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 1-3 of 3 results.

A289807 p-INVERT of (1,2,2,3,3,4,4,...) (A080513), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 13, 42, 133, 424, 1348, 4291, 13653, 43449, 138261, 439979, 1400101, 4455420, 14178073, 45117606, 143573662, 456881476, 1453892534, 4626590576, 14722780217, 46850970327, 149089600359, 474434334814, 1509749422360, 4804338875098, 15288412556740
Offset: 0

Views

Author

Clark Kimberling, Aug 12 2017

Keywords

Comments

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
See A289780 for a guide to related sequences.

Links

Crossrefs

Cf. A080513, A289780.

Programs

  • Mathematica
    z = 60; s = x (1 + x - x^2)/((1 - x)^2*(1 + x)); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A080513 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289807 *)
LinearRecurrence[{3,2,-5,1,2,-1},{1,4,13,42,133,424},30] (* Harvey P. Dale, Aug 20 2024 *)

Formula

G.f.: (1 + x - x^2)/(1 - 3 x - 2 x^2 + 5 x^3 - x^4 - 2 x^5 + x^6).
a(n) = 3*a(n-1) + 2*a(n-2) - 5*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).

A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 14, 47, 156, 517, 1714, 5684, 18851, 62520, 207349, 687676, 2280686, 7563923, 25085844, 83197513, 275925586, 915110636, 3034975799, 10065534960, 33382471801, 110713382644, 367182309614, 1217764693607, 4038731742156, 13394504020957, 44423039068114
Offset: 0

Views

Author

Clark Kimberling, Aug 10 2017

Keywords

Comments

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).
Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).
Guide to p-INVERT sequences using p(S) = 1 - S - S^2:
t(A000012) = t(1,1,1,1,1,1,1,...) = A001906
t(A000290) = t(1,4,9,16,25,36,...) = A289779
t(A000027) = t(1,2,3,4,5,6,7,8,...) = A289780
t(A000045) = t(1,2,3,5,8,13,21,...) = A289781
t(A000032) = t(2,1,3,4,7,11,14,...) = A289782
t(A000244) = t(1,3,9,27,81,243,...) = A289783
t(A000302) = t(1,4,16,64,256,...) = A289784
t(A000351) = t(1,5,25,125,625,...) = A289785
t(A005408) = t(1,3,5,7,9,11,13,...) = A289786
t(A005843) = t(2,4,6,8,10,12,14,...) = A289787
t(A016777) = t(1,4,7,10,13,16,...) = A289789
t(A016789) = t(2,5,8,11,14,17,...) = A289790
t(A008585) = t(3,6,9,12,15,18,...) = A289795
t(A000217) = t(1,3,6,10,15,21,...) = A289797
t(A000225) = t(1,3,7,15,31,63,...) = A289798
t(A000578) = t(1,8,27,64,625,...) = A289799
t(A000984) = t(1,2,6,20,70,252,...) = A289800
t(A000292) = t(1,4,10,20,35,56,...) = A289801
t(A002620) = t(1,2,4,6,9,12,16,...) = A289802
t(A001906) = t(1,3,8,21,55,144,...) = A289803
t(A001519) = t(1,1,2,5,13,34,...) = A289804
t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805
t(A008619) = t(1,1,2,2,3,3,4,4,...) = A289806
t(A080513) = t(1,2,2,3,3,4,4,5,...) = A289807
t(A133622) = t(1,2,1,3,1,4,1,5,...) = A289809
t(A000108) = t(1,1,2,5,14,42,...) = A081696
t(A081696) = t(1,1,3,9,29,97,...) = A289810
t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843
t(A175676) = t(1,0,0,2,0,0,3,0,...) = A289844
t(A079977) = t(1,0,1,0,2,0,3,...) = A289845
t(A059841) = t(1,0,1,0,1,0,1,...) = A289846
t(A000040) = t(2,3,5,7,11,13,...) = A289847
t(A008578) = t(1,2,3,5,7,11,13,...) = A289828
t(A000142) = t(1!, 2!, 3!, 4!, ...) = A289924
t(A000201) = t(1,3,4,6,8,9,11,...) = A289925
t(A001950) = t(2,5,7,10,13,15,...) = A289926
t(A014217) = t(1,2,4,6,11,17,29,...) = A289927
t(A000045*) = t(0,1,1,2,3,5,...) = A289975 (* indicates prepended 0's)
t(A000045*) = t(0,0,1,1,2,3,5,...) = A289976
t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977
t(A290990*) = t(0,1,2,3,4,5,...) = A290990
t(A290990*) = t(0,0,1,2,3,4,5,...) = A290991
t(A290990*) = t(0,0,01,2,3,4,5,...) = A290992

Examples

			Example 1:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S.
S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - (x + 2x^2 + 3x^3 + 4x^4 + ... )
- p(0) + 1/p(S(x)) = -1 + 1 + x + 3x^2 + 8x^3 + 21x^4 + ...
T(x) = 1 + 3x + 8x^2 + 21x^3 + ...
t(s) = (1,3,8,21,...) = A001906.
***
Example 2:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S - S^2.
S(x) =  x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - ( x + 2x^2 + 3x^3 + 4x^4 + ...) - ( x + 2x^2 + 3x^3 + 4x^4 + ...)^2
- p(0) + 1/p(S(x)) = -1 + 1 + x + 4x^2 + 14x^3 + 47x^4 + ...
T(x) = 1 + 4x + 14x^2 + 47x^3 + ...
t(s) = (1,4,14,47,...) = A289780.

