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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A101493 Triangle read by rows: T(n,k) = (n+1)*(2*(n+1)-1) - k*(2*k-1).

Original entry on oeis.org

1, 6, 5, 15, 14, 9, 28, 27, 22, 13, 45, 44, 39, 30, 17, 66, 65, 60, 51, 38, 21, 91, 90, 85, 76, 63, 46, 25, 120, 119, 114, 105, 92, 75, 54, 29, 153, 152, 147, 138, 125, 108, 87, 62, 33, 190, 189, 184, 175, 162, 145, 124, 99, 70, 37, 231, 230, 225, 216, 203, 186, 165, 140, 111, 78, 41
Offset: 0

Views

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

Keywords

Comments

The triangle is generated from the product B*A of the infinite lower triangular matrices A =
1 0 0 0 ...
1 1 0 0 ...
1 1 1 0 ...
1 1 1 1 ...
... and B =
1 0 0 0 ...
1 5 0 0 ...
1 5 9 0 ...
1 5 9 13 ...
...
T(n+0,0) = n*(2*n-1) = A000384(n) (Hexagonal numbers)
since T(n,n) = 4*n+1 = A016813(n).
T(n,n) = 4*n + 1 = A016813(n);
T(n+1,n) = 8*n + 6 = A017137(n);
T(n+2,n) = 12*n + 3 = A017557(n);
T(n,n)*T(n,0) = (n+1)*(2*n+1)*(4*n+1) = A079588(n).

Examples

			Triangle begins:
   1;
   6,  5;
  15, 14,  9;
  28, 27, 22, 13;
  45, 44, 39, 30, 17;
  66, 65, 60, 51, 38, 21;

Links

Crossrefs

Row sums give 10-gonal pyramidal numbers: n(n+1)(8n-5)/6 = A007585(n+1).
Cf. A101492 (for product A*B), A007585, A000384.

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->(n+1)*(2*n+1)-k*(2*k-1)))); # Muniru A Asiru, Mar 05 2019
  • PARI
    T(n,k)=if(k>n,0,(n+1)*(2*(n+1)-1)-k*(2*k-1))
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

A017141 a(n) = (8*n+6)^5.

Original entry on oeis.org

7776, 537824, 5153632, 24300000, 79235168, 205962976, 459165024, 916132832, 1680700000, 2887174368, 4704270176, 7339040224, 11040808032, 16105100000, 22877577568, 31757969376, 43204003424, 57735339232, 75937500000, 98465804768, 126049300576, 159494694624, 199690286432
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013663, A016841, A017137.

Programs

Formula

G.f.: 32*(x^5+3119*x^4+40314*x^3+63854*x^2+15349*x+243)/(x-1)^6. - Colin Barker, Sep 17 2012
From Amiram Eldar, Apr 26 2023: (Start)
a(n) = A017137(n)^5.
a(n) = 2^5*A016841(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/2048 - 5*Pi^5/98304. (End)

A017143 a(n) = (8*n+6)^7.

Original entry on oeis.org

279936, 105413504, 2494357888, 21870000000, 114415582592, 435817657216, 1338925209984, 3521614606208, 8235430000000, 17565568854912, 34792782221696, 64847759419264, 114868566764928, 194871710000000, 318547390056832, 504189521813376, 775771085481344, 1164175380274048
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013665, A016843, A017137.

Programs

Formula

a(n) = 128*A016843(n). - R. J. Mathar, Aug 26 2015
From Amiram Eldar, Apr 26 2023: (Start)
a(n) = A017137(n)^7.
Sum_{n>=0} 1/a(n) = 127*zeta(7)/32768 - 61*Pi^7/47185920. (End)

A017145 a(n) = (8*n+6)^9.

Original entry on oeis.org

10077696, 20661046784, 1207269217792, 19683000000000, 165216101262848, 922190162669056, 3904305912313344, 13537086546263552, 40353607000000000, 106868920913284608, 257327417311663616, 572994802228616704, 1195092568622310912, 2357947691000000000, 4435453859151328768
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013667, A016845, A017137.

