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.

Previous Showing 11-15 of 15 results.

A026625 a(n) = Sum_{k=0..floor(n/2)} A026615(n-k,k).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 21, 36, 58, 96, 155, 253, 409, 664, 1074, 1740, 2815, 4557, 7373, 11932, 19306, 31240, 50547, 81789, 132337, 214128, 346466, 560596, 907063, 1467661, 2374725, 3842388, 6217114, 10059504, 16276619, 26336125, 42612745, 68948872, 111561618
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000032, A000045, A026615, A026616, A026617, A026618, A026619.
Cf. A026620, A026621, A026622, A026623, A026624, A026956, A026957.
Cf. A026958, A026959, A026960.

Programs

  • Magma
    [n eq 1 select 1 else 3*Fibonacci(n+1) - 2*Fibonacci(n) - (3+(-1)^n)/2: n in [0..40]]; // G. C. Greubel, Jun 16 2024
  • Mathematica
    Join[{1,1},Table[Fibonacci[n-1]+LucasL[n]-(3+(-1)^n)/2,{n,2,40}]] (* or *) Join[{1,1},LinearRecurrence[{1,2,-1,-1},{2,4,7,13},40]] (* Harvey P. Dale, Sep 27 2011 *)
  • PARI
    Vec((1 - x^2 + x^3 + x^4 + x^5) / ((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017
  • SageMath
    [3*fibonacci(n+1) -2*fibonacci(n) -(3+(-1)^n)//2 + int(n==1) for n in range(41)] # G. C. Greubel, Jun 16 2024

Formula

For n>1, a(n) = Fibonacci(n-1) + Lucas(n) - (3 + (-1)^n)/2. - Ralf Stephan, May 13 2004
From Colin Barker, Jul 12 2017: (Start)
G.f.: (1 - x^2 + x^3 + x^4 + x^5) / ((1 - x)*(1 + x)*(1 - x - x^2)).
a(n) = 2^(-1-n)*(-5*((-2)^n + 3*2^n) - (-15+sqrt(5))*(1+sqrt(5))^n + (1-sqrt(5))^n*(15+sqrt(5))) / 5 for n>1.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>5.
(End)

A026956 Self-convolution of array T given by A026615.

Original entry on oeis.org

1, 2, 11, 52, 200, 742, 2752, 10278, 38670, 146426, 557408, 2131318, 8179646, 31491202, 121568150, 470404274, 1823968074, 7085220834, 27567196704, 107414120214, 419080195374, 1636990646274, 6401210885934, 25055584929954, 98160790785714, 384885441746202, 1510279309724502
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000108, A026615, A026616, A026617, A026618, A026619, A026620.
Cf. A026621, A026622, A026623, A026624, A026625, A026957, A026958.
Cf. A026959, A026960.

Programs

  • Magma
    [n le 1 select n+1 else Catalan(n-2)*(49*n^2-105*n+48)/n - 6: n in [0..40]]; // G. C. Greubel, Jun 17 2024
  • Mathematica
    Table[If[n==0, 1, CatalanNumber[n-2]*(49*n^2-105*n+48)/n -6], {n,0,40}] (* G. C. Greubel, Jun 17 2024 *)
  • SageMath
    [1,2]+[catalan_number(n-2)*(49*n^2-105*n+48)/n -6 for n in range(2,41)] # G. C. Greubel, Jun 17 2024

Formula

From G. C. Greubel, Jun 17 2024: (Start)
a(n) = Sum_{k=0..n} A026615(n, k) * A026615(n, n-k).
a(n) = A000108(n-2)*(49*n^2 - 105*n + 48)/n - 6, for n >= 1, with a(0) = 1.
G.f.: (4 - 8*x + 5*x^2 - x^3 - (3 - x + 4*x^2)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)).
E.g.f.: (1/6)*( 18 + 24*x - 36*exp(x) + 4*exp(2*x)*(6 - 6*x + x^2) * BesselI(0, 2*x) + x*exp(2*x)*(23 - 4*x)*BesselI(1, 2*x) ). (End)

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026957 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026615.

Original entry on oeis.org

1, 6, 35, 154, 613, 2362, 9028, 34510, 132241, 508210, 1958460, 7565906, 29292820, 113633930, 441579702, 1718642278, 6698377449, 26139863330, 102125977396, 399415127682, 1563614796608, 6126581578954, 24024810462810, 94281930087290, 370254213115948, 1454967778894282
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026624, A026625, A026956, A026958, A026959.
Cf. A026960.

