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 1-6 of 6 results.

A131051 Row sums of triangle A133805.

Original entry on oeis.org

1, 3, 8, 18, 38, 78, 158, 318, 638, 1278, 2558, 5118, 10238, 20478, 40958, 81918, 163838, 327678, 655358, 1310718, 2621438, 5242878, 10485758, 20971518, 41943038, 83886078, 167772158, 335544318, 671088638, 1342177278, 2684354558
Offset: 1

Views

Author

Gary W. Adamson, Sep 23 2007

Keywords

Comments

Last digit of a(n) is 8 for n > 2. - Jon Perry, Nov 19 2012

Examples

			a(4) = 18 = sum of row 4 terms of triangle A133805: (7 + 6 + 4 + 1).
a(4) = 18 = (1, 3, 3, 1), dot (1, 2, 3, 2) = (1 + 6 + 9 + 2).

Crossrefs

Essentially a duplicate of A051633.
Cf. A133805.
Cf. A083329, A053209.

Programs

  • Magma
    a:=[1]; for n in [2..31] do Append(~a,2*n-2+&+[a[i]:i in [1..n-1]]); end for; a; // Marius A. Burtea, Oct 15 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 31); Coefficients(R!( (1+x^2)/((1-x)*(1-2*x)))); // Marius A. Burtea, Oct 15 2019

Formula

Binomial transform of [1, 2, 3, 2, 3, 2, 3, ...].
O.g.f.: (1+x^2)/((1-x)(1-2*x)). a(n)=A051633(n-2). - R. J. Mathar, Jun 13 2008
a(n) = 5*2^(n-2)-2, n>1. - Gary Detlefs, Jun 22 2010
a(n) = 2(n-1) + Sum_{i=1..n-1} a(i). - Jon Perry, Nov 19 2012

Extensions

More terms from R. J. Mathar, Jun 13 2008

A048491 a(n) = T(8,n), array T given by A048483.

Original entry on oeis.org

1, 10, 28, 64, 136, 280, 568, 1144, 2296, 4600, 9208, 18424, 36856, 73720, 147448, 294904, 589816, 1179640, 2359288, 4718584, 9437176, 18874360, 37748728, 75497464, 150994936, 301989880, 603979768, 1207959544, 2415919096
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (9, 9, 9, ...).

Links

Programs

Formula

a(n) = 9 * 2^n - 8. - Ralf Stephan
Equals binomial transform of [1, 9, 9, 9, ...]. - Gary W. Adamson, Apr 29 2008
a(n) = 2*a(n-1) + 8, with a(0)=1. - Vincenzo Librandi, Aug 06 2010
a(n) = 2*A053209(n), n>0. - Philippe Deléham, Apr 15 2013
a(n) = 3*a(n-1) - 2*a(n-2) with a(0)=1, a(1)=10. - Philippe Deléham, Apr 15 2013
G.f.: (1+7*x)/((1-x)*(1-2*x)). - Philippe Deléham, Apr 15 2013

A053208 Row sums of A053207.

Original entry on oeis.org

0, 3, 10, 24, 52, 108, 220, 444, 892, 1788, 3580, 7164, 14332, 28668, 57340, 114684, 229372, 458748, 917500, 1835004, 3670012, 7340028, 14680060, 29360124, 58720252, 117440508, 234881020, 469762044, 939524092, 1879048188
Offset: 0

Views

Author

Asher Auel, Dec 13 1999

Keywords

Links

Crossrefs

Cf. A053207, A053209.

Programs

  • Magma
    [0] cat [7*2^(n-1) -4: n in [1..50]]; // G. C. Greubel, Sep 03 2018
  • Mathematica
    Join[{0},NestList[2#+4&,3,30]] (* Harvey P. Dale, Nov 08 2013 *)
Join[{0}, Table[7*2^(n-1) -4, {n,50}]] (* G. C. Greubel, Sep 03 2018 *)
  • PARI
    concat([0], vector(50, n, 7*2^(n-1) -4)) \\ G. C. Greubel, Sep 03 2018

Formula

a(0) = 0, a(1) = 3, a(n+1) = 2*a(n) + 4, for n >= 1.
a(n) = 7*2^(n-1) - 4, n >= 1.
G.f.: x*(x + 3)/((x - 1)*(2*x - 1)). - Chai Wah Wu, Jul 24 2020

A224692 Expansion of (1+5*x+7*x^2-x^3)/((1-2*x^2)*(1-x)*(1+x)).

Original entry on oeis.org

1, 5, 10, 14, 28, 32, 64, 68, 136, 140, 280, 284, 568, 572, 1144, 1148, 2296, 2300, 4600, 4604, 9208, 9212, 18424, 18428, 36856, 36860, 73720, 73724, 147448, 147452, 294904, 294908, 589816, 589820, 1179640, 1179644, 2359288, 2359292, 4718584, 4718588, 9437176
Offset: 0

Views

Author

Philippe Deléham, Apr 15 2013

Keywords

Links

Crossrefs

Cf. A048491, A053209.

