A016131 -id:A016131 - cp's OEIS frontend

A078121 Infinite lower triangular matrix, M, that satisfies [M^2](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 16, 8, 1, 1, 36, 84, 64, 16, 1, 1, 202, 656, 680, 256, 32, 1, 1, 1828, 8148, 10816, 5456, 1024, 64, 1, 1, 27338, 167568, 274856, 174336, 43680, 4096, 128, 1, 1, 692004, 5866452, 11622976, 8909648, 2794496, 349504, 16384, 256, 1

Offset: 0

Paul D. Hanna, Nov 18 2002

M also satisfies: [M^(2k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(2^n)](i,j) = M(i+n,j+n) for all n>=0.

			The square of the matrix is the same matrix excluding the first row and column:
  [1, 0, 0, 0, 0]^2 = [ 1, 0, 0, 0, 0]
  [1, 1, 0, 0, 0]     [ 2, 1, 0, 0, 0]
  [1, 2, 1, 0, 0]     [ 4, 4, 1, 0, 0]
  [1, 4, 4, 1, 0]     [10,16, 8, 1, 0]
  [1,10,16, 8, 1]     [36,84,64,16, 1]

Alois P. Heinz, Rows n = 0..80, flattened
MathOverflow, Closed form for diagonals of A078121, (2025).

Cf. A002577, A016131, A078122, A089177.

M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
       add(M(i-1, k)*M(k, j-1), k=0..i-1))
    end:
seq(seq(M(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 27 2015

rows = 10; M[k_] := Table[ Which[j == 1, 1, i == j, 1, 1 < j < i, m[i, j], True, 0], {i, 1, k}, {j, 1, k}]; m2[i_, j_] := m[i+1, j+1]; M2[k_] := Table[ Which[jJean-François Alcover, Feb 27 2015 *)
M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[M[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[M[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 27 2015, after Alois P. Heinz *)

rows_upto(n) = my(A, v1); v1 = vector(n+1, i, vector(i, j, 0)); v1[1][1] = 1; for(i=1, n, v1[i+1][1] = 1; v1[i+1][i+1] = 1); for(i=2, n, for(j=1, i-1, A = (i+j+1)%2; v1[i+1][j+1] = 2*sum(k=0, (i-j-1)\2, v1[i-j+1][2*k+A+1]*v1[j+2*k+A+1][j]))); v1 \\ Mikhail Kurkov, Aug 27 2025

M(1,j) = A002577(j) (partitions of 2^j into powers of 2), M(j+1,j) = 2^j, M(j+2,j) = 4^j, M(j+3,j) = A016131(j).
M(n,k) = the coefficient of x^(2^n - 2^(n-k)) in the power series expansion of 1/Product_{j=0..n-k} (1-x^(2^j)) whenever 0<=k0 (conjecture).
M(n,k) = Sum_{j=0..n-k-1} M(n-k,j)*M(k+j,k-1)*(1+(-1)^(n+k+j+1)) for 0 < k < n with M(n,0) = M(n,n) = 1. - Mikhail Kurkov, Jun 01 2025
From Mikhail Kurkov, Jul 01 2025: (Start)
Conjecture 1: let R(n,x) be the n-th row polynomial, then R(n,x) = x*R(n-1,x) + Sum_{k=1..n-1} M(n-1,k-1)*R(k,x)*(-1)^(n+k+1) = R(n-1,x) + x*Sum_{k=1..n-1} (M(n-1,k) - M(n-2,k))*R(k,x) for n > 1 with R(0,x) = 1, R(1,x) = x + 1.
Conjecture 2: M(n+m,n) ~ 2^(m*(2*n+m-1)/2)/m! as n -> oo. More generally, it also looks like that M(n+m,n) for m > 0 can be represented as (Sum_{j=0..flooor((m-1)/2)} 2^((m-2*j)*(2*(n-j)+m-1)/2)*P(m,j)*(-1)^j)/m! where P(m,j) are some positive integer coefficients. (End)

A016140 Expansion of 1/((1-3x)(1-8*x)).

Original entry on oeis.org

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

Offset: 0

N. J. A. Sloane

In general, for expansion of 1/((1-b*x)*(1-c*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - b*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

8*a(n) gives the number of edges in the n-th-order Sierpiński carpet graph. - Eric W. Weisstein, Aug 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

Cf. A190543.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11,-24},{1,11},30] (* Harvey P. Dale, Feb 03 2022 *)

Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,11,24) for n in range(1, 30)] # Zerinvary Lajos, Apr 27 2009

a(n) = (8^(n+1) - 3^(n+1))/5.
a(n) = 11*a(n-1) - 24*a(n-2).
a(n) = 3*a(n-1) + 8^n.
a(n) = 8*a(n-1) + 3^n.
a(n) = Sum_{i=0..n} 3^i*8^(n-i).
E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024

A016133 Expansion of 1/((1-2x)(1-9*x)).

