A014825 -id:A014825 - cp's OEIS frontend

A108286 Triangle read by rows; columns are simple recursive sequences.

1, 3, 1, 6, 4, 1, 10, 11, 5, 1, 15, 26, 18, 6, 1, 21, 57, 58, 27, 7, 1, 28, 120, 179, 112, 38, 8, 1, 36, 247, 543, 453, 194, 51, 9, 1, 45, 502, 1636, 1818, 975, 310, 66, 10, 1

Offset: 1

Gary W. Adamson, May 31 2005

Left column = triangular numbers; Col. 2, (1, 4, 11...) = A000295; Col. 3, (1, 5, 18...) = A000340; Col. 4, (1, 6, 27...) = A014825; Col.5, (1, 7, 38...) = A014827.

			First few rows of the triangle are:
1;
3, 1;
6, 4, 1;
10, 11; 5, 1;
15, 26, 18, 6, 1;
21, 57, 58, 27, 7, 1;
...
3rd offset column: (1, 5, 18, 58...) = "1", then a(r) = 3*a(r-1) + r; e.g. 58 = 3*18 + 4 since 58 is the fourth term in the third column.

Cf. A108285, A000295, A000340, A014825, A014827.

r-th term in n-th column: initial "1", then a(r) = n*a(r-1) + r. Diagonals of A108285 become the columns of A108286.

A145766 Partial sums of A020988.

Original entry on oeis.org

0, 2, 12, 54, 224, 906, 3636, 14558, 58248, 233010, 932060, 3728262, 14913072, 59652314, 238609284, 954437166, 3817748696, 15270994818, 61083979308, 244335917270, 977343669120, 3909374676522, 15637498706132, 62549994824574

Offset: 0

Vladimir Joseph Stephan Orlovsky, Oct 18 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hacène Belbachir and El-Mehdi Mehiri, Enumerating moves in the optimal solution of the Tower of Hanoi, arXiv:2210.08657 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A014825, A020988.

lst={}; s=0; Do[s+=(s+=n+s); AppendTo[lst,s],{n,0,5!}]; lst
Accumulate[LinearRecurrence[{5,-4},{0,2},30]] (* or *) LinearRecurrence[ {6,-9,4},{0,2,12},30] (* Harvey P. Dale, Sep 25 2013 *)

a(n) = Sum_{i=0..n} A020988(i). a(n+1)-a(n)=A020988(n+1).
a(n) = 2*(4^(n+1)-3n-4)/9 = 2*A014825(n). - R. J. Mathar, Oct 21 2008
G.f.: 2*x/((1-x)^2*(1-4*x)). [Colin Barker, Jan 11 2012]
a(n) = 6*a(n-1)-9*a(n-2)+4*a(n-3), for n>2, with {a(k)}={0,2,12}, k=0,1,2. - L. Edson Jeffery, Mar 01 2012

Edited by R. J. Mathar, Oct 21 2008

A014851 Numbers k that divide s(k), where s(1)=1, s(j)=4*s(j-1)+j.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 10, 12, 18, 20, 21, 27, 30, 36, 42, 50, 54, 60, 63, 68, 78, 81, 84, 90, 100, 108, 110, 126, 147, 150, 156, 162, 171, 180, 189, 204, 210, 220, 234, 243, 250, 252, 270, 294, 300, 310, 324, 330, 340, 342, 378, 390, 410, 420, 441, 450, 468, 486

Offset: 1

N. J. A. Sloane, Olivier Gérard

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

s(n) = A014825(n).

nxt[{k_,a_}]:={k+1,4a+k+1}; Transpose[Select[NestList[nxt,{1,1},500], Divisible[#[[2]],#[[1]]]&]][[1]] (* Harvey P. Dale, May 29 2013 *)

A089000 Square table, read by antidiagonals, of coefficients T(k,n) (row k; column n) defined by: T(k,n) = k*T(k,n-1)+ n; T(k,0) = 0.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 11, 5, 1, 0, 6, 15, 26, 18, 6, 1, 0, 7, 21, 57, 58, 27, 7, 1, 0, 8, 28, 120, 179, 112, 38, 8, 1, 0, 9, 36, 247, 543, 453, 194, 51, 9, 1, 0, 10, 45, 502, 1636, 1818, 975, 310, 66, 10, 1, 0

Offset: 0

Philippe Deléham, Nov 02 2003

Rows begin:
{0, 1, 2, 3, 4, 5, 6, 7, 8, ...}:see A001477
{0, 1, 3, 6, 10, 15, 21, 28, ...} : see A000217
{0, 1, 4, 11, 26, 57, 120, 247, 502, ...} : see A000295
{0, 1, 5, 18, 58, 179, 543, 1636, ...} : see A000340
{0, 1, 6, 27, 112, 453, 1818, 7279, ...} : see A014825
{0, 1, 7, 38, 194, 975, 4881, 24412, ...} : see A014827
{0, 1, 8, 51, 310, 1865, 11196, 67183, ...}: see diagonals of triangle A088990
Diagonal begin:
{0, 1, 4, 18, 112, 975, 11196, ... } :see A062805
{0, 1, 5, 27, 194, 1865, ...} : see A023811
Column {3, 6, 11, 18, 27, 38, 51, ...} : see A010000

Unprotect[Power]; 0^0=1; T[n_,k_]:=Sum[j*k^(n-j),{j,0,n}]; Table[T[n-k,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Apr 19 2025 *)

T(k, n)= (k^(n+1)- (k-1)*n - k)/(k-1)^2. T(k, n) = Sum(j, 0<=j<=n; j*k^(n-j)).