Programs

  • Magma
    [(8*n+6)^9: n in [0..20]]; // Vincenzo Librandi, Jul 21 2011
  • Mathematica
    (8*Range[0,20]+6)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{10077696,20661046784,1207269217792,19683000000000,165216101262848,922190162669056,3904305912313344,13537086546263552,40353607000000000,106868920913284608},20] (* Harvey P. Dale, Jun 04 2016 *)

Formula

a(n) = 512*A016845(n). - R. J. Mathar, Aug 26 2015
From Amiram Eldar, Apr 26 2023: (Start)
a(n) = A017137(n)^9.
Sum_{n>=0} 1/a(n) = 511*zeta(9)/524288 - 277*Pi^9/8455716864. (End)

A017147 a(n) = (8*n+6)^11.

Original entry on oeis.org

362797056, 4049565169664, 584318301411328, 17714700000000000, 238572050223552512, 1951354384207722496, 11384956040305711104, 52036560683837093888, 197732674300000000000, 650190514836423555072, 1903193578437064103936, 5062982072492057196544, 12433743083946522728448
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013669, A016847, A017137.

Programs

Formula

a(n) = 2048*A016847(n). - R. J. Mathar, Aug 26 2015
From Amiram Eldar, Apr 26 2023: (Start)
a(n) = A017137(n)^11.
Sum_{n>=0} 1/a(n) = 2047*zeta(11)/8388608 - 50521*Pi^11/60881161420800. (End)

A047400 Numbers that are congruent to {1, 3, 6} mod 8.

Original entry on oeis.org

1, 3, 6, 9, 11, 14, 17, 19, 22, 25, 27, 30, 33, 35, 38, 41, 43, 46, 49, 51, 54, 57, 59, 62, 65, 67, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 97, 99, 102, 105, 107, 110, 113, 115, 118, 121, 123, 126, 129, 131, 134, 137, 139, 142, 145, 147, 150, 153, 155, 158
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Union of A017077, A017101 and A017137. - R. J. Mathar, Apr 14 2008

Links

Crossrefs

Cf. A004773.
Cf. A017077, A017101, A017137.

Programs

  • Magma
    [n: n in [1..300] | n mod 8 in [1, 3, 6]]; // Vincenzo Librandi, Mar 27 2011
  • Maple
    A047400:=n->2*(12*n-9+sqrt(3)*sin(2*n*Pi/3))/9: seq(A047400(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016
  • Mathematica
    Select[Range[0, 150], MemberQ[{1, 3, 6}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 10 2016 *)
  • PARI
    a(n) = {x=8*floor((n-1)/3);if(n%3==1,x=x+1);if(n%3==2,x=x+3);if(n%3==0,x=x+6);x} \\ Michael B. Porter, Oct 02 2009

Formula

a(n) = A004773(n-1) + A004773(n). - Gary W. Adamson, Sep 13 2007
G.f.: x*(1+x)*(2x^2+x+1)/((-1+x)^2*(x^2+x+1)). a(n) = a(n-3)+8 for n>3. - R. J. Mathar, Apr 14 2008
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*(12*n-9+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-5, a(3k-2) = 8k-7. (End)

A047589 Numbers that are congruent to {6, 7} mod 8.

Original entry on oeis.org

6, 7, 14, 15, 22, 23, 30, 31, 38, 39, 46, 47, 54, 55, 62, 63, 70, 71, 78, 79, 86, 87, 94, 95, 102, 103, 110, 111, 118, 119, 126, 127, 134, 135, 142, 143, 150, 151, 158, 159, 166, 167, 174, 175, 182, 183, 190, 191, 198, 199, 206, 207, 214, 215, 222, 223, 230, 231
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the values of n for which binomial(n,6) is odd. See Maple code. - Gary Detlefs, Nov 29 2011

Links

Crossrefs

Union of A017137 and A004771.
Cf. A047451, A047521.

Programs

  • Maple
    for i from 1 to 240 do if(floor((i mod 8)/6) <>0) then print(i) fi od; # Gary Detlefs, Nov 30 2011
  • Mathematica
    LinearRecurrence[{1,1,-1},{6,7,14},60] (* Harvey P. Dale, Sep 11 2017 *)

Formula

a(n) = 8*n-a(n-1)-3 with n>1, a(1)=6. - Vincenzo Librandi, Aug 06 2010
a(n) = 6*floor((n-1)/2) + n + 5. - Gary Detlefs, Nov 29 2011
a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: x*(6+x+x^2)/((1-x)^2*(1+x)). - Colin Barker, Mar 18 2012
a(n) = (1-3*(-1)^n+8*n)/2. - Colin Barker, May 14 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 - log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A144583 Value of concatenation of m and e in binary representation, where n=m*2^e and m odd.

