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.

A289788 a(n) = (1/2)*A289787(n).

Original entry on oeis.org

1, 6, 31, 156, 785, 3954, 19919, 100344, 505489, 2546430, 12827791, 64620756, 325530881, 1639881066, 8260997855, 41615265264, 209639359969, 1056070674294, 5320018479679, 26799907726860, 135006120168881, 680101314855906, 3426050596003631, 17258932500172776
Offset: 0

Views

Author

Clark Kimberling, Aug 10 2017

Keywords

Links

Crossrefs

Cf. A289780, A289787.

Programs

  • Mathematica
    z = 60; s = 2*x/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005843 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289787 *)
u/2 (* A289788 *)

Formula

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

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).

A292480 p-INVERT of the odd positive integers, where p(S) = 1 - S^2.

Original entry on oeis.org

0, 1, 6, 20, 56, 160, 480, 1456, 4384, 13136, 39360, 118064, 354272, 1062928, 3188736, 9565936, 28697632, 86093264, 258280512, 774841520, 2324523104, 6973567888, 20920705152, 62762119792, 188286360736, 564859074896, 1694577214656, 5083731648560
Offset: 0

Views

Author

Clark Kimberling, Oct 02 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).
In the following guide to p-INVERT sequences using s = (1,3,5,7,9,...) = A005408, in some cases t(1,3,5,7,9,...) is a shifted (or differently indexed) version of the cited sequence:
p(S) *********** t(1,3,5,7,9,...)
1 - S A003946
1 - S^2 A292480
1 - S^3 (not yet in OEIS)
(1 - S)^2 (not yet in OEIS)
(1 - S)^3 (not yet in OEIS)
1 - S - S^2 A289786
1 + S - S^2 A289484
1 - S - 2 S^2 A289785
1 - S - 3 S^2 A289786
1 - S - 4 S^2 A289787
1 - S - 5 S^2 A289788
1 - S - 6 S^2 A289789
1 - S - 7 S^2 A289790
1 + S - 2 S^2 A289791
1 - S + S^2 - S^3 A289792
1 + S - 3 S^2 A289793
1 - S - S^2 - S^3 A289794

Examples

			s = (1,3,5,7,9,...), S(x) = x + 3 x^2 + 5 x^3 + 7 x^4 + ...,
p(S(x)) = 1 - ( x + 3 x^2 + 5 x^3 + 7 x^4 + ...)^2,
1/p(S(x)) = 1 + x^2 + 6 x^3 + 20 x^4 + 56 x^5 + ...
T(x) = (-1 + 1/p(S(x)))/x = x + 6 x^2 + 20 x^3 + 56 x^4 + ...
t(s) = (0,1,2,20,56,...).

Links

Crossrefs

Cf. A005408, A292479.

Programs

  • Magma
    I:=[0,1,6,20]; [n le 4 select I[n] else 4*Self(n-1)- 5*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 03 2017
  • Mathematica
    z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292480 *)
Join[{0}, LinearRecurrence[{4, -5, 6}, {1, 6, 20}, 30]] (* Vincenzo Librandi, Oct 03 2017 *)

Formula

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

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