cf. A138894.

LinearRecurrence[{2,-3,4,-3,2,-1},{0, 11, 36, 29, 8, 47},56] (* or *) CoefficientList[Series[x(11+14x-10x^2+14x^3+7x^4)/((1-x)^2(1+x^2)^2),{x,0,55}],x] (* James C. McMahon, Jun 24 2025 *)

G.f.: x(11+14x-10x^2+14x^3+7x^4)/((1-x)^2(1+x^2)^2);
a(n) = 9n+2-(7n+2)cos(Pi*n/2);
a(n) = 6*((n/3)*(cos(Pi*n/4))^2+(2/3)*(4n+1)*(sin(Pi*n/4))^2);
a(4n) = 8n; a(4n+1) = 11+36n; a(4n+2) = 4*(9+16n); a(4n+3) = 29+36n;

a(47)-a(55) from James C. McMahon, Jun 24 2025

A198962 5*9^n-1.

Original entry on oeis.org

4, 44, 404, 3644, 32804, 295244, 2657204, 23914844, 215233604, 1937102444, 17433922004, 156905298044, 1412147682404, 12709329141644, 114383962274804, 1029455660473244, 9265100944259204, 83385908498332844, 750473176484995604

Offset: 0

Vincenzo Librandi, Nov 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Magma
```
[5*9^n-1: n in [0..20]]
```

5*9^Range[0,30]-1 (* or *) LinearRecurrence[{10,-9},{4,44},30] (* Harvey P. Dale, Oct 20 2019 *)

a(n) = 4*A138894(n).
a(n) = 9*a(n-1)+8, n>0.
a(n) = 10*a(n-1)-9*a(n-2), n>1.
G.f. ( 4+4*x ) / ( (9*x-1)*(x-1) ). - R. J. Mathar, Nov 01 2011

cp's OEIS Frontend

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