A023001 -id:A023001 - cp's OEIS frontend

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

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

A125118 Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

Original entry on oeis.org

1, 3, 4, 7, 13, 21, 15, 40, 85, 156, 31, 121, 341, 781, 1555, 63, 364, 1365, 3906, 9331, 19608, 127, 1093, 5461, 19531, 55987, 137257, 299593, 255, 3280, 21845, 97656, 335923, 960800, 2396745, 5380840, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111

Offset: 1

Reinhard Zumkeller, Nov 21 2006

			First 4 rows:
1: [1]_2
2: [11]_2 ........ [11]_3
3: [111]_2 ....... [111]_3 ....... [111]_4
4: [1111]_2 ...... [1111]_3 ...... [1111]_4 ...... [1111]_5
_
1: 1
2: 2+1 ........... 3+1
3: (2+1)*2+1 ..... (3+1)*3+1 ..... (4+1)*4+1
4: ((2+1)*2+1)*2+1 ((3+1)*3+1)*3+1 ((4+1)*4+1)*4+1 ((5+1)*5+1)*5+1.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Repunit

This triangle shares some features with triangle A104878.
This triangle is a portion of rectangle A055129.
Each term of A110737 comes from the corresponding row of this triangle.
Diagonals (adjusting offset as necessary): A060072, A023037, A031973, A173468.
Columns (adjusting offset as necessary): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
Cf. A023037, A031973, A125119, A125120 (row sums).

[((k+1)^n -1)/k : k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 15 2022

Table[((k+1)^n -1)/k, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Aug 15 2022 *)

def A125118(n,k): return ((k+1)^n -1)/k
flatten([[A125118(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Aug 15 2022

T(n, k) = Sum_{i=0..n-1} (k+1)^i.
T(n+1, k) = (k+1)*T(n, k) + 1.
Sum_{k=1..n} T(n, k) = A125120(n).
T(2*n-1, n) = A125119(n).
T(n, 1) = A000225(n).
T(n, 2) = A003462(n) for n>1.
T(n, 3) = A002450(n) for n>2.
T(n, 4) = A003463(n) for n>3.
T(n, 5) = A003464(n) for n>4.
T(n, 9) = A002275(n) for n>8.
T(n, n) = A060072(n+1).
T(n, n-1) = A023037(n) for n>1.
T(n, n-2) = A031973(n) for n>2.
T(n, k) = A055129(n, k+1) = A104878(n+k, k+1), 1<=k<=n. - Mathew Englander, Dec 19 2020

A295235 Numbers k such that the positions of the ones in the binary representation of k are in arithmetic progression.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 28, 30, 31, 32, 33, 34, 36, 40, 42, 48, 56, 60, 62, 63, 64, 65, 66, 68, 72, 73, 80, 84, 85, 96, 112, 120, 124, 126, 127, 128, 129, 130, 132, 136, 144, 146, 160, 168, 170, 192, 224, 240, 248

Offset: 1

Rémy Sigrist, Nov 18 2017

Also numbers k of the form Sum_{b=0..h-1} 2^(i+j*b) for some h >= 0, i >= 0, j > 0 (in fact, h = A000120(k), and if k > 0, i = A007814(k)).
There is a simple bijection between the finite sets of nonnegative integers in arithmetic progression and the terms of this sequence: s -> Sum_{i in s} 2^i; the term 0 corresponds to the empty set.
For any n > 0, A054519(n) gives the numbers of terms with n+1 digits in binary representation.
For any n >= 0, n is in the sequence iff 2*n is in the sequence.
For any n > 0, A000695(a(n)) is in the sequence.
The first prime numbers in the sequence are: 2, 3, 5, 7, 17, 31, 73, 127, 257, 8191, 65537, 131071, 262657, 524287, ...
This sequence contains the following sequences: A000051, A000079, A000225, A000668, A002450, A019434, A023001, A048645.
For any k > 0, 2^k - 2, 2^k - 1, 2^k, 2^k + 1 and 2^k + 2 are in the sequence (e.g., 14, 15, 16, 17, and 18).
Every odd term is a binary palindrome (and thus belongs to A006995).
Odd terms are A064896. - Robert Israel, Nov 20 2017

