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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.

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A143326 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words with length less than or equal to n (n,k >= 1).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 10, 1, 5, 16, 33, 22, 1, 6, 25, 76, 105, 52, 1, 7, 36, 145, 316, 345, 106, 1, 8, 49, 246, 745, 1336, 1041, 232, 1, 9, 64, 385, 1506, 3865, 5356, 3225, 472, 1, 10, 81, 568, 2737, 9276, 19345, 21736, 9705, 976, 1, 11, 100, 801, 4600, 19537, 55686
Offset: 1

Views

Author

Alois P. Heinz, Aug 07 2008

Keywords

Comments

The coefficients of the polynomial of row n are given by the n-th row of triangle A134541; for example row 4 has polynomial -k+k^3+k^4.

Examples

			T(2,3) = 9, because there are 9 primitive words of length less than or equal to 2 over 3-letter alphabet {a,b,c}: a, b, c, ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
  1,   2,    3,     4,      5,       6,       7,        8, ...
  1,   4,    9,    16,     25,      36,      49,       64, ...
  1,  10,   33,    76,    145,     246,     385,      568, ...
  1,  22,  105,   316,    745,    1506,    2737,     4600, ...
  1,  52,  345,  1336,   3865,    9276,   19537,    37360, ...
  1, 106, 1041,  5356,  19345,   55686,  136801,   298936, ...
  1, 232, 3225, 21736,  97465,  335616,  960337,  2396080, ...
  1, 472, 9705, 87016, 487465, 2013936, 6722737, 19169200, ...
  ...
From _Wolfdieter Lang_, Feb 01 2014: (Start)
The triangle Tri(n,m) := T(m,n-(m-1)) begins:
n\m  1   2    3     4     5      6      7     8    9  10 ...
1:   1
2:   2   1
3:   3   4    1
4:   4   9   10     1
5:   5  16   33    22     1
6:   6  25   76   105    52      1
7:   7  36  145   316   345    106      1
8:   8  49  246   745  1336   1041    232     1
9:   9  64  385  1506  3865   5356   3225   472    1
10: 10  81  568  2737  9276  19345  21736  9705  976   1
...
For the columns see A000027, A000290, A081437, ... (End)

Links

Crossrefs

Column 1: A000012. Rows 1-3: A000027, A000290, A081437 and A085490. See also A143324, A143327, A134541, A008683.

Programs

  • Maple
    with(numtheory):
f0:= proc(n) option remember;
       unapply(k^n-add(f0(d)(k), d=divisors(n) minus {n}), k)
     end:
g0:= proc(n) option remember; unapply(add(f0(j)(x), j=1..n), x) end:
T:= (n, k)-> g0(n)(k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);
  • Mathematica
    f0[n_] := f0[n] = Function[k, k^n-Sum[f0[d][k], {d, Divisors[n] // Most}]]; g0[n_] := g0[n] = Function[x, Sum[f0[j][x], {j, 1, n}]]; T[n_, k_] := g0[n][k]; Table[T[n, 1+d-n], {d, 1, 12}, {n, 1, d}]//Flatten (* Jean-François Alcover, Feb 12 2014, translated from Maple *)

Formula

T(n,k) = Sum_{1<=j<=n} Sum_{d|j} k^d * mu(j/d).
T(n,k) = Sum_{1<=j<=n} A143324(j,k).
T(n,k) = A143327(n,k) * k.

A270109 a(n) = n^3 + (n+1)*(n+2).

Original entry on oeis.org

2, 7, 20, 47, 94, 167, 272, 415, 602, 839, 1132, 1487, 1910, 2407, 2984, 3647, 4402, 5255, 6212, 7279, 8462, 9767, 11200, 12767, 14474, 16327, 18332, 20495, 22822, 25319, 27992, 30847, 33890, 37127, 40564, 44207, 48062, 52135, 56432, 60959, 65722, 70727, 75980, 81487, 87254
Offset: 0

Views

Author

Bruno Berselli, Mar 11 2016, at the suggestion of Giuseppe Amoruso in BASE Cinque forum

Keywords

Comments

For n>1, many consecutive terms of the sequence are generated by floor(sqrt(n^2 + 2)^3) + n^2 + 2.
It appears that this is a subsequence of A000037 (the nonsquares).
The primes in the sequence belong to A045326.
Inverse binomial transform is 2, 5, 8, 6, 0, 0, 0, ... (0 continued).

Links

Crossrefs

Subsequence of A001651, A047212.
Cf. A000037, A045326.
Cf. A027444: numbers of the form n^3+n*(n+1); A085490: numbers of the form n^3+(n-1)*n.
Cf. A008865: numbers of the form n+(n+1)*(n+2); A130883: numbers of the form n^2+(n+1)*(n+2).

