A000383 -id:A000383 - cp's OEIS frontend

A249169 Fibonacci 16-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-16).

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65535, 131069, 262136, 524268, 1048528, 2097040, 4194048, 8388032, 16775936, 33551616, 67102720, 134204416, 268406784, 536809472, 1073610752, 2147205120, 4294377472, 8588689409

Offset: 15

Alan N. Inglis, Oct 22 2014

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).

Other k-step Fibonacci sequences: Cf. A000045, A000213, A000288, A000322, A000383, A060455, A123526, A127193, A127194, A168083, A207539, A168084, A220469, A220493.

a:= proc(n) option remember; `if`(n<15, 0,
      `if`(n=15, 1, add(a(n-j), j=1..16)))
    end:
seq(a(n), n=15..50);  # Alois P. Heinz, Oct 23 2014

CoefficientList[Series[-1 /(x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)

a(n) = a(n-1) + a(n-2) + ... + a(n-16).

G.f.: -x^15 / (x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5 +x^4+x^3+x^2+x-1). - Alois P. Heinz, Oct 23 2014

A345372 a(n) = Sum_{i=1..n} nac(i,n) where nac(i,n) is the n-th i-bonacci number. The n-th i-bonacci number here is equal to 1 for the first i terms, with subsequent terms equaling the sum of the previous n terms.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 60, 114, 217, 411, 780, 1481, 2820, 5379, 10288, 19720, 37884, 72924, 140640, 271695, 525698, 1018611, 1976276, 3838889, 7465191, 14531683, 28313776, 55214993, 107762464, 210477611, 411387724, 804609206, 1574671586, 3083549861, 6041628460

Offset: 1

Christoph B. Kassir, Jun 16 2021

nonn

a(n) is the sum of the first n elements of the n-th column of the following array:
  1, 1, 1, 1, 1, ... (1-bonacci numbers)
  1, 1, 2, 3, 5, ... (2-bonacci or Fibonacci numbers)
  1, 1, 1, 3, 5, ... (3-bonacci or tribonacci numbers)
  1, 1, 1, 1, 4, ... (4-bonacci or tetranacci numbers)
  ...
For n >= 3, this sequence is 2 + antidiagonal sums of A061451.

Cf. A000045, A000213, A000288, A000322, A000383, A060455, A061451.

b:= proc(i, n) option remember; `if`(n=0, 0,
      `if`(n<=i, 1, add(b(i, n-j), j=1..i)))
    end:
a:= n-> add(b(i, n), i=1..n):
seq(a(n), n=1..36);  # Alois P. Heinz, Jun 16 2021

b[i_, n_] := b[i, n] = If[n==0, 0,
     If[n<=i, 1, Sum[b[i, n-j], {j, 1, i}]]];
a[n_] := Sum[b[i, n], {i, 1, n}];
Table[a[n], {n, 1, 36}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

a(n) = Sum_{i=1..n} nac(i,n) where nac(i,n) = 1 if 1 <= n <= i, Sum_{k=1..i} nac(i,n-k) if n > i.

cp's OEIS Frontend

A249169 Fibonacci 16-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-16).

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