Original entry on oeis.org

2, 3, 6, 6, 10, 7, 14, 7, 18, 11, 22, 14, 26, 15, 30, 12, 34, 19, 38, 22, 42, 23, 46, 15, 50, 27, 54, 30, 58, 31, 62, 13, 66, 35, 70, 38, 74, 39, 78, 23, 82, 43, 86, 46, 90, 47, 94, 28, 98, 51, 102, 54, 106, 55, 110, 31, 114, 59, 118, 62, 122, 63, 126, 14, 130, 67, 134, 70
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 19 2008

Keywords

Comments

A007088(a(n)) = concatenation of A007088(A000265(n)) and A007088(A007814(n));
a(A005408(n)) = A016825(n);
a(A004767(n)-1) = A004767(n);
a(A017113(n)) = A017137(n).

Links

Extensions

Edited by Charles R Greathouse IV, Apr 26 2010

A173773 a(3*n) = 8*n+2, a(3*n+1) = 2*n+1, a(3*n+2) = 8*n+6.

Original entry on oeis.org

2, 1, 6, 10, 3, 14, 18, 5, 22, 26, 7, 30, 34, 9, 38, 42, 11, 46, 50, 13, 54, 58, 15, 62, 66, 17, 70, 74, 19, 78, 82, 21, 86, 90, 23, 94, 98, 25, 102, 106, 27, 110, 114, 29, 118, 122, 31, 126, 130, 33, 134, 138, 35, 142, 146, 37, 150, 154, 39, 158, 162, 41, 166
Offset: 0

Views

Author

Paul Curtz, Nov 26 2010

Keywords

Links

Crossrefs

Cf. A005408, A017089, A017137.

Programs

  • Mathematica
    Table[Sequence @@ {8*n+2, 2*n+1,  8*n+6}, {n, 0, 20}] (* Amiram Eldar, Oct 08 2023 *)

Formula

G.f.: (1+x)*(2-x+7*x^2-x^3+2*x^4)/(1-x^3)^2.
Sum_{n>=0} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/8. - Amiram Eldar, Oct 08 2023

A365372 Array read by ascending antidiagonals: A(n, k) = n*(k*n^2 - 1) with k > 0.

Original entry on oeis.org

0, 6, 1, 24, 14, 2, 60, 51, 22, 3, 120, 124, 78, 30, 4, 210, 245, 188, 105, 38, 5, 336, 426, 370, 252, 132, 46, 6, 504, 679, 642, 495, 316, 159, 54, 7, 720, 1016, 1022, 858, 620, 380, 186, 62, 8, 990, 1449, 1528, 1365, 1074, 745, 444, 213, 70, 9, 1320, 1990, 2178, 2040, 1708, 1290, 870, 508, 240, 78, 10
Offset: 1

Views

Author

Stefano Spezia, Sep 02 2023

Keywords

Examples

			The array begins:
    0,   1,   2,   3,    4,    5, ...
    6,  14,  22,  30,   38,   46, ...
   24,  51,  78, 105,  132,  159, ...
   60, 124, 188, 252,  316,  380, ...
  120, 245, 370, 495,  620,  745, ...
  210, 426, 642, 858, 1074, 1290, ...
  ...

Crossrefs

Cf. A007531, A017137, A035328 (k=4), A058895 (main diagonal), A365373 (antidiagonal sums).

Programs

  • Mathematica
    A[n_,k_]:=n(k n^2-1); Table[A[n-k+1,k],{n,11},{k,n}]//Flatten

Formula

G.f.: x*y*(x^2*y + y - 2*x*(y - 3))/((1 - x)^4*(1 - y)^2).
1st column: A(n, 1) = A007531(n+1).
2nd row: A(2, n) = A017137(n-1).
Previous Showing 21-30 of 31 results. Next

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

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