A113444 -id:A113444 - cp's OEIS frontend

A113435 a(n) = a(n-1) + Sum_{k=0..n/3} a(n-3k) with a(0)=1.

1, 1, 1, 2, 3, 4, 7, 11, 16, 26, 41, 62, 98, 154, 237, 371, 581, 901, 1406, 2197, 3418, 5329, 8317, 12956, 20196, 31501, 49096, 76532, 119338, 186029, 289997, 452141, 704861, 1098826, 1713111, 2670692, 4163483, 6490879, 10119152, 15775426

Offset: 0

Floor van Lamoen, Nov 04 2005

If presented in three rows a(3n), a(3n+1) and a(3n+2) each term is the sum of the previous term in the sequence and the partial sum of its row.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 19.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-1).

Cf. A113439, A113444, A028495.

Partial sums of A176848.

CoefficientList[Series[(1 - x^3)/(1 - x - 2*x^3 + x^4), {x,0,50}], x] (* G. C. Greubel, Mar 10 2017 *)
LinearRecurrence[{1,0,2,-1},{1,1,1,2},40] (* Harvey P. Dale, Dec 17 2023 *)

x='x+O(x^50); Vec((1 - x^3)/(1 - x - 2*x^3 + x^4)) \\ G. C. Greubel, Mar 10 2017

a(n) = a(n-1) + 2*a(n-3) - a(n-4) = 7*a(n-3) - 5*a(n-6) + 11*a(n-9) - a(n-12).
G.f.: (1-x^3)/(1-x-2*x^3+x^4).
G.f.: 1/(1-x) + x^3*Q(0)/(2-2*x) , where Q(k) = 1 + 1/(1 - x*(4*k+1 + 2*x^2 - x^3)/( x*(4*k+3 + 2*x^2 - x^3 ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 11 2013

A113439 a(n) = a(n-1) + Sum_{k=1..floor(n/4)} a(n-4k), with a(0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 12, 17, 23, 34, 50, 72, 101, 146, 212, 306, 436, 627, 905, 1305, 1871, 2689, 3872, 5577, 8014, 11521, 16576, 23858, 34309, 49337, 70968, 102108, 146868, 211233, 303832, 437080, 628708, 904306, 1300737, 1871065, 2691401

Offset: 0

Floor van Lamoen, Nov 04 2005

If presented in four rows a(4k), a(4k+1), a(4k+2) and a(4k+3), each term is the sum of the previous term in the sequence and the partial sum of its row; see Example section.

			From _Jon E. Schoenfield_, Mar 11 2017: (Start)
Table of values T(j,k) = a(4k+j) in 4 rows:
.
  j | k=0     1     2     3     4     5     6     7
----+--------------------------------------------------
  0 |   1     2     8    34   146   627  2689 11521 ...
  1 |   1     3    12    50   212   905  3872 16576 ...
  2 |   1     4    17    72   306  1305  5577 23858 ...
  3 |   1     5    23   101   436  1871  8014 34309 ...
.
    T(2,4) = T(1,4) + T(2,0) + T(2,1) + T(2,2) + T(2,3)
      306  =   212  +    1   +    4   +   17   +   72
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-1).

Cf. A028495, A113435, A113444.

CoefficientList[Series[(1 - x^4)/(1 - x - 2*x^4 + x^5), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
LinearRecurrence[{1,0,0,2,-1},{1,1,1,1,2},50] (* Harvey P. Dale, Nov 10 2019 *)

x='x+O('x^50); Vec((1-x^4)/(1-x-2*x^4+x^5)) \\ G. C. Greubel, Mar 11 2017

a(n) = a(n-1) + 2*a(n-4) - a(n-5).
a(n) = 9*a(n-4) - 28*a(n-8) + 38*a(n-12) - 20*a(n-16) +a(n-20).
G.f.: (1-x^4)/(1-x-2*x^4+x^5).

A113445 Triangle T(n,k) read by rows: consider the sequence a(m) = a(m-1) + sum_{0

Original entry on oeis.org

1, 3, -1, 5, -6, 1, 7, -15, 11, -1, 9, -28, 38, -20, 1, 11, -45, 90, -90, 37, -1, 13, -66, 175, -260, 207, -70, 1, 15, -91, 301, -595, 707, -469, 135, -1, 17, -120, 476, -1176, 1862, -1848, 1052, -264, 1, 19, -153, 708, -2100, 4158, -5502, 4692, -2340, 521, -1

Offset: 0

Floor van Lamoen, Nov 04 2005

The sequence a(m) is also the expansion of (1-x^n)/(1-x-2x^n+x^{n+1}).

Instead of b(i) = a(n*i) one can take b(i) = a(n*i+p) for p=1..n-1.

			For n=5 (A113444) the recurrence relation is b(i) = 11b(i-1)-45b(i-2) +90b(i-3)-90b(i-4)+37b(i-5)-b(i-6), so the fifth row reads 11, -45, 90, -90, 37, -1.

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0..2 give: A005408, -A000384, A007585(n-1) for n>=1. - Alois P. Heinz, Jul 16 2009

T:= (n,k)-> (-1)^k /(k+1)! *(1+k +(n-k) *2^(k+1)) *mul (n+j-k, j=1..k):
seq(seq(T(n,k), k=0..n), n=0..11);  # Alois P. Heinz, Jul 16 2009

From Alois P. Heinz, Jul 16 2009: (Start)
T(n,k) = (-1)^k/(k+1)! * (1+k+(n-k)*2^(k+1)) * Product_{j=1..k}(n+j-k).
G.f. of column k: (-1)^k * x^k * (1+(2^(k+1)-1)*x)/(1-x)^(k+2). (End)

More terms from Alois P. Heinz, Jul 16 2009

cp's OEIS Frontend

A113435 a(n) = a(n-1) + Sum_{k=0..n/3} a(n-3k) with a(0)=1.

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A113439 a(n) = a(n-1) + Sum_{k=1..floor(n/4)} a(n-4k), with a(0)=1.

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A113445 Triangle T(n,k) read by rows: consider the sequence a(m) = a(m-1) + sum_{0

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