			The binary representation of the number 42 is "101010" and has ones evenly spaced, hence 42 appears in the sequence.
The first terms, alongside their binary representations, are:
   n  a(n)  a(n) in binary
  --  ----  --------------
   1    0           0
   2    1           1
   3    2          10
   4    3          11
   5    4         100
   6    5         101
   7    6         110
   8    7         111
   9    8        1000
  10    9        1001
  11   10        1010
  12   12        1100
  13   14        1110
  14   15        1111
  15   16       10000
  16   17       10001
  17   18       10010
  18   20       10100
  19   21       10101
  20   24       11000

Rémy Sigrist, Table of n, a(n) for n = 1..1000

Cf. A000051, A000079, A000120, A000225, A000668, A000695, A002450, A006995, A007814, A019434, A023001, A048645, A054519, A064896.

Cf. A029931, A048793 (binary indices triangle), A070939, A291166, A325328 (prime indices rather than binary indices), A326669, A326675.

f:= proc(d) local i,j,k;
  op(sort([seq(seq(add(2^(d-j*k),k=0..m),m=1..d/j),j=1..d),2^(d+1)]))
end proc:
0,1,seq(f(d),d=0..10); # Robert Israel, Nov 20 2017

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],SameQ@@Differences[bpe[#]]&] (* Gus Wiseman, Jul 22 2019 *)

is(n) = my(h=hammingweight(n)); if(h<3, return(1), my(i=valuation(n,2),w=#binary(n)); if((w-i-1)%(h-1)==0, my(j=(w-i-1)/(h-1)); return(sum(k=0,h-1,2^(i+j*k))==n), return(0)))

A033138 a(n) = floor(2^(n+2)/7).

Original entry on oeis.org

1, 2, 4, 9, 18, 36, 73, 146, 292, 585, 1170, 2340, 4681, 9362, 18724, 37449, 74898, 149796, 299593, 599186, 1198372, 2396745, 4793490, 9586980, 19173961, 38347922, 76695844, 153391689, 306783378, 613566756, 1227133513, 2454267026

Offset: 1

Clark Kimberling

Previous name was: "Base 2 digits are, in order, the first n terms of the periodic sequence with initial period 1,0,0".
Here we let p = 3 to produce the above sequence, but p can be an arbitrary natural number. By letting p = 2, 4, 6, 7 we produce A000975, A083593, A195904 and A117302. We denote by U[p,n,m] the number of cases in which the first player in a game of Russian roulette gets killed when p players use a gun with n chambers and m bullets. They never rotate the cylinder after the game starts.
The chambers can be represented by the list {1,2,...,n}. We are going to calculate the following (0), (1),...(t) separately. (0) The first player gets killed when one bullet is in the first chamber and the remaining (m-1)- bullets are in {2,3,...,n}. We have binomial(n-1,m-1) cases for this. (1) The first gets killed when one bullet is in the (p+1)st chamber and the rest of the bullets are in {p+2,..,n}. We have binomial(n-p-1,m-1) cases for this. We continue to calculate and the last is (t), where t = floor((n-m)/p). (t) The first gets killed when one bullet is in the (pt+1)st chamber and the remaining bullets are in {pt+2,...,n}. We have binomial(n-pt-1,m-1) cases for this.
Therefore U[p,n,m] = Sum_{z=0..t} binomial(n-pz-1,m-1), where t = floor((n-m)/p). Let A[p,n] be the number of the cases in which the first player gets killed when p players use a gun with n chambers and the number of bullets can be from 1 to n. Then A[p,n] = Sum_{m=1..n} U[p,n,m]. - Ryohei Miyadera, Tomohide Hashiba, Yuta Nakagawa, Hiroshi Matsui, Jun 04 2006
Partial sums of A077947. - Mircea Merca, Dec 28 2010
a(n+1) is the number of partitions of n into two kinds of part 1 and one kind of part 2. - Joerg Arndt, Mar 10 2015
A078010(n) = b(n+1) - 2*b(n) + b(n-1) where b=A078010. - Michael Somos, Nov 18 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 926
S. Klavzar, Structure of Fibonacci cubes: a survey, Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia; Preprint series Vol. 49 (2011), 1150 ISSN 2232-2094.
Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Cf. A000975, A078010, A083593, A195904, A117302, A023001, A111662, A077947.

