cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A122542 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 8, 6, 1, 0, 2, 12, 18, 8, 1, 0, 2, 16, 38, 32, 10, 1, 0, 2, 20, 66, 88, 50, 12, 1, 0, 2, 24, 102, 192, 170, 72, 14, 1, 0, 2, 28, 146, 360, 450, 292, 98, 16, 1, 0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1
Offset: 0

Views

Author

Philippe Deléham, Sep 19 2006, May 28 2007

Keywords

Comments

Riordan array (1, x*(1+x)/(1-x)). Rising and falling diagonals are the tribonacci numbers A000213, A001590.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 2,  1;
  0, 2,  4,   1;
  0, 2,  8,   6,   1;
  0, 2, 12,  18,   8,    1;
  0, 2, 16,  38,  32,   10,   1;
  0, 2, 20,  66,  88,   50,  12,   1;
  0, 2, 24, 102, 192,  170,  72,  14,   1;
  0, 2, 28, 146, 360,  450, 292,  98,  16,  1;
  0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

Links

Crossrefs

Other versions: A035607, A113413, A119800, A266213.
Diagonals: A000012, A005843, A001105, A035597 - A035606.
Columns: A000007, A040000, A008575, A005899, A008412-A008416, A008418, A008420, A035706 - A035745.
Sums include: A000007, A001333 (row), A001590 (diagonal), A007483, A057077 (signed row), A078016 (signed diagonal), A086901, A091928, A104934, A122558, A122690.
Cf. A155161, A059283.

Programs

  • Haskell
    a122542 n k = a122542_tabl !! n !! k
a122542_row n = a122542_tabl !! n
a122542_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [0, 1])
-- Reinhard Zumkeller, Jul 20 2013, Apr 17 2013
  • Magma
    function T(n, k) // T = A122542
  if k eq 0 then return 0^n;
  elif k eq n then return 1;
  else return T(n-1,k) + T(n-1,k-1) + T(n-2,k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024
  • Mathematica
    CoefficientList[#, y]& /@ CoefficientList[(1-x)/(1 - (1+y)x - y x^2) + O[x]^11, x] // Flatten (* Jean-François Alcover, Sep 09 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, 0, T[n-1,k-1] +T[n-1,k] +T[n-2,k- 1] ]]; (* T = A122542 *)
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 27 2024 *)
  • Sage
    def A122542_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (0..n)]
for n in (0..10): print(A122542_row(n)) # Peter Luschny, Mar 16 2016

Formula

Sum_{k=0..n} x^k*T(n,k) = A000007(n), A001333(n), A104934(n), A122558(n), A122690(n), A091928(n) for x = 0, 1, 2, 3, 4, 5. - Philippe Deléham, Jan 25 2012
Sum_{k=0..n} 3^(n-k)*T(n,k) = A086901(n).
Sum_{k=0..n} 2^(n-k)*T(n,k) = A007483(n-1), n >= 1. - Philippe Deléham, Oct 08 2006
T(2*n, n) = A123164(n).
T(n, k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), n > 1. - Philippe Deléham, Jan 25 2012
G.f.: (1-x)/(1-(1+y)*x-y*x^2). - Philippe Deléham, Mar 02 2012
From G. C. Greubel, Oct 27 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A057077(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001590(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A078016(n). (End)

A126528 Number of base 7 n-digit numbers with adjacent digits differing by five or less.

Original entry on oeis.org

1, 7, 47, 317, 2137, 14407, 97127, 654797, 4414417, 29760487, 200635007, 1352612477, 9118849897, 61476161767, 414451220087, 2794088129357, 18836784876577, 126991149906247, 856130823820367, 5771740692453437, 38911098273822457, 262325293105201927
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+11^(n-1) for base>=5n-4; a(base,n)=a(base-1,n)+11^(n-1)-2 when base=5n-5.
For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,6} containing no subwords 00 and 11. - Milan Janjic, Jan 31 2015

Links

Crossrefs

Cf. Base 7 differing by four or less A126502, three or less A126475, two or less A126394, one or less A126361.

