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.

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).

A289976 p-INVERT of (0,0,1,2,3,5,8,...), the Fibonacci numbers preceded by two zeros, where p(S) = 1 - S - S^2.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 9, 18, 36, 70, 137, 268, 522, 1017, 1980, 3852, 7492, 14568, 28321, 55051, 106999, 207952, 404134, 785366, 1526186, 2965752, 5763103, 11198858, 21761463, 42286357, 82169547, 159668921, 310262351, 602888757, 1171506956, 2276419286
Offset: 0

Views

Author

Clark Kimberling, Aug 21 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 A290890 for a guide to related sequences.

Links

Crossrefs

Cf. A000045, A289975, A289780.

Programs

  • Mathematica
    z = 60; s = x^3/(1 - x - x^2); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* 0,0,1,2,3,5,... *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A289976 *)

Formula

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

A289781 p-INVERT of the positive Fibonacci numbers (A000045), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 3, 9, 27, 80, 237, 701, 2073, 6129, 18120, 53569, 158367, 468181, 1384083, 4091760, 12096453, 35760689, 105719157, 312537041, 923951760, 2731474161, 8075043963, 23872213729, 70573310907, 208635540400, 616788246957, 1823408134821, 5390532719313
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).
See A289780 for a guide to related sequences.

Links

Crossrefs

Cf. A000045, A289780, A289975, A289976, A289977.

Programs

  • Mathematica
    z = 60; s = x/(1 - x - x^2); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000045 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289781 *)

Formula

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

A289977 p-INVERT of (0,0,0,1,2,3,5,8,...), the Fibonacci numbers preceded by three zeros, where p(S) = 1 - S - S^2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 7, 12, 23, 41, 77, 140, 258, 470, 861, 1570, 2867, 5225, 9526, 17352, 31607, 57547, 104766, 190684, 347029, 631476, 1148985, 2090427, 3803044, 6918379, 12585209, 22892932, 41641932, 75744383, 137772396, 250592150, 455792833, 829016539
Offset: 0

Views

Author

Clark Kimberling, Aug 21 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 A290890 for a guide to related sequences.

Links

Crossrefs

Cf. A000045, A289975, A289780.

Programs

  • Mathematica
    z = 60; s = x^4/(1 - x - x^2); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* 0,0,0,1,2,3,5,... *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A289977 *)
LinearRecurrence[{2,1,-2,0,-1,-1,0,1},{0,0,0,1,1,2,3,7},40] (* Harvey P. Dale, Jul 14 2018 *)
  • PARI
    concat(vector(3), Vec(x^3*(1 - x)*(1 - x^2 - x^3) / (1 - 2*x - x^2 + 2*x^3 + x^5 + x^6 - x^8) + O(x^50))) \\ Colin Barker, Aug 24 2017

Formula

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-5) - a(n-6) + a(n-8).
G.f.: x^3*(1 - x)*(1 - x^2 - x^3) / (1 - 2*x - x^2 + 2*x^3 + x^5 + x^6 - x^8). - Colin Barker, Aug 24 2017
Showing 1-4 of 4 results.

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