[Round((4*2^n-2)/7): n in [1..40]]; // Vincenzo Librandi, Jun 25 2011

seq(iquo(2^n,7),n=3..34); # Zerinvary Lajos, Apr 20 2008

U[p_, n_, m_, v_]:=Block[{t}, t=Floor[(1+p-m+n-v)/p];Sum[Binomial[n-v-p*z,m-1],{z,0,t-1}]]; A[p_,n_,v_]:=Sum[U[p,n,k,v],{k,1,n}]; (* Here we let p = 3 to produce the above sequence, but this code can produce A000975, A083593, A195904, A117302 for p = 2,4,6,7. *) Table[A[3,n,1], {n,1,20}] (* Ryohei Miyadera, Tomohide Hashiba, Yuta Nakagawa, Hiroshi Matsui, Jun 04 2006 *)

a(n)=2^(n+2)\7 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*a(n-1) + a(n-3) - 2*a(n-4). -John W. Layman
G.f.: 1/((1-x^3)*(1-2*x)); a(n) = sum{k=0..floor(n/3), 2^(n-3*k)}; a(n) = Sum_{k=0..n} 2^k*( cos(2*Pi*(n-k)/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3 ). - Paul Barry, Apr 16 2005
a(n) = floor(2^(n+2)/7). - Gary Detlefs, Sep 06 2010
a(n) = floor((4*2^n - 1)/7) = ceiling((4*2^n - 4)/7) = round((4*2^n - 2)/7) = round((8*2^n - 5)/14); a(n) = a(n-3) + 2^(n-1), n>3. - Mircea Merca, Dec 28 2010
a(n) = 4/7*2^n - 5/21*cos(2/3*Pi*n) + 1/21*3^(1/2)*sin(2/3*Pi*n)-1/3. - Leonid Bedratyuk, May 13 2012

Edited by Jeremy Gardiner, Oct 08 2011

New name (using formula form Gary Detlefs) from Joerg Arndt, Mar 10 2015

A014827 a(1)=1, a(n) = 5*a(n-1) + n.

Original entry on oeis.org

1, 7, 38, 194, 975, 4881, 24412, 122068, 610349, 3051755, 15258786, 76293942, 381469723, 1907348629, 9536743160, 47683715816, 238418579097, 1192092895503, 5960464477534, 29802322387690, 149011611938471, 745058059692377, 3725290298461908, 18626451492309564, 93132257461547845

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. See also Proc. Amer. Math. Soc. 148 (2020), pp. 461-469.
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016218, A016256, A023000, A023001.

[(5^(n+1)-4*n-5)/16: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

a:=n->sum((5^(n-j)-1^(n-j))/4,j=0..n): seq(a(n), n=1..21); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=7},Table[c=6*b-5*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

[(gaussian_binomial(n,1,5)-n)/4 for n in range(2,23)] # Zerinvary Lajos, May 29 2009

a(n) = (5^(n+1) - 4*n - 5)/16.
G.f.: x/((1-5*x)*(1-x)^2).
From Paul Barry, Jul 30 2004: (Start)
a(n) = Sum_{k=0..n} (n-k)*5^k = Sum_{k=0..n} k*5^(n-k).
a(n) = Sum_{k=0..n} binomial(n+2,k+2)*4^k [Offset 0]. (End)
From Elmo R. Oliveira, Mar 29 2025: (Start)
E.g.f.: exp(x)*(5*exp(4*x) - 4*x - 5)/16.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 3. (End)

A104878 A sum-of-powers number triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 15, 13, 5, 1, 1, 6, 31, 40, 21, 6, 1, 1, 7, 63, 121, 85, 31, 7, 1, 1, 8, 127, 364, 341, 156, 43, 8, 1, 1, 9, 255, 1093, 1365, 781, 259, 57, 9, 1, 1, 10, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 1, 11, 1023, 9841, 21845

Offset: 0

Paul Barry, Mar 28 2005

Columns are partial sums of the columns of A004248. Row sums are A104879. Diagonal sums are A104880.

The rows of this triangle (apart from the initial "1" in each row) are the antidiagonals of rectangle A055129. The diagonals of this triangle (apart from the initial "1") are the rows of rectangle A055129. The columns of this triangle (apart from the leftmost column) are the same as the columns of rectangle A055129 but shifted downward. - Mathew Englander, Dec 21 2020

			Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 15, 13,  5,  1;
  1,  6, 31, 40, 21,  6,  1;
  ...

