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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A219190 Numbers of the form k*(5*k+1), where k = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

0, 4, 6, 18, 22, 42, 48, 76, 84, 120, 130, 174, 186, 238, 252, 312, 328, 396, 414, 490, 510, 594, 616, 708, 732, 832, 858, 966, 994, 1110, 1140, 1264, 1296, 1428, 1462, 1602, 1638, 1786, 1824, 1980, 2020, 2184, 2226, 2398, 2442, 2622, 2668, 2856, 2904, 3100
Offset: 1

Views

Author

Bruno Berselli, Nov 14 2012

Keywords

Comments

Equivalently, numbers m such that 20*m+1 is a square.
Also, integer values of h*(h+1)/5.
More generally, for the numbers of the form n*(k*n+1) with n in A001057, we have:
. generating function (offset 1): x^2*(k-1+2*x+(k-1)*x^2)/((1+x)^2*(1-x)^3);
. n-th term: b(n) = (2*k*n*(n-1)+(k-2)*(-1)^n*(2*n-1)+k-2)/8;
. first differences: (n-1)*((-1)^n*(k-2)+k)/2;
. b(2n+1)-b(2n) = 2*n (independent from k);
. (4*k)*b(n)+1 = (2*k*n+(k-2)*(-1)^n-k)^2/4.

Links

Crossrefs

Subsequence of A011858.
Cf. A090771: square roots of 20*a(n)+1 (see the first comment).
Cf. numbers of the form n*(k*n+1) with n in A001057: k=0, A001057; k=1, A110660; k=2, A000217; k=3, A152749; k=4, A074378; k=5, this sequence; k=6, A036498; k=7, A219191; k=8, A154260.
Cf. similar sequences listed in A219257.

Programs

  • Magma
    k:=5; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];
  • Magma
    I:=[0,4,6,18,22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    Rest[Flatten[{# (5 # - 1), # (5 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (2 + x + 2 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,6,18,22},50] (* Harvey P. Dale, Jan 21 2015 *)

Formula

G.f.: 2*x^2*(2 + x + 2*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (10*n*(n-1) + 3*(-1)^n*(2*n - 1) + 3)/8.
a(n) = 2*A057569(n) = A008851(n+1)*A047208(n)/5.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Harvey P. Dale, Jan 21 2015
Sum_{n>=2} 1/a(n) = 5 - sqrt(1+2/sqrt(5))*Pi. - Amiram Eldar, Mar 15 2022
a(n) = A132356(n-1)/2, n >= 1. - Bernard Schott, Mar 15 2022

A081055 Number of partitions of 2n in which no parts are multiples of 4.

Original entry on oeis.org

1, 2, 4, 9, 16, 29, 50, 82, 132, 208, 320, 484, 722, 1060, 1539, 2210, 3138, 4416, 6163, 8528, 11716, 15986, 21666, 29190, 39104, 52098, 69060, 91106, 119634, 156416, 203664, 264128, 341256, 439321, 563600, 720648, 918530, 1167154, 1478720
Offset: 0

Views

Author

Michael Somos, Mar 03 2003

Keywords

Comments

Euler transform of period 16 sequence [2,1,3,1,3,0,2,0,2,0,3,1,3,1,2,0,...].

Links

Crossrefs

Cf. A001935, A074378, A081056.

Programs

  • Mathematica
    Table[Count[IntegerPartitions[2n], x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 38}] (* Robert Price, Jul 28 2020 *)
  • PARI
    a(n)=local(X); if(n<0,0,X=x+x*O(x^(2*n)); polcoeff(eta(X^4)/eta(X),2*n))

Formula

G.f.: (sum_{n>=0} x^A074378(n))/(sum_n (-x)^n^2).
a(n) = A001935(2n).
a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

A154292 Integers of the form m*(6*m -+ 1)/2.

Original entry on oeis.org

11, 13, 46, 50, 105, 111, 188, 196, 295, 305, 426, 438, 581, 595, 760, 776, 963, 981, 1190, 1210, 1441, 1463, 1716, 1740, 2015, 2041, 2338, 2366, 2685, 2715, 3056, 3088, 3451, 3485, 3870, 3906, 4313, 4351, 4780, 4820, 5271, 5313, 5786, 5830, 6325, 6371
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Links

Crossrefs

Cf. A001318, A074378, A057569, A057570.

Programs

  • Magma
    &cat[[n*(6*n-1) div 2, n*(6*n+1) div 2]: n in [2..60 by 2]]; // Vincenzo Librandi, Sep 10 2016
  • Mathematica
    Flatten[Table[{n (6n-1)/2,n (6n+1)/2},{n,2,50,2}]] (* Harvey P. Dale, Jan 19 2013 *)
  • PARI
    Vec(x*(11+2*x+11*x^2)/((1-x)^3*(1+x)^2) + O(x^60)) \\ Colin Barker, Feb 26 2016

Formula

From Colin Barker, Feb 26 2016: (Start)
a(n) = (12*n^2 - 10*(-1)^n*n + 12*n - 5*(-1)^n + 5)/4.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
G.f.: x*(11 + 2*x + 11*x^2) / ((1-x)^3*(1+x)^2). (End)
E.g.f.: (1/4)*(-5 + 10*x + (5 + 24*x + 12*x^2)*exp(2*x))*exp(-x). - G. C. Greubel, Sep 10 2016
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=1} 1/a(n) = 131/11 - (2+sqrt(3))*Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = 133/11 - 3*log(12) - 2*sqrt(3)*log(2+sqrt(3)). (End)

A259161 Positive pentagonal numbers (A000326) that are triangular numbers (A000217) divided by 2.

Original entry on oeis.org

5, 48510, 465793515, 4472549283020, 42945417749765025, 412361896760694487530, 3959498889750770719498535, 38019107927025003687930446040, 365059470355795195660737423378045, 3505300996337237541709397051345542550, 33657899801770684519698434826282476187555
Offset: 1

Views

Author

Colin Barker, Jun 19 2015

Keywords

Examples

			5 is in the sequence because 5 is the 2nd pentagonal number, and 2*5 is the 4th triangular number.

Links

Crossrefs

Cf. A000217, A000326, A074378, A259156-A259160, A259162-A259167.

Programs

  • Mathematica
    LinearRecurrence[{9603, -9603, 1}, {5, 48510, 465793515}, 20] (* Vincenzo Librandi, Jun 20 2015 *)
  • PARI
    Vec(-5*x*(99*x+1)/((x-1)*(x^2-9602*x+1)) + O(x^20))

Formula

G.f.: -5*x*(99*x+1) / ((x-1)*(x^2-9602*x+1)).

A259163 Positive heptagonal numbers (A000566) that are triangular numbers (A000217) divided by 2.

Original entry on oeis.org

18, 189, 37727235, 393298308, 78448579122960, 817809556618215, 163122994382238923193, 1700522115268371779430, 339191755844562643229618814, 3536001066647854270462804353, 705302447816298343956844397692383, 7352626249945315029422809413582264
Offset: 1

Views

Author

Colin Barker, Jun 19