A108285 Triangle read by rows, generated from (1, 2, 3, ...).

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 11, 10, 1, 6, 18, 26, 15, 1, 7, 27, 58, 57, 21, 1, 8, 38, 112, 179, 120, 28, 1, 9, 51, 194, 453, 543, 247, 36

Offset: 0

Gary W. Adamson, May 30 2005

By diagonals (d=1,2,3,...) going to the left with (1,3,6,...) = d(1), these are sequences of the form (k-th term a(k) = d*a(k-1) + k). Example: 1, 7, 38, 194, ... (the 5th diagonal) = A014827, is generated by a(k) = 5*a(k-1) + k. Diagonal 2 = (1, 4, 11, 26, ...) = A000295; Diagonal 3 = (1, 5, 18, ...) = A000340; Diagonal 4 = (1, 6, 27, ...) = A014825.

Triangle A108243 is generated by analogous operations from (..., 3, 2, 1) instead of (1, 2, 3, ...).

			4th column (offset) = 10, 26, 58, 112, ...= f(x), x = 1, 2, 3; x^3 + 2x^2 + 3x + 4.
First few rows of the triangle are:
  1;
  1, 3;
  1, 4, 6;
  1, 5, 11, 10;
  1, 6, 18, 26, 15;
  1, 7, 27, 58, 57, 21;
  1, 8, 38, 112, 179, 120, 28;
  ...

Cf. A108286, A000295, A000349, A014825, A000217, A108283, A108284, A108286.

n-th column = f(x), x = 1, 2, 3, ...; x^(n) + 2*x^(n-1) + 3*x^(n-2) + ... + (n+1).

A145655 Partial sums of A080674.

Original entry on oeis.org

4, 24, 108, 448, 1812, 7272, 29116, 116496, 466020, 1864120, 7456524, 29826144, 119304628, 477218568, 1908874332, 7635497392, 30541989636, 122167958616, 488671834540, 1954687338240, 7818749353044, 31274997412264

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 16 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (6, -9, 4).

Cf. A080674.

lst={};s=0;Do[s+=(s+=(s+=n));AppendTo[lst,s],{n,3*4!}];lst
Accumulate[Table[4(4^n-1)/3,{n,0,40}]] (* or *) LinearRecurrence[{6,-9,4},{0,4,24},40] (* Harvey P. Dale, Nov 27 2013 *)

a(n) = sum_{i=0..n} A080674(i). a(n+1)-a(n) = A080674(n+1).
a(n) = 4*(4^(n+1)-3n-4)/9 = 4*A014825(n). - R. J. Mathar, Oct 21 2008
G.f.: 4x/((1-x)^2(1-4x)). - R. J. Mathar, Oct 21 2008
a(1)=4, a(2)=24, a(3)=108, a(n)=6*a(n-1)-9*a(n-2)+4*a(n-3). - Harvey P. Dale, Nov 27 2013

Edited by R. J. Mathar, Oct 21 2008

A364378 Numbers whose representation in Jacobsthal greedy base (A265747) is palindromic.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 20, 22, 27, 36, 41, 44, 60, 68, 84, 86, 97, 112, 123, 132, 143, 158, 169, 172, 204, 220, 252, 260, 292, 308, 340, 342, 363, 396, 417, 432, 453, 486, 507, 516, 537, 570, 591, 606, 627, 660, 681, 684, 748, 780, 844, 860, 924, 956, 1020, 1028

Offset: 1

Amiram Eldar, Jul 21 2023

A128209(n) = A001045(n) + 1 is a term for n >= 3, since its representation is two 1's with n-3 0's between them.
A084639(n) is a term for n >= 1 since its representation is n 1's.
A014825(n) is a term for n >= 1 since its representation is n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A265747(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     2              2
   4     4             11
   5     6            101
   6     9            111
   7    12           1001
   8    20           1111
   9    22          10001
  10    27          10101

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001045, A014825, A084639, A128209, A265747.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A364214.

palJacobQ[n_] := PalindromeQ[A265747[n]]; Select[Range[0, 1000], palJacobQ] (* using A265747[n] *)

is(n) = {my(dig = digits(A265747(n))); dig == Vecrev(dig);} \\ using A265747(n)

A368530 a(n) = Sum_{k=1..n} k^3 * 4^(n-k).