Links

Crossrefs

Cf. A000027.

Programs

  • GAP
    P:=[1,4,14,47];; for n in [5..10^2] do P[n]:=5*P[n-1]-7*P[n-2]+5*P[n-3]-P[n-4]; od; P; # Muniru A Asiru, Sep 03 2017
  • Mathematica
    z = 60; s = x/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289780 *)
  • PARI
    x='x+O('x^99); Vec((1-x+x^2)/(1-5*x+7*x^2-5*x^3+x^4)) \\ Altug Alkan, Aug 13 2017

Formula

G.f.: (1 - x + x^2)/(1 - 5 x + 7 x^2 - 5 x^3 + x^4).
a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).

A341148 Triangle read by rows: T(n,k) is number of cubes in the k-th vertical slice of the polycube called "tower" described in A221529 where n is the longest side of its base, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 4, 3, 2, 7, 6, 4, 3, 12, 10, 7, 3, 3, 19, 17, 12, 9, 5, 4, 30, 26, 20, 13, 8, 4, 4, 45, 41, 31, 23, 16, 10, 5, 5, 67, 60, 48, 34, 25, 15, 11, 5, 5, 97, 89, 71, 55, 39, 28, 17, 12, 6, 6, 139, 127, 104, 78, 60, 40, 28, 17, 11, 6, 6, 195, 181, 149, 118, 89, 65, 45, 32, 21, 15, 7, 7
Offset: 1

Views

Author

Omar E. Pol, Feb 06 2021

Keywords

Comments

The row sums of triangle give A066186 because the correspondence divisor/part. For more information see A338156.
For further information about the tower see A221529.

Examples

			Triangle begins:
    1;
    2,   2;
    4,   3,   2;
    7,   6,   4,   3;
   12,  10,   7,   3,  3;
   19,  17,  12,   9,  5,  4;
   30,  26,  20,  13,  8,  4,  4;
   45,  41,  31,  23, 16, 10,  5,  5;
   67,  60,  48,  34, 25, 15, 11,  5,  5;
   97,  89,  71,  55, 39, 28, 17, 12,  6,  6;
  139, 127, 104,  78, 60, 40, 28, 17, 11,  6,  6;
  195, 181, 149, 118, 89, 65, 45, 32, 21, 15,  7,  7;
...
Illustration of initial terms:
              Top view
  n   k       of the tower       Heights        T(n,k)
               _
  1   1       |_|                1                 1
.              _ _
  2   1       |   |              1 1               2
  2   2       |_ _|              1 1               2
.              _ _ _
  3   1       |_|   |            2 1 1             4
  3   2       |    _|            1 1 1             3
  3   3       |_ _|              1 1               2
.              _ _ _ _
  4   1       |_| |   |          3 2 1 1           7
  4   2       |_ _|   |          2 2 1 1           6
  4   3       |      _|          1 1 1 1           4
  4   4       |_ _ _|            1 1 1             3
.              _ _ _ _ _
  5   1       |_| | |   |        5 3 2 1 1        12
  5   2       |_ _|_|   |        3 3 2 1 1        10
  5   3       |_ _|  _ _|        2 2 1 1 1         7
  5   4       |     |            1 1 1             3
  5   5       |_ _ _|            1 1 1             3
.              _ _ _ _ _ _
  6   1       |_| | | |   |      7 5 3 2 1 1      19
  6   2       |_ _|_| |   |      5 5 3 2 1 1      17
  6   3       |_ _|  _|   |      3 3 2 2 1 1      12
  6   4       |_ _ _|    _|      2 2 2 1 1 1       9
  6   5       |        _|        1 1 1 1 1         5
  6   6       |_ _ _ _|          1 1 1 1           4
.
The levels of the terraces of the tower are the partition numbers A000041 starting from the base.
Note that the top view of the tower is essentially the same as the top view of the stepped pyramid described in A245092 except that in the tower both the symmetric representation of sigma(n) and the symmetric representation of sigma(n-1) are unified in the level 1 of the structure because the first two partitions numbers A000041 are [1, 1].

Crossrefs

Column 1 gives A000070.
Leading diagonal gives A080513.
Row sums give A066186.
Cf. A000041, A024916, A236104, A237270, A237271, A237593, A221529, A245092, A262626, A336811, A336812, A338156, A340035.
Showing 1-3 of 3 results.

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