Cf. A004248 (first differences by column), A104879 (row sums), A104880 (antidiagonal sums), A125118 (version of this triangle with fewer terms).
This triangle (ignoring the leftmost column) is a rotation of rectangle A055129.
Columns (adjusting offset as necessary): A000012, A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
T(2n,n) gives A031973.

A104878 :=proc(n,k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: for n from 0 to 7 do seq(A104878(n,k), k=0..n) od; seq(seq(A104878(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Aug 21 2011

T(n, k) = if(k=1, n, if(k<=n, (k^(n-k+1)-1)/(k-1), 0));
G.f. of column k: x^k/((1-x)(1-k*x)). [corrected by Werner Schulte, Jun 05 2019]
T(n, k) = A069777(n+1,k)/A069777(n,k). [Johannes W. Meijer, Aug 21 2011]
T(n, k) = A055129(n+1-k, k) for n >= k > 0. - Mathew Englander, Dec 19 2020

A015007 q-factorial numbers for q=8.

Original entry on oeis.org

1, 1, 9, 657, 384345, 1799118945, 67375205371305, 20185139902805378865, 48378633136349277767794425, 927610024989668734297857360967425, 142287668466497494704440569679875994730825, 174605966461872393482359052970987514818406771638225

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015005, A015006, A015008, A015009, A015011.
Column q=8 of A069777.
Cf. A023001, A132036.

[n le 1 select 1 else (8^n-1)*Self(n-1)/7: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==((8^n - 1) * a[n-1])/7}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 8], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} ((q^k - 1) / (q - 1)), with q=8.
a(0) = 1, a(n) = (8^n-1)*a(n-1)/7. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A023001(k).
a(n) ~ c * 8^(n*(n+1)/2)/7^n, where c = A132036. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A052379 Number of integers from 1 to 10^(n+1)-1 that lack 0 and 1 as a digit.

Original entry on oeis.org

8, 72, 584, 4680, 37448, 299592, 2396744, 19173960, 153391688, 1227133512, 9817068104, 78536544840, 628292358728, 5026338869832, 40210710958664, 321685687669320, 2573485501354568, 20587884010836552, 164703072086692424, 1317624576693539400, 10540996613548315208

Offset: 0

Odimar Fabeny, Mar 12 2000

			For n=1, the numbers from 1 to 99 which have 0 or 1 as a digit are the numbers 1 and 10, 20, 30, ..., 90 and 11, 12, ..., 18, 19 and 21, 31, ..., 91. So a(1) = 99 - 27 = 72.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A023001, A024101, A052386.

[(8^(n+2)-1)/7-1: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

A052379:=n->(8^(n+2)-1)/7-1: seq(A052379(n), n=0..20); # Wesley Ivan Hurt, Sep 26 2014

(8^(Range[0,20]+2)-1)/7-1 (* or *) LinearRecurrence[{9,-8},{8,72},20] (* Harvey P. Dale, Sep 22 2013 *)

a(n)=8^(n+2)\7 - 1 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = (8^(n+2) - 1)/7 - 1.
G.f.: 8/((1-x)*(1-8*x)). - R. J. Mathar, Nov 19 2007
a(n) = 8*a(n-1) + 8.
a(n) = Sum_{k=1..n} 8^k. - corrected by Michel Marcus, Sep 25 2014
Conjecture: a(n) = A023001(n+2)-1. - R. J. Mathar, May 18 2007. Comment from Vim Wenders, Mar 26 2008: The conjecture is true: the g.f. leads to the closed form a(n) = -(8/7)*(1^n) + (64/7)*(8^n) = (-8 + 64*8^n)/7 = (-8 + 8^(n+2))/7 = (8^(n+2) - 1)/7 - 1 = A023001(n+2) - 1.
a(n) = 9*a(n-1) - 8*a(n-2); a(0)=8, a(1)=72. - Harvey P. Dale, Sep 22 2013
a(n) = 8*A023001(n+1). - Alois P. Heinz, Feb 15 2023

More terms and revised description from James Sellers, Mar 13 2000

A016208 Expansion of 1/((1-x)(1-3x)(1-4x)).