Programs

  • Magma
    [1] cat [(n-1)*Binomial(2*n,n-1)*(49*n^3 -105*n^2 +62*n -24)/( 24*Binomial(2*n,4)) -2*(2*n-1): n in [2..40]]; // G. C. Greubel, Jun 17 2024
  • Mathematica
    Table[If[n==1, 1, (n-1)*Binomial[2*n,n-1]*(49*n^3 -105*n^2 +62*n -24 )/(24*Binomial[2*n,4]) - 2*(2*n-1)], {n,40}] (* G. C. Greubel, Jun 17 2024 *)
  • SageMath
    [1]+[(n-1)*binomial(2*n,n-1)*(49*n^3-105*n^2+62*n-24 )/( 24*binomial(2*n, 4)) -2*(2*n-1) for n in range(2,41)] # G. C. Greubel, Jun 17 2024

Formula

a(n) = (n-1)*binomial(2*n, n-1)*(49*n^3 - 105*n^2 + 62*n - 24 )/( 24*binomial(2*n, 4)) - 2*(2*n-1), for n >= 2, with a(1) = 1. - G. C. Greubel, Jun 17 2024

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026958 a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A026615.

Original entry on oeis.org

1, 10, 69, 340, 1476, 6074, 24419, 97136, 384428, 1517422, 5981070, 23556746, 92743296, 365078146, 1437124303, 5657887016, 22279053380, 87749051950, 345704345066, 1362361338578, 5370436417996, 21176724230654, 83529562154498, 329573910914930, 1300752571946396
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026624, A026625, A026956, A026957, A026959.
Cf. A026960.

Programs

  • Magma
    [n eq 2 select 1 else Binomial(2*n,n+2)*(49*n^4 -154*n^3 + 209*n^2 -200*n +108)/(24*Binomial(2*n,4)) -2*(n^2-2*n+2): n in [2..40]]; // G. C. Greubel, Jun 17 2024
  • Mathematica
    Table[Binomial[2*n,n+2]*(49*n^4 -154*n^3 +209*n^2 -200*n +108)/(24* Binomial[2*n,4]) -2*(n^2-2*n+2) + Boole[n==2], {n,2,40}] (* G. C. Greubel, Jun 17 2024 *)
  • SageMath
    [binomial(2*n,n+2)*(49*n^4 -154*n^3 +209*n^2 -200*n +108 )/(24*binomial(2*n,4)) -2*(n^2-2*n+2) +int(n==2) for n in range(2,41)] # G. C. Greubel, Jun 17 2024

Formula

a(n) = binomial(2*n, n+2)*(49*n^4 - 154*n^3 + 209*n^2 - 200*n + 108)/(24*binomial(2*n, 4)) -2*(n^2 - 2*n + 2) + [n=2]. - G. C. Greubel, Jun 17 2024

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026959 a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A026615.

Original entry on oeis.org

1, 14, 115, 640, 3049, 13494, 57491, 239768, 986976, 4027666, 16335660, 65955960, 265386251, 1064993622, 4264898875, 17051078256, 68080259516, 271537515786, 1082098938452, 4309269809044, 17151303222746, 68232856509950, 271350536990740, 1078796298028680, 4287906741748940
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026624, A026625, A026956, A026957, A026958.
Cf. A026960.

Programs

  • Magma
    [n eq 3 select 1 else Binomial(2*n,n+3)*(49*n^4 -154*n^3 +279*n^2 -390*n +288)/(24* Binomial(2*n,4)) -(n-2)*(2*n^2-5*n+9)/3: n in [3..40]]; // G. C. Greubel, Jun 17 2024
  • Mathematica
    Table[(2*n-4)!*(49*n^4 -154*n^3 +279*n^2 -390*n +288)/((n-3)!*(n+3)!) - (n-2)*(2*n^2-5*n+9)/3 +Boole[n==3], {n,3,40}] (* G. C. Greubel, Jun 17 2024 *)
  • SageMath
    [binomial(2*n,n+3)*(49*n^4 -154*n^3 +279*n^2 -390*n +288)/(24*binomial(2*n,4)) -(1/3)*(n-2)*(2*n^2-5*n+9) +int(n==3) for n in range(3,41)] # G. C. Greubel, Jun 17 2024

Formula

a(n) = binomial(2*n, n+3)*(49*n^4 - 154*n^3 + 279*n^2 - 390*n + 288)/(4! * binomial(2*n, 4)) - (1/3)*(n-2)*(2*n^2 - 5*n + 9) + [n=3]. - G. C. Greubel, Jun 17 2024

Extensions

More terms from Sean A. Irvine, Oct 20 2019
Previous Showing 11-15 of 15 results.

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