Original entry on oeis.org

1, 11, 103, 935, 8431, 75911, 683263, 6149495, 55345711, 498111911, 4483008223, 40347076055, 363123688591, 3268113205511, 29413018865983, 264717169826615, 2382454528505071, 21442090756676711, 192978816810352543, 1736809351293697175, 15631284161644323151

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, The sequence of higher order Mersenne numbers and associated binomial transforms, arXiv:2307.08073 [math.NT], 2023.
Index entries for linear recurrences with constant coefficients, signature (11,-18).

Cf. A016131, A016136.

Cf. A016204 (partial sums); A191465 (this sequence times 7).

[+9^(n+1)/7 -2^(n+1)/7 : n in [0..20]]; // Vincenzo Librandi, Aug 14 2011

CoefficientList[Series[1/((1-2x)(1-9x)),{x,0,30}],x] (* or *) LinearRecurrence[ {11,-18},{1,11},30] (* Harvey P. Dale, Apr 19 2020 *)

Vec(1/((1-2*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,11,18) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

a(n) = 11*a(n-1) - 18*a(n-2).
a(n) = a(n) = 9*a(n-1) + 2^n. - Paul Curtz, Feb 14 2008
E.g.f.: exp(2*x)*(9*exp(7*x) - 2)/7. - Stefano Spezia, Jul 30 2022

A016311 Expansion of 1/((1-2x)(1-7x)(1-8*x)).

Original entry on oeis.org

1, 17, 203, 2101, 20163, 184821, 1643251, 14298917, 122461955, 1036190485, 8684988819, 72248167173, 597363137827, 4914549713909, 40265910006707, 328773866154469, 2676717032006979, 21739418975585493

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (17,-86,112).

Cf. A016130, A016131. - Zerinvary Lajos, Jun 05 2009

[(160*8^n-147*7^n+2*2^n)/15: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

CoefficientList[Series[1/((1-2x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[{17,-86,112},{1,17,203},30] (* Harvey P. Dale, Jul 12 2012 *)

[(8^n - 2^n)/6-(7^n - 2^n)/5 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = A016131(n+1) - A016130(n+1). - Zerinvary Lajos, Jun 05 2009
a(n) = 4*8^(n+1)/3 - 7^(n+2)/5 + 2^(n+1)/15. - R. J. Mathar, Mar 14 2011
From Vincenzo Librandi, Sep 02 2011: (Start)
a(n) = (160*8^n - 147*7^n + 2*2^n)/15;
a(n) = 15*a(n-1) - 56*a(n-2) + 2^n. (End)
a(n) = 17*a(n-1) - 86*a(n-2) + 112*a(n-3), with a(0)=1, a(1)=17, a(2)=203. - Harvey P. Dale, Jul 12 2012

A016316 Expansion of 1/((1-2x)(1-8x)(1-9x)).

Original entry on oeis.org

1, 19, 255, 2975, 32231, 333759, 3353335, 32976175, 319155111, 3051352799, 28893830615, 271497720975, 2535105456391, 23548956856639, 217804673719095, 2007154559579375, 18439691005140071, 168959618797797279, 1544655767192730775, 14094055488835543375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (19,-106,144).

Cf. A016131, A016133.

I:=[1,19,255]; [n le 3 select I[n] else 19*Self(n-1)-106*Self(n-2)+144*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 26 2013

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-8*x)*(1-9*x)))); // Vincenzo Librandi, Jun 26 2013

CoefficientList[Series[1 / ((1 - 2 x) (1 - 8 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 26 2013 *)
LinearRecurrence[{19,-106,144},{1,19,255},30] (* Harvey P. Dale, Dec 29 2021 *)

a(n)=(9^n-2^n)/7-(8^n-2^n)/6 \\ Charles R Greathouse IV, Sep 24 2012

[(9^n - 2^n)/7-(8^n - 2^n)/6 for n in range(2,20)] # Zerinvary Lajos, Jun 05 2009

a(n) = 2^(n+1)/21 - 4*8^(n+1)/3 + 9^(n+2)/7; a(n) = A016133(n+1) - A016131(n+1). - Zerinvary Lajos, Jun 05 2009 [corrected by R. J. Mathar, Mar 14 2011]
From Vincenzo Librandi, Jun 26 2013: (Start)
a(n) = 19*a(n-1) - 106*a(n-2) + 144*a(n-3).
a(n) = 17*a(n-1) - 72*a(n-2) + 2^n. (End)
E.g.f.: exp(2*x)*(2 - 224*exp(6*x) + 243*exp(7*x))/21. - Stefano Spezia, Jul 30 2022

A248217 a(n) = 8^n - 2^n.

