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 1-10 of 17 results. Next

A027014 a(n) = T(2*n, n+3), T given by A027011.

Original entry on oeis.org

1, 10, 203, 1719, 9484, 40615, 147760, 481849, 1458335, 4194686, 11658332, 31674918, 84807212, 224985864, 593525255, 1560542957, 4095205128, 10735046293, 28123686540, 73654666767, 192865240859, 504973920796, 1322099323816, 3461379173004, 9062108456296
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • PARI
    Vec(x^3*(1+146*x^2+14*x^3+134*x^4-174*x^5+105*x^6-43*x^7+10*x^8-x^9)/((1-x)^7*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

Formula

G.f.: x^3*(1+146*x^2+14*x^3+134*x^4-174*x^5+105*x^6-43*x^7+10*x^8-x^9) / ((1-x)^7*(1-3*x+x^2)). - Colin Barker, Feb 19 2016

A027018 a(n) = T(2*n+1, n+3), T given by A027011.

Original entry on oeis.org

1, 9, 150, 1085, 5283, 20495, 69007, 212020, 613633, 1708508, 4640978, 12414802, 32903418, 86731043, 227905816, 597838223, 1566763325, 4103989113, 10747219441, 28140274566, 73676929931, 192894712070, 505012447636, 1322149114676, 3461442847524, 9062189100301
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000032, A027011.

Programs

  • Magma
    A027018:= func< n | n eq 2 select 1 else Lucas(2*n+8) -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)/30 >;
[A027018(n): n in [2..50]]; // G. C. Greubel, Jun 16 2025
  • Mathematica
    Table[LucasL[2*n+8] -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)/30 + Boole[n==2], {n,2,50}] (* G. C. Greubel, Jun 16 2025 *)
  • PARI
    Vec(x^2*(1+103*x^2-30*x^3+69*x^4-73*x^5+34*x^6-9*x^7+x^8)/((1-x)^6*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016
  • SageMath
    def A027018(n): return lucas_number2(2*n+8,1,-1) -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)//30 + int(n==2)
print([A027018(n) for n in range(2,51)]) # G. C. Greubel, Jun 16 2025

Formula

G.f.: x^2*(1+103*x^2-30*x^3+69*x^4-73*x^5+34*x^6-9*x^7+x^8) / ((1-x)^6*(1-3*x+x^2)). - Colin Barker, Feb 19 2016
From G. C. Greubel, Jun 16 2025: (Start)
a(n) = A000032(2*n+8) - (1/30)*(1410 + 1351*n + 655*n^2 + 230*n^3 + 20*n^4 + 24*n^5) + [n=2].
E.g.f.: exp(3*x/2)*( 47*cosh(sqrt(5)*x/2) + 21*sqrt(5)*sinh(sqrt(5)*x/2) ) + x^2/2 - (1/30)*(1410 + 2280*x + 1845*x^2 + 950*x^3 + 260*x^4 + 24*x^5)*exp(x). (End)

A027012 a(n) = T(2*n, n+1), T given by A027011.

Original entry on oeis.org

1, 6, 47, 199, 661, 1954, 5442, 14696, 39065, 103025, 270655, 709716, 1859412, 4869594, 12750611, 33383659, 87401977, 228824086, 599072310, 1568395100, 4106115485, 10749954101, 28143749827, 73681298664, 192900149736, 505019154414, 1322157317687
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Mathematica
    Join[{1},LinearRecurrence[{6,-13,13,-6,1},{6,47,199,661,1954},30]] (* Harvey P. Dale, Nov 17 2013 *)
  • PARI
    Vec(x*(1+24*x^2-18*x^3+6*x^4-x^5)/((1-x)^3*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

Formula

a(1)=1, a(n) = Lucas(2*n+6) - (6*n^2+17*n+18). - Ralf Stephan, May 05 2005
From Colin Barker, Feb 19 2016: (Start)
a(n) = -8 + (2^(-1-n)*((3-sqrt(5))^n*(-15+7*sqrt(5))+(3+sqrt(5))^n*(15+7*sqrt(5))))/sqrt(5) + 13*(1+n) - 6*(1+n)*(2+n) for n>1.
a(n) = 6*a(n-1)-13*a(n-2)+13*a(n-3)-6*a(n-4)+a(n-5) for n>6.
G.f.: x*(1+24*x^2-18*x^3+6*x^4-x^5) / ((1-x)^3*(1-3*x+x^2)).
(End)

Extensions

More terms from Harvey P. Dale, Nov 17 2013

A027013 a(n) = T(2*n, n+2), T given by A027011.