Programs

  • Mathematica
    LinearRecurrence[{6, 5}, {1, 7}, 25] (* Paolo Xausa, Aug 08 2024 *)
  • PARI
    Vec((1+x)/(1-6*x-5*x^2) + O(x^30)) \\ Colin Barker, Sep 08 2016

Formula

From Philippe Deléham, Mar 24 2012: (Start)
G.f.: (1+x)/(1-6*x-5*x^2).
a(n) = 6*a(n-1) + 5*a(n-2), a(0) = 1, a(1) = 7 .
a(n) = Sum_{k=0..=n} A054458(n,k)*4^k.
(End)
a(n) = A091928(n+1)/5. - Philippe Deléham, Mar 27 2012
a(n) = (((3-sqrt(14))^n * (-4+sqrt(14)) + (3+sqrt(14))^n * (4+sqrt(14)))) / (2*sqrt(14)). - Colin Barker, Sep 08 2016

A182436 Triangle T(n,k), read by rows, given by (2, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 1, 2, 5, 2, 4, 8, 11, 4, 4, 20, 25, 24, 8, 8, 28, 70, 69, 52, 16, 8, 60, 126, 213, 178, 112, 32, 16, 80, 288, 460, 599, 440, 240, 64, 16, 160, 472, 1128, 1489, 1600, 1056, 512, 128, 32, 208, 976, 2152, 3914, 4457, 4120, 2480, 1088, 256
Offset: 0

Views

Author

Philippe Deléham, Apr 28 2012

Keywords

Comments

Row sums are the powers of 3.

Examples

			Triangle begins :
1
2, 1
2, 5, 2
4, 8, 11, 4
4, 20, 25, 24, 8
8, 28, 70, 69, 52, 16
8, 60, 126, 213, 178, 112, 32
16, 80, 288, 460, 599, 440, 240, 64
16, 160, 472, 1128, 1489, 1600, 1056, 512, 128
32, 208, 976, 2152, 3914, 4457, 4120, 2480, 1088, 256

Crossrefs

Cf. A016116, A011782, A111297, A000244

Formula

G.f.: (1+2*x-y*x)/(1-2*y*x-(2+y)*x^2).
T(n,k) = 2*T(n-1,k-1) + 2*T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = T(2,0) = T(2,2) = 2, T(2,1) = 5 and T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A123335(n-1), A016116(n+1), A000244(n), A057087(n), A091928(n) for x = -2, -1, 0, 1, 2, 3 respectively.

A210636 Riordan array ((1-x)/(1-2*x-x^2), x*(1+x)/(1-2*x-x^2)).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 7, 13, 7, 1, 17, 40, 32, 10, 1, 41, 117, 124, 60, 13, 1, 99, 332, 437, 286, 97, 16, 1, 239, 921, 1447, 1193, 553, 143, 19, 1, 577, 2512, 4584, 4556, 2682, 952, 198, 22, 1, 1393, 6761, 14048, 16336, 11666, 5282, 1510, 262, 25, 1
Offset: 0

Views

Author

Philippe Deléham, Mar 26 2012

Keywords

Comments

Triangle T(n,k), 0<=k<=n, read by rows, given by (1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Product of A122542 and A007318 (Pascal's triangle) as lower triangular matrices .

Examples

			Triangle begins :
1
1, 1
3, 4, 1
7, 13, 7, 1
17, 40, 32, 10, 1
41, 117, 124, 60, 13, 1
99, 332, 437, 286, 97, 16, 1
239, 921, 1447, 1193, 553, 143, 19, 1
577, 2512, 4584, 4556, 2682, 952, 198, 22, 1
1393, 6761, 14048, 16336, 11666, 5282, 1510, 262, 25, 1

Crossrefs

Cf. Columns :A001333, A119915, Diagonals : A000012, A016777, Antidiagonal sums : A077995

Formula

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-x)/(1-2*x-y*x-x^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A001333(n), A104934(n), A122958(n), A122690(n), A091928(n) for x = -1, 0, 1, 2, 3, 4 respectively.
Showing 1-4 of 4 results.

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