Original entry on oeis.org

0, 6, 60, 504, 4080, 32736, 262080, 2097024, 16776960, 134217216, 1073740800, 8589932544, 68719472640, 549755805696, 4398046494720, 35184372056064, 281474976645120, 2251799813554176, 18014398509219840, 144115188075331584, 1152921504605798400

Offset: 0

Vincenzo Librandi, Oct 04 2014

If 2^(n+1) is the length of the even leg of a primitive Pythagorean triangle (PPT) then it constrains the odd leg to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n+2^n, an area of 8^n-2^n and an inradius of 2^n-1. For n > 0, a(n) is the area of such triangles. - Frank M Jackson, Sep 07 2018

Maximum anomalous cancellation multiplicity of (2n+1)-digit integers: number of (2n+1)-digit integers which can be anomalously canceled with a fixed (2n+1)-digit integer. The maximum is obtained at 88...88911...11 containing n 8's and n 1's (see Example below). Anomalous cancellation is a "canceling" of digits of a and b in the numerator and denominator of a fraction a/b which results in a fraction equal to the original, and no 0 or digits that appear different times in a and b are canceled. For example, 49/98 = 4/8, 138/184 = 3/4, 1985/5955 = 185/555, 88911/43956 = 8811/4356, but 120/340 is not because canceling the 0's is not an anomalous cancellation. - Xiaohan Zhang, Nov 21 2019

			For n=1, there are 6 numbers with 3 digits that can be anomalously canceled with 891: 297, 396, 495, 594, 693, 792. For n=2 there are 60 numbers with 88911: 12987, 13986, 14985, 15984, 16983, 17982, 21978, 22977, 23976, 24975, 25974, 26973, 27972, 28971, 31968, 32967, 33966, 34965, 35964, 36963, 37962, 38961, 41958, 42957, 43956, 44955, 45954, 46953, 47952, 48951, 51948, 52947, 53946, 54945, 55944, 56943, 57942, 58941, 61938, 62937, 63936, 64935, 65934, 66933, 67932, 68931, 71928, 72927, 73926, 74925, 75924, 76923, 77922, 78921, 82917, 83916, 84915, 85914, 86913, 87912. For n=3 504 numbers with 8889111, and no other (2n+1)-digit number has greater multiplicity. There seems to be a pattern of integer partitions in these examples, because the sum of the digits of numbers above are all multiples of 9. - _Xiaohan Zhang_, Nov 21 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Anomalous Cancellation
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. similar sequences listed in A248216.

Cf. A000079, A024036.

Magma
```
[8^n-2^n: n in [0..25]];
```

Table[8^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 2 x) (1 - 8 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-16},{0,6},30] (* Harvey P. Dale, Mar 29 2015 *)

a(n) = 8^n-2^n; \\ Altug Alkan, Sep 07 2018

def A248217(n): return 6*binomial(pow(2,n) +1, 3)
print([A248217(n) for n in range(41)]) # G. C. Greubel, Dec 26 2024

G.f.: 6*x/((1-2*x)*(1-8*x)).
a(n) = 10*a(n-1) - 16*a(n-2).
a(n) = 2^n*(4^n-1) = A000079(n) * A024036(n) = A001018(n) - A000079(n).
E.g.f.: exp(2*x)*(-1 + exp(6*x)). - Stefano Spezia, Sep 07 2018
a(n) = 6*A016131(n-1). - R. J. Mathar, Mar 10 2022

A120689 a(n) = 10a(n-1) - 16a(n-2), with a(0) = 0 and a(1) = 3.

Original entry on oeis.org

0, 3, 30, 252, 2040, 16368, 131040, 1048512, 8388480, 67108608, 536870400, 4294966272, 34359736320, 274877902848, 2199023247360, 17592186028032, 140737488322560, 1125899906777088, 9007199254609920, 72057594037665792

Offset: 0

Gary W. Adamson, Jun 25 2006

a(n) is a leg in a Pythagorean triangle along with A081342(n) (the hypotenuse) and 4^n. Example: a(4) = 2040, A081342(4) = 2056; then sqrt(2056^2 - 2040^2) = 256 = 4^4. Characteristic polynomial of M = x^2 -10*x + 16.

Order of modular group of degree 2^(n-1)+1. - Artur Jasinski, Aug 04 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Mathieu, Mémoire sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses variables de toutes les manières possibles, Journ. de math. pures et appliquées (2) 5 (1860), 9-42 (see p. 39).
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A016131, A081342.