Programs

  • Mathematica
    CoefficientList[Series[(1+5x+7x^2-x^3)/((1-2x^2)(1-x)(1+x)),{x,0,40}],x] (* or *) LinearRecurrence[{0,3,0,-2},{1,5,10,14},50] (* Harvey P. Dale, Sep 17 2016 *)

Formula

G.f.: (1+5*x+7*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).
a(n) = a(n-1)+4 if n odd.
a(n) = a(n-1)*2 if n even.
a(2n) = 9*2^n - 8 = A048491(n).
a(2n+1) = 9*2^n - 4 = A053209(n+1).
a(n) = 3*a(n-2) - 2*a(n-4) with n>3, a(0)=1, a(1)=5, a(2)=10, a(3)=14.
a(n) = 9*2^floor(n/2)-2*(-1)^n-6. [Bruno Berselli, Apr 27 2013]

A370882 Square array T(n,k) = 9*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

9, 8, 18, 7, 17, 36, 6, 16, 35, 72, 5, 15, 34, 71, 144, 4, 14, 33, 70, 143, 288, 3, 13, 32, 69, 142, 287, 576, 2, 12, 31, 68, 141, 286, 575, 1152, 1, 11, 30, 67, 140, 285, 574, 1151, 2304, 0, 10, 29, 66, 139, 284, 573, 1150, 2303, 4608, -1, 9, 28, 65, 138, 283, 572, 1149, 2302, 4607, 9216
Offset: 0

Views

Author

Paul Curtz, Mar 05 2024

Keywords

Comments

Just after A367559 and A368826.

Examples

			Table begins:
       k=0  1  2  3   4   5
  n=0:   9 18 36 72 144 288 ...
  n=1:   8 17 35 71 143 287 ...
  n=2:   7 16 34 70 142 286 ...
  n=3:   6 15 33 69 141 285 ...
  n=4:   5 14 32 68 140 284 ...
  n=5:   4 13 31 67 139 283 ...
Every line has the signature (3,-2). For n=1: 3*17 - 2*8 = 35.
Main diagonal's difference table:
  9   17   34   69   140   283   570  1145  ...  =  b(n)
  8   17   35   71   143   287   575  1151  ...  =  A052996(n+2)
  9   18   36   72   144   288   576  1152  ...  =  A005010(n)
  ...
b(n+1) - 2*b(n) = A023443(n).

Crossrefs

Cf. A000225, A033484, A048491, A005010, A052996, A053209, A083329, A154251, A176449, A304383, A367559, A368826.
Cf. A023443.

Programs

  • Mathematica
    T[n_, k_] := 9*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)

Formula

T(0,k) = 9*2^k = A005010(k);
T(1,k) = 9*2^k - 1 = A052996(k+2);
T(2,k) = 9*2^k - 2 = A176449(k);
T(3,k) = 9*2^k - 3 = 3*A083329(k);
T(4,k) = 9*2^k - 4 = A053209(k);
T(5,k) = 9*2^k - 5 = A304383(k+3);
T(6,k) = 9*2^k - 6 = 3*A033484(k);
T(7,k) = 9*2^k - 7 = A154251(k+1);
T(8,k) = 9*2^k - 8 = A048491(k);
T(9,k) = 9*2^k - 9 = 3*A000225(k).
G.f.: (9 - 9*y + x*(11*y - 10))/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Mar 17 2024

A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 7, 10, 14, 19, 15, 22, 32, 46, 65, 31, 46, 68, 100, 146, 211, 63, 94, 140, 208, 308, 454, 665, 127, 190, 284, 424, 632, 940, 1394, 2059, 255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305, 511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171
Offset: 0

Views

Author

Paul Curtz, Jun 20 2025

Keywords

Examples

			Triangle begins:
    0;
    1,   1;
    3,   4,    5;
    7,  10,   14,   19;
   15,  22,   32,   46,   65;
   31,  46,   68,  100,  146,  211;
   63,  94,  140,  208,  308,  454,  665;
  127, 190,  284,  424,  632,  940, 1394, 2059;
  255, 382,  572,  856, 1280, 1912, 2852, 4246,  6305;
  511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
  ...

Crossrefs

Columns k=0..2: A000225, A033484, A053209 (sans 1).
Diagonals: A001047, A027649, A053581 (sans 1), A291012 (sans 2).
Cf. A028243 (row sums), A036561, A059268, A081656, A248216, A321003.

Programs

  • Magma
    /* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jun 27 2025
  • Maple
    T:= proc(n,k) option remember;
     `if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2025
  • Mathematica
    t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 20 2025 *)

Formula

T(n,n) = 3^n - 2^n = A001047(n).
T(n,k) = T(n,k+1) - T(n-1,k) for 0 <= k < n.
T(n,k) = 2^(n-k)*3^k - 2^k = A036561(n,k) - A059268(n,k).
T(2n,n) = A248216(n+1).
Showing 1-6 of 6 results.

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