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.

A199300 a(n) = (2*n + 1)*7^n.

Original entry on oeis.org

1, 21, 245, 2401, 21609, 184877, 1529437, 12353145, 98001617, 766718533, 5931980229, 45478515089, 346032180025, 2616003280989, 19668469112621, 147174406808233, 1096686708796833, 8142067989552245, 60251303122686613, 444556912229552577, 3271482918202092041
Offset: 0

Views

Author

Philippe Deléham, Nov 04 2011

Keywords

Links

Crossrefs

Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199301.

Programs

  • Magma
    [(2*n+1)*7^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011
  • Mathematica
    a[n_] := (2*n + 1)*7^n; Array[a, 25, 0] (* Amiram Eldar, Dec 10 2022 *)
LinearRecurrence[{14,-49},{1,21},30] (* Harvey P. Dale, Mar 26 2025 *)
  • PARI
    a(n) = (2*n+1)*7^n \\ Amiram Eldar, Dec 10 2022

Formula

a(n) = 14*a(n-1) - 49*a(n-2).
G.f.: (1+7*x)/(1-7*x)^2.
a(n) = 7*a(n-1) + 2*7^n. - Vincenzo Librandi, Nov 05 2011
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(7)*arccoth(sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(7)*arccot(sqrt(7)). (End)
E.g.f.: exp(7*x)*(1 + 14*x). - Stefano Spezia, May 09 2023

Extensions

a(15) corrected by Vincenzo Librandi, Nov 05 2011

A199301 a(n) = (2n+1)*8^n.

Original entry on oeis.org

1, 24, 320, 3584, 36864, 360448, 3407872, 31457280, 285212672, 2550136832, 22548578304, 197568495616, 1717986918400, 14843406974976, 127543348822016, 1090715534753792, 9288674231451648, 78812993478983680, 666532744850833408, 5620492334958379008, 47269781688880726016
Offset: 0

Views

Author

Philippe Deléham, Nov 04 2011

Keywords

Links

Crossrefs

Cf. A001018 (Powers of 8), A005408 (2n+1).
Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199300.

Programs

Formula

a(n) = 16*a(n-1)-64*a(n-2).
G.f.: (1+8*x)/(1-8*x)^2.
a(n) = 8*(a(n-1)+2^(3*n-2)). - Vincenzo Librandi, Nov 05 2011
a(n) = A005408(n) * A001018(n). - Wesley Ivan Hurt, Oct 30 2014
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(8)*arccoth(sqrt(8)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(8)*arccot(sqrt(8)). (End)
E.g.f.: exp(8*x)*(1 + 16*x). - Stefano Spezia, May 09 2023

Extensions

a(18) corrected by Vincenzo Librandi, Nov 05 2011

A027271 a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.

Original entry on oeis.org

1, 4, 18, 48, 180, 432, 1512, 3456, 11664, 25920, 85536, 186624, 606528, 1306368, 4199040, 8957952, 28553472, 60466176, 191476224, 403107840, 1269789696, 2660511744, 8344332288, 17414258688, 54419558400, 113192681472, 352638738432, 731398864896, 2272560758784
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026536, A053469, A199299 (bisection).

Programs

  • Magma
    [Round(6^(n/2)*( 3*((n+1) mod 2) + Sqrt(6)*(n mod 2) )*(n+1)/3): n in [0..40]]; // G. C. Greubel, Apr 12 2022
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
A027271[n_]:= A027271[n]= Sum[(k+1)*T[n,k], {k,0,2*n}];
Table[A027271[n], {n,0,40}] (* G. C. Greubel, Apr 12 2022 *)
  • PARI
    A027271(n)=my(b(n)=if(!bittest(n,0),n\2*6^(n\2-1)));4*b(n+1)+b(n+2)+6*b(n) \\ could be made more efficient and explicit by simplifying the formula for n even and for n odd separately. - M. F. Hasler, Sep 29 2012
  • SageMath
    [6^(n/2)*( 3*((n+1)%2) + sqrt(6)*(n%2) )*(n+1)/3 for n in (0..40)] # G. C. Greubel, Apr 12 2022

Formula

From Paul Barry, Mar 03 2004: (Start)
G.f.: (1+4*x+6*x^2)/(1-6*x^2)^2 = (d/dx)((1+3*x)/(1-6*x^2)).
a(n) = 6^(n/2)*((3-sqrt(6))*(-1)^n + (3+sqrt(6)))*(n+1)/6. (End)
a(n) = 4*b(n) + b(n+1) + 6*b(n-1) with b(n)= 0, 1, 0, 12, 0, 108, 0, 864, ... (aerated A053469). - R. J. Mathar, Sep 29 2012
E.g.f.: (1 + 2*x)*cosh(sqrt(6)*x) + sqrt(2/3)*(1 + 3*x)*sinh(sqrt(6)*x). - Stefano Spezia, May 07 2023

A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2*n)*k^n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 6, 1, 0, 7, 20, 9, 1, 0, 9, 56, 45, 12, 1, 0, 11, 144, 189, 80, 15, 1, 0, 13, 352, 729, 448, 125, 18, 1, 0, 15, 832, 2673, 2304, 875, 180, 21, 1, 0, 17, 1920, 9477, 11264, 5625, 1512, 245, 24, 1, 0, 19, 4352, 32805, 53248, 34375, 11664, 2401, 320, 27, 1
Offset: 0

Views

Author

Stefano Spezia, May 08 2023

Keywords

Examples

			The array begins:
    1,  1,   1,    1,     1,     1, ...
    0,  3,   6,    9,    12,    15, ...
    0,  5,  20,   45,    80,   125, ...
    0,  7,  56,  189,   448,   875, ...
    0,  9, 144,  729,  2304,  5625, ...
    0, 11, 352, 2673, 11264, 34375, ...
    ...

Crossrefs

Cf. A000007 (k=0), A000012 (n=0), A004248, A005408 (k=1), A008585 (n=1), A014480 (k=2), A033429 (n=2), A058962 (k=4), A124647 (k=3), A155988 (k=9), A171220 (k=5), A176043, A199299 (k=6), A199300 (k=7), A199301 (k=8), A244727 (n=3), A362886 (antidiagonal sums).

Programs

  • Mathematica
    A[n_,k_]:=(1+2n)k^n; Join[{1}, Table[A[n-k,k],{n,10},{k,0,n}]]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1+k*x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+2k*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

A(n, k) = A005408(n)*A004248(n, k).
O.g.f. of column k: (1 + k*x)/(1 - k*x)^2.
E.g.f. of column k: exp(k*x)*(1 + 2*k*x).
A(n, n) = A176043(n+1).
Showing 1-4 of 4 results.

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