[2^(n-1)*(4^n-1): n in [0..30]]; // G. C. Greubel, Dec 27 2022

a[0]:=0: a[1]:=3; for n from 2 to 20 do a[n]:=10*a[n-1]-16*a[n-2] end do: seq(a[n], n = 0 .. 20); # Emeric Deutsch, Aug 16 2007
seq(binomial(2^n,2)*(2^n + 1),n=0..19); # Zerinvary Lajos, Jan 07 2008

Table[2^(n-1) (4^n-1), {n,0,20}] (* Artur Jasinski, Aug 04 2007 *)

A120689=BinaryRecurrenceSequence(10,-16,0,3)
[A120689(n) for n in range(31)] # G. C. Greubel, Dec 27 2022

a(n) = 8^n - A081342(n).
Given M = 2 X 2 matrix [5,3; 3,5]; M^n * [1,0] = [A081342(a), a(n)]. E.g. a(4) = 2040, right term in = M^4 * [1,0] = [2056, 2040] = [A081342(4), a(4)].
a(n) = 2^(n-1)*(4^n - 1). - Artur Jasinski, Aug 04 2007
From R. J. Mathar, Feb 16 2011: (Start)
a(n) = 3*A016131(n-1).
G.f.: 3*x / ( (1-2*x)*(1-8*x) ). (End)
E.g.f.: (1/2)*(exp(8*x) - exp(2*x)). - G. C. Greubel, Dec 27 2022

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 13 2007

More terms from Emeric Deutsch, Aug 16 2007

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

Original entry on oeis.org

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

A016203 Expansion of g.f. 1/((1-x)(1-2x)(1-8x)).

Original entry on oeis.org

1, 11, 95, 775, 6231, 49911, 399415, 3195575, 25565111, 204521911, 1636177335, 13089422775, 104715390391, 837723139511, 6701785148855, 53614281256375, 428914250182071, 3431314001718711, 27450512014273975, 219604096115240375, 1756832768924020151, 14054662151396355511

Offset: 0

N. J. A. Sloane

4*a(n) is the total number of holes in a certain box fractal (start with 8 boxes, 0 hole) after n iterations. See illustration in link. - Kival Ngaokrajang, Jan 27 2015

Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (11,-26,16).

Cf. A016131, A023001.

a:=n->sum((8^(n-j)-2^(n-j))/6,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007

Vec(1/((1-x)*(1-2*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (4*8^(n+1) - 7*2^(n+1) + 3)/21. - Mitch Harris, Jun 27 2005; corrected by Yahia Kahloune, May 06 2013
a(0) = 1, a(n) = 8*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 09 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(32*exp(7*x) - 14*exp(x) + 3)/21.
a(n) = 11*a(n-1) - 26*a(n-2) + 16*a(n-3).
a(n) = A016131(n+1) - A023001(n+2). (End)

More terms from Elmo R. Oliveira, Mar 26 2025

A092123 Largest coefficient in expansion of P(0)=x, P(n+1)=P(n)*[1+P(n)].

Original entry on oeis.org

1, 1, 2, 10, 302, 391232, 912140731560, 6870302396056798235043564, 552249828443015013351729477795257932661645918815144

Offset: 0

Ralf Stephan, Apr 02 2004

nonn

It would be nice to have a formula for the 2^(n-1)th coefficient.

The exponent of x in P(n) with a(n) as coefficient is {1,1,2,5,9,19,37,74,147,294,587,1175,...}.

Cf. A016131.

PARI
```
P(n)=if(n<1,x,P(n-1)*(P(n-1)+1))
```
PARI
```
a(n)=vecmax(Vec(P(n)))
```

cp's OEIS Frontend

A078121 Infinite lower triangular matrix, M, that satisfies [M^2](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

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Maple

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A016140 Expansion of 1/((1-3*x)*(1-8*x)).

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Magma

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A016133 Expansion of 1/((1-2*x)*(1-9*x)).

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A016311 Expansion of 1/((1-2*x)*(1-7*x)*(1-8*x)).

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A016316 Expansion of 1/((1-2x)*(1-8x)*(1-9x)).

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Magma

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A248217 a(n) = 8^n - 2^n.

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A120689 a(n) = 10*a(n-1) - 16*a(n-2), with a(0) = 0 and a(1) = 3.

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A016140 Expansion of 1/((1-3x)(1-8*x)).

A016133 Expansion of 1/((1-2x)(1-9*x)).

A016311 Expansion of 1/((1-2x)(1-7x)(1-8*x)).

A016316 Expansion of 1/((1-2x)(1-8x)(1-9x)).

A120689 a(n) = 10a(n-1) - 16a(n-2), with a(0) = 0 and a(1) = 3.

A016203 Expansion of g.f. 1/((1-x)(1-2x)(1-8x)).