A106596 -id:A106596 - cp's OEIS frontend

A106580 Triangle T(n, k) = T(n, k-1) + Sum_{i >= 1} T(n-2i, k-i) with T(n,0)=1, read by rows.

1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 1, 2, 5, 7, 7, 1, 2, 5, 9, 12, 12, 1, 2, 5, 13, 22, 29, 29, 1, 2, 5, 13, 26, 41, 53, 53, 1, 2, 5, 13, 34, 65, 101, 130, 130, 1, 2, 5, 13, 34, 73, 129, 194, 247, 247, 1, 2, 5, 13, 34, 89, 185, 322, 481, 611, 611, 1, 2, 5, 13, 34, 89, 201, 386, 645, 945, 1192, 1192, 1, 2, 5, 13, 34, 89, 233, 514, 973, 1613, 2354, 2965, 2965

Offset: 0

N. J. A. Sloane, May 30 2005

Next term is previous term + terms directly above you on a vertical line.

An intermingling of two independent triangles, A106595 and A106596.

			Triangle begins as:
  1;
  1, 1;
  1, 2, 2;
  1, 2, 3,  3;
  1, 2, 5,  7,  7;
  1, 2, 5,  9, 12, 12;
  1, 2, 5, 13, 22, 29,  29;
  1, 2, 5, 13, 26, 41,  53,  53;
  1, 2, 5, 13, 34, 65, 101, 130, 130;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A106595, A106596.

A106580:= proc(n,k) option remember; if k=0 then 1; else A106580(n,k-1) + add(A106580(n-2*i, k-i), i=1..min(k, floor(n/2), n-k)); fi ; end: for n from 0 to 12 do for k from 0 to n do printf("%d, ", A106580(n,k)); od; od; # R. J. Mathar, May 02 2007

t[, 0]= 1; t[n, k_]:= t[n, k] = t[n, k-1] + Sum[t[n-2j, k-j], {j, 1, Min[k, Floor[n/2], n-k]}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jan 08 2014, after Maple *)

@CachedFunction
def T(n, k):
    if (k<0): return 0
    elif (k==0): return 1
    else: return T(n, k-1) + sum( T(n-2*j, k-j) for j in (1..min(k, n//2, n-k)))
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Sep 07 2021

T(n, k) = T(n, k-1) + Sum_{i>=1} T(n-2*i, k-i), with T(n, 0) = 1.

More terms from R. J. Mathar, May 02 2007

A107062 Union of set of Belgian-k numbers for k = 0..9 which begin with k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 20, 22, 25, 26, 30, 31, 33, 34, 35, 39, 40, 43, 44, 50, 52, 53, 55, 60, 61, 62, 66, 68, 70, 71, 77, 80, 86, 88, 90, 93, 99, 100, 101, 106, 110, 111, 113, 115, 121, 122, 130, 131, 137, 142, 155, 157, 161, 170, 171, 172, 178

Offset: 0

Eric Angelini, Jun 07 2005

See A106039 for definition and link.

Cf. A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043, A107070.

belgianKQ[n_] := Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - id[[2]], Plus @@ id]]]; Select[ Range@ 178, belgianKQ] (* Robert G. Wilson v, May 06 2011 *)

A107070 Numbers m with the following property. Suppose m = d1 d2 ... dk in base 10. Construct the sequence with first term d1 and successive differences d1 d2 ... dk d1 d2 ... dk d1 d2 ...; then this sequence has as its initial k digits d1 d2 ... dk and also contains the number m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 61, 71, 918, 3612, 5101, 8161, 12481, 51011, 248161, 361213, 5101111, 7141519, 8161723, 481617232, 2481617232, 4816172324, 5101111121, 24816172324, 51011111213, 71415192025, 612131516192, 816172324313, 3612131516192, 5101111121314, 6121315161920, 9181927283739

Offset: 1

Eric Angelini, Jun 07 2005

These are sometimes called Eric numbers or Belgian numbers. - N. J. A. Sloane, May 06 2011
Each digit {1..9} will produce a quasi-automorphic sequence. Thus this sequence is infinite. - Robert G. Wilson v, May 06 2011
The existence of the nine templates upon which the quasi-automorphic sequences are decided guarantees that no more than nine solutions exist for a given digit-length. The equidistribution of the ten base-ten digits within these templates predicts a long-term average of two solutions per digit-length. All nine solutions happen trivially for digit-length 1 (terms 1-9) and not again until digit-length 1899283 (terms 3594728-3594736). - Hans Havermann, May 27 2011, Aug 15 2011
The n-th term is prime for: n= 2, 3, 5, 7, 10, 11, 14, 15, 18, 19, 51, 55, 238, 907, 979, 1814, ..., . - Robert G. Wilson v, May 06 2011

			The following example shows why 61 is a member:
6.12.13.19.20.26.27.33.34.40.41.47.48.54.55.61... (sequence)
.6..1..6..1..6..1..6..1..6..1..6..1..6..1..6... (first differences)

E. Angelini, Belgian numbers.
E. Angelini, Belgian Numbers [Cached copy with permission]
J.-P. Davalan, Nombres belges [Includes applets to generate sequence]

Cf. A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043, A107062.

belgianDQ[n_] := Block[{id = IntegerDigits@ n, id1}, id1 = id[[1]]; MemberQ[ Accumulate@ Join[{0}, id], Mod[n - id1, Plus @@ id]] && id == Take[ Flatten[ IntegerDigits[ FoldList[#1 + #2 &, id1, id]]], Length@ id]] (* Robert G. Wilson v, May 06 2011 *)

Minor edits by N. J. A. Sloane, May 06 2011

cp's OEIS Frontend

A106580 Triangle T(n, k) = T(n, k-1) + Sum_{i >= 1} T(n-2i, k-i) with T(n,0)=1, read by rows.

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A107062 Union of set of Belgian-k numbers for k = 0..9 which begin with k.

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Author

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Mathematica

A107070 Numbers m with the following property. Suppose m = d1 d2 ... dk in base 10. Construct the sequence with first term d1 and successive differences d1 d2 ... dk d1 d2 ... dk d1 d2 ...; then this sequence has as its initial k digits d1 d2 ... dk and also contains the number m.

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Mathematica

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