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.

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A135856 A007318 * a bidiagonal matrix with all 1's in the main diagonal and all 4's in the subdiagonal.

Original entry on oeis.org

1, 5, 1, 9, 6, 1, 13, 15, 7, 1, 17, 28, 22, 8, 1, 21, 45, 50, 30, 9, 1, 25, 66, 95, 80, 39, 10, 1, 29, 91, 161, 175, 119, 49, 11, 1, 33, 120, 252, 336, 294, 168, 60, 12, 1, 37, 153, 372, 588, 630, 462, 228, 72, 13, 1
Offset: 1

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Author

Gary W. Adamson, Dec 01 2007

Keywords

Comments

Row sums = A048487.
When the first column is removed from this triangle, the result is A125233. - Georg Fischer, Jul 26 2023

Examples

			First few rows of the triangle:
   1;
   5,  1;
   9,  6,  1;
  13, 15,  7,  1;
  17, 28, 22,  8,  1;
  21, 45, 50, 30,  9,  1;
  25, 66, 95, 80, 39, 10,  1;
  ...

Crossrefs

Cf. A048487, A125233.

Formula

Binomial transform of an infinite lower triangular matrix with all 1's in the main diagonal and all 4's in the subdiagonal (by columns, (1, 4, 0, 0, 0, ...) in every column.

A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182
Offset: 0

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Author

Ctibor O. Zizka, Mar 06 2008

Keywords

Comments

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

Examples

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

Links

Crossrefs

Cf. A002275, A011557.

Programs

  • Mathematica
    Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)
  • PARI
    a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020
  • Sage
    [3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

Formula

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.
O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

Extensions

More terms from R. J. Mathar, Mar 16 2008
More terms from Jinyuan Wang, Feb 27 2020

A268896 Start at a(0)=1. a(n) = a(n-1)+2 if n == 1,2 (mod 3) and a(n)=a(n-1)+a(n-3) if n == 0 (mod 3).

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 16, 18, 20, 36, 38, 40, 76, 78, 80, 156, 158, 160, 316, 318, 320, 636, 638, 640, 1276, 1278, 1280, 2556, 2558, 2560, 5116, 5118, 5120, 10236, 10238, 10240, 20476, 20478, 20480, 40956, 40958
Offset: 0

Views

Author

Ravesh Sukhram, Feb 27 2016

Keywords

Comments

See Mathematica section for an explicit formula for the n-th term. - Benedict W. J. Irwin, May 30 2016

Links

Programs

  • Mathematica
    Simplify[Table[1/6 (10 (2^n)^(1/3) + 4 (-3 + 5 2^(n/3)) Cos[(2 n Pi)/3] + 5 2^((4 + n)/3)Sin[(n Pi)/3] (Sqrt[3] (-1 + 2^(1/3)) Cos[(n Pi)/3] + (1 + 2^(1/3)) Sin[(n Pi)/3]) - 4 (3 + Sqrt[3] Sin[(2 n Pi)/3])), {n, 0, 20}]] (* Benedict W. J. Irwin, May 30 2016 *)

Formula

G.f.: ( 1+3*x+5*x^2+3*x^3-x^4-5*x^5 ) / ( (x-1)*(2*x^3-1)*(1+x+x^2) ). - R. J. Mathar, Apr 16 2016
a(3n) = A048487(n). a(3n+1) = A131051(n+1). a(3n+2)=A020714(n). - R. J. Mathar, Apr 16 2016
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