Programs

  • Magma
    [n^3+(n+1)*(n+2): n in [0..50]];
  • Mathematica
    Table[n^3 + (n + 1) (n + 2), {n, 0, 50}]
  • Maxima
    makelist(n^3+(n+1)*(n+2), n, 0, 50);
  • PARI
    vector(50, n, n--; n^3+(n+1)*(n+2))
  • Sage
    [n^3+(n+1)*(n+2) for n in (0..50)]

Formula

O.g.f.: (2 - x + 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (2 + x)*(1 + x)^2*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3.
a(n+h) - a(n) + a(n-h) = n^3 + n^2 + (6*h^2+3)*n + (2*h^2+2) for any h. This identity becomes a(n) = n^3 + n^2 + 3*n + 2 if h=0.
a(h*a(n) + n) = (h*a(n))^3 + (3*n+1)*(h*a(n))^2 + (3*n^2+2*n+3)*(h*a(n)) + a(n) for any h, therefore a(h*a(n) + n) is always a multiple of a(n).
a(n) + a(-n) = 2*A059100(n) = A255843(n).
a(n) - a(-n) = 4*A229183(n).

A247236 Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+k)^k.

Original entry on oeis.org

1, -1, 2, -1, -10, 3, -1, 26, -33, 4, -1, -54, 207, -76, 5, -1, 96, -993, 824, -145, 6, -1, -156, 4047, -6736, 2375, -246, 7, -1, 236, -14769, 46184, -28985, 5634, -385, 8, -1, -340, 49743, -280408, 293575, -95166, 11711, -568, 9, -1, 470, -157617, 1556672, -2609465, 1322334, -260449, 22112, -801, 10
Offset: 0

Views

Author

Derek Orr, Nov 27 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^0 + T(n,1)*(x+1)^1 + T(n,2)*(x+2)^2 + ... + T(n,n)*(x+n)^n, for n >= 0.

Examples

			From _Wolfdieter Lang_, Jan 12 2015: (Start)
The triangle T(n,k) starts:
n\k 0    1       2       3        4       5       6     7    8  9 ...
0:  1
1: -1    2
2: -1  -10       3
3: -1   26     -33       4
4: -1  -54     207     -76        5
5: -1   96    -993     824     -145       6
6: -1 -156    4047   -6736     2375    -246       7
7: -1  236  -14769   46184   -28985    5634    -385     8
8: -1 -340   49743 -280408   293575  -95166   11711  -568    9
9: -1  470 -157617 1556672 -2609465 1322334 -260449 22112 -801 10
... Reformatted.
---------------------------------------------------------------------
n=3: 1 + 2*x + 3*x^2 + 4*x^3 = -1*(x+0)^0 + 26*(x+1)^1 - 33*(x+2)^2 + 4*(x+3)^3. (End)

Crossrefs

Cf. A248345, A000027, A002717, A085490.

Programs

  • PARI
    T(n,k)=(k+1)-sum(i=k+1,n,i^(i-k)*binomial(i,k)*T(n,i))
for(n=0,10,for(k=0,n,print1(T(n,k),", ")))

Formula

T(n,n) = n+1 = A000027(n+1), n >= 0.
T(n,1) = ((-1)^n*(1-4*n^3-10*n^2-4*n)-1)/8 = 2*(-1)^(n+1)*A002717(n), for n >= 1.
T(n,n-1) = n - n^2 - n^3 (A085490), for n >= 1.
T(n,n-2) = (n^5-2*n^4-3*n^3+6*n^2-2)/2, for n >= 2.

Extensions

Edited. - Wolfdieter Lang, Jan 12 2015

A214731 a(n) = n^3 - 2*n^2 - 1.

Original entry on oeis.org

-2, -1, 8, 31, 74, 143, 244, 383, 566, 799, 1088, 1439, 1858, 2351, 2924, 3583, 4334, 5183, 6136, 7199, 8378, 9679, 11108, 12671, 14374, 16223, 18224, 20383, 22706, 25199, 27868, 30719, 33758, 36991, 40424, 44063, 47914, 51983, 56276, 60799, 65558, 70559
Offset: 1

Views

Author

Marco Piazzalunga, Jul 27 2012

Keywords

Links

Crossrefs

Cf. A080859, A085490, A144390 (first differences), A152619.
Similar sequences: A152015 (of the type m^3+2m^2-1), A081437 (m^3-2m^2+1).

Programs

Formula

From Bruno Berselli, Jul 27 2012: (Start)
G.f.: -x*(2-7*x-x^3)/(1-x)^4.
a(n) = A085490(n-1) + 2.
a(n) = A152619(n-2) - 1 for n>1.
a(n) - a(n-2) = A080859(n-2) - 1 for n>2. (End)
E.g.f.: 1 - (1-x)*(1+x)^2*exp(x). - G. C. Greubel, Dec 31 2023

Extensions

a(3) corrected by Charles R Greathouse IV, Jul 27 2012
Showing 1-4 of 4 results.

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