Original entry on oeis.org

1, 8, 107, 654, 2801, 9859, 30869, 89951, 250780, 680665, 1818310, 4813018, 12674542, 33283434, 87272241, 228658744, 598864479, 1568137061, 4105798635, 10749568905, 28143285770, 73680744203, 192899492252, 505018380164, 1322156411756, 3461451749404
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{8,-26,45,-45,26,-8,1},{1,8,107,654,2801,9859,30869,89951},30] (* Harvey P. Dale, Oct 08 2018 *)
  • PARI
    Vec(x^2*(1+69*x^2-39*x^3+36*x^4-26*x^5+8*x^6-x^7)/((1-x)^5*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

Formula

G.f.: x^2*(1+69*x^2-39*x^3+36*x^4-26*x^5+8*x^6-x^7) / ((1-x)^5*(1-3*x+x^2)). - Colin Barker, Feb 19 2016

A027016 T(2n+1,n+1), T given by A027011.

Original entry on oeis.org

1, 5, 28, 98, 291, 806, 2164, 5729, 15072, 39542, 103615, 271370, 710568, 1860413, 4870756, 12751946, 33385179, 87403694, 228826012, 599074457, 1568397480, 4106118110, 10749956983, 28143752978, 73681302096, 192900153461, 505019158444, 1322157322034
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{5,-8,5,-1},{1,5,28,98,291},30] (* Harvey P. Dale, Aug 08 2019 *)
  • PARI
    Vec((1+11*x^2-7*x^3+x^4)/((1-x)^2*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 17 2016

Formula

For n>0, a(n) = Fibonacci(2n+6) - Fibonacci(2n+2) - 6n - 7.
From Colin Barker, Feb 17 2016: (Start)
a(n) = 5*a(n-1)-8*a(n-2)+5*a(n-3)-a(n-4) for n>4.
G.f.: (1+11*x^2-7*x^3+x^4) / ((1-x)^2*(1-3*x+x^2)).
(End)

Extensions

a(22) - a(25) from Vincenzo Librandi, Apr 18 2011

A027017 a(n) = T(2*n+1, n+2), T given by A027011.

Original entry on oeis.org

1, 7, 73, 373, 1404, 4506, 13226, 36889, 99947, 266455, 704150, 1852212, 4860468, 12739243, 33369709, 87385081, 228803856, 599048334, 1568366942, 4106082685, 10749916175, 28143706267, 73681248938, 192900093288, 505019090664, 1322157246031, 3461452722961
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • PARI
    Vec(x*(1+43*x^2-31*x^3+17*x^4-7*x^5+x^6)/((1-x)^4*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

Formula

G.f.: x*(1+43*x^2-31*x^3+17*x^4-7*x^5+x^6) / ((1-x)^4*(1-3*x+x^2)). - Colin Barker, Feb 19 2016

A027019 a(n) = A027011(2n+1, n+4).

Original entry on oeis.org

1, 11, 267, 2620, 16305, 77040, 303553, 1055231, 3358268, 10039132, 28714818, 79649852, 216360858, 579601543, 1538746237, 4061993928, 10685640859, 28051744332, 73551897971, 192720956939, 504774508726, 1321827638248, 3461013838244, 9061623069576, 23724397134644
Offset: 3

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 21 2019

A027022 a(n) = Sum_{k=floor((n+1)/2)..n} T(k,n-k); i.e., a(n) is n-th diagonal sum of left-justified array T given by A027011.

Original entry on oeis.org

1, 3, 4, 7, 11, 15, 26, 34, 57, 79, 123, 181, 269, 406, 597, 900, 1332, 1991, 2968, 4414, 6596, 9805, 14639, 21792, 32488, 48418, 72130, 107532, 160191, 238776, 355785, 530211, 790156, 1177431, 1754739, 2614807, 3896754, 5806922, 8653577, 12895791
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027011.

Programs

  • Mathematica
    CoefficientList[Series[(-x^4 - 3 x^3 + x^2 + 3 x + 1) / ((1 - x^2) (1 - 2 x^2 - x^3 + x^4)), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 03 2017 *)

Formula

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

A027015 a(n) = A027011(2n, n+4).

Original entry on oeis.org

1, 12, 343, 3864, 26990, 140455, 599801, 2228414, 7486615, 23370193, 69141497, 196766283, 544680787, 1478898009, 3962865228, 10526310127, 27802386717, 73170829088, 192150938490, 503938185601, 1320621962194, 3459303317894, 9059231670566, 23721098636714
Offset: 4

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 21 2019

A027020 a(n) = greatest number in row n of array T given by A027011.

Original entry on oeis.org

1, 3, 4, 7, 15, 28, 47, 98, 199, 373, 661, 1404, 2801, 5283, 9859, 20495, 40615, 77040, 147760, 303553, 599801, 1143134, 2228414, 4544731, 8968421, 17160232, 33801192, 68602923, 135308317, 259763268, 515250948, 1042217402, 2055373383
Offset: 1

Views

Author

Clark Kimberling

Keywords

Showing 1-10 of 17 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.