Original entry on oeis.org

0, 1, 12, 75, 364, 1581, 6540, 26503, 106524, 426825, 1708300, 6834531, 27339852, 109361605, 437449164, 1749800031, 6999204220, 27996821793, 111987293004, 447949178875, 1791796723500, 7167186903261, 28668747623692, 114674990506935, 458699962041564

Offset: 0

Seiichi Manyama, Dec 29 2023

Index entries for linear recurrences with constant coefficients, signature (8,-22,28,-17,4).

Cf. A000578, A000537, A213575, A066999.
Cf. A097788, A368525.
Cf. A014825, A368529.

PARI
```
a(n) = sum(k=1, n, k^3*4^(n-k));
```

G.f.: x * (1+4*x+x^2)/((1-4*x) * (1-x)^4).
a(n) = 8*a(n-1) - 22*a(n-2) + 28*a(n-3) - 17*a(n-4) + 4*a(n-5).
a(n) = (11*4^(n+1) - (9*n^3 + 36*n^2 + 60*n + 44))/27.
a(0) = 0; a(n) = 4*a(n-1) + n^3.

A229463 Expansion of g.f. 1/((1-x)^2(1-26x)).

Original entry on oeis.org

1, 28, 731, 19010, 494265, 12850896, 334123303, 8687205886, 225867353045, 5872551179180, 152686330658691, 3969844597125978, 103215959525275441, 2683614947657161480, 69773988639086198495, 1814123704616241160886, 47167216320022270183053, 1226347624320579024759396

Offset: 0

Yahia Kahloune, Sep 24 2013

This sequence was chosen to illustrate a method of solution.

			a(3) = (26^5 - 25*3 - 51)/625 = 19010.

Index entries for linear recurrences with constant coefficients, signature (28,-53,26).

Cf. A000295, A000340, A014825, A014827, A014936.

my(x='x+O('x^18)); Vec(1/((1-26*x)*(1-x)^2)) \\ Elmo R. Oliveira, May 24 2025

a(n) = (26^(n+2) - 25*n - 51)/625.
In general, for the expansion of 1/((1-s*x)^2*(1-r*x)) with r>s>=1 we have the formula: a(n) = (r^(n+2)- s^(n+1)*((r-s)*n +(2*r-s)))/(r-s)^2.
From Elmo R. Oliveira, May 24 2025: (Start)
E.g.f.: exp(x)*(-51 - 25*x + 676*exp(25*x))/625.
a(n) = 28*a(n-1) - 53*a(n-2) + 26*a(n-3). (End)

A350717 a(n) = 4*a(n-1) - n - 1, for n > 0, a(0) = 1.

Original entry on oeis.org

1, 2, 5, 16, 59, 230, 913, 3644, 14567, 58258, 233021, 932072, 3728275, 14913086, 59652329, 238609300, 954437183, 3817748714, 15270994837, 61083979328, 244335917291, 977343669142, 3909374676545, 15637498706156, 62549994824599, 250199979298370, 1000799917193453, 4003199668773784

Offset: 0

Paul Curtz, Feb 03 2022

Last digit (using 0 to 9) is of period 10: repeat [1, 2, 5, 6, 9, 0, 3, 4, 7, 8].

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A007583 (first differences), A014825, A160156.

LinearRecurrence[{6, -9, 4}, {1, 2, 5}, 28] (* Amiram Eldar, Feb 03 2022 *)

a(n) = if (n, 4*a(n-1) - n - 1, 1); \\ Michel Marcus, Feb 03 2022

print([(2**(2*n+1) + 3*n + 7)//9 for n in range(30)])
# Gennady Eremin, Feb 05 2022

a(n) = (2^(2*n+1) + 3*n + 7)/9.
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3), n >= 3.
a(n) = a(n-1) + A007583(n-1).
a(n) = 2*a(n-1) + A014825(n-1).
G.f.: (-2*x^2 + 4*x - 1)/((x - 1)^2*(4*x - 1)). - Thomas Scheuerle, Feb 03 2022
a(n) = -1 + 5*a(n-1) - 4*a(n-2), n >= 2.
a(n) = 1 + A160156(n-1), n >= 1.

More terms from Michel Marcus, Feb 03 2022

cp's OEIS Frontend

A108286 Triangle read by rows; columns are simple recursive sequences.

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A145766 Partial sums of A020988.

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A014851 Numbers k that divide s(k), where s(1)=1, s(j)=4*s(j-1)+j.

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A089000 Square table, read by antidiagonals, of coefficients T(k,n) (row k; column n) defined by: T(k,n) = k*T(k,n-1)+ n; T(k,0) = 0.

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A108285 Triangle read by rows, generated from (1, 2, 3, ...).

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A145655 Partial sums of A080674.

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A364378 Numbers whose representation in Jacobsthal greedy base (A265747) is palindromic.

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PARI

A368530 a(n) = Sum_{k=1..n} k^3 * 4^(n-k).

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A229463 Expansion of g.f. 1/((1-x)^2*(1-26*x)).

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A229463 Expansion of g.f. 1/((1-x)^2(1-26x)).