Original entry on oeis.org

1, 8, 45, 220, 1001, 4368, 18565, 77540, 320001, 1309528, 5326685, 21572460, 87087001, 350739488, 1410132405, 5662052980, 22712782001, 91044838248, 364760483725, 1460785327100, 5848371485001, 23409176469808, 93683777468645, 374876324642820, 1499928942876001

Offset: 0

N. J. A. Sloane

Binomial transform of A085277. - Paul Barry, Jun 25 2003

Number of walks of length 2n+5 between two nodes at distance 5 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
Index entries for linear recurrences with constant coefficients, signature (8,-19,12)

Cf. A000225, A000295, A000392, A002275, A003462, A003463, A003464, A023000, A023001, A002452, A016123, A016125, A016256.

a:=[1,8,45];; for n in [4..30] do a[n]:=8*a[n-1]-19*a[n-2]+12*a[n-3]; od; Print(a); # Muniru A Asiru, Apr 19 2019

Table[(2^(2*n + 3) - 3^(n + 2) + 1)/6, {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-x)(1-3x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[ {8,-19,12},{1,8,45},30] (* Harvey P. Dale, Apr 09 2012 *)

Vec(1/((1-x)*(1-3*x)*(1-4*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = 16*4^n/3 + 1/6 - 9*3^n/2. - Paul Barry, Jun 25 2003
a(0) = 0, a(1) = 8, a(n) = 7*a(n-1) - 12*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
a(0) = 1, a(1) = 8, a(2) = 45, a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3). - Harvey P. Dale, Apr 09 2012

A016209 Expansion of 1/((1-x)(1-3x)(1-5x)).

Original entry on oeis.org

1, 9, 58, 330, 1771, 9219, 47188, 239220, 1205941, 6059229, 30384718, 152189310, 761743711, 3811110039, 19062724648, 95335146600, 476740303081, 2383895225649, 11920057258978, 59602029687090

Offset: 0

N. J. A. Sloane

For a combinatorial interpretation following from a(n) = A039755(n+2,2) = h^{(3)}A039755.%20-%20_Wolfdieter%20Lang">n, the complete homogeneous symmetric function of degree n in the symbols {1, 3, 5} see A039755. - _Wolfdieter Lang, May 26 2017

			a(2) = h^{(3)}_2 = 1^2 + 3^2 + 5^2 + 1^1*(3^1 + 5^1) + 3^1*5^1 = 58. - _Wolfdieter Lang_, May 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A016218, A016208, A000392, A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256, A039755, A021424.

[(5^(n+2)-2*3^(n+2)+1)/8: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

A016209 := proc(n) (5^(n+2)-2*3^(n+2)+1)/8; end proc: # R. J. Mathar, Mar 22 2011

Join[{a=1,b=9},Table[c=8*b-15*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2011 *)
CoefficientList[Series[1/((1-x)(1-3x)(1-5x)),{x,0,30}],x] (* or *) LinearRecurrence[ {9,-23,15},{1,9,58},30] (* Harvey P. Dale, Feb 20 2020 *)

PARI
```
a(n)=if(n<0,0,n+=2; (5^n-2*3^n+1)/8)
```

a(n) = A039755(n+2, 2).
a(n) = (5^(n+2) - 2*3^(n+2)+1)/8 = a(n-1) + A005059(n+1) = 8*a(n-1) - 15*a(n-2) + 1 = (A003463(n+2) - A003462(n+2))/2. - Henry Bottomley, Jun 06 2000
G.f.: 1/((1-x)(1-3*x)(1-5*x)). See the name.
E.g.f.: (25*exp(5*x) - 18*exp(3*x) + exp(x))/8, from the e.g.f. of the third column (k=2) of A039755. - Wolfdieter Lang, May 26 2017

cp's OEIS Frontend

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

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A125118 Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

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A295235 Numbers k such that the positions of the ones in the binary representation of k are in arithmetic progression.

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A033138 a(n) = floor(2^(n+2)/7).

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A014827 a(1)=1, a(n) = 5*a(n-1) + n.

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A104878 A sum-of-powers number triangle.

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A015007 q-factorial numbers for q=8.

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

A016208 Expansion of 1/((1-x)(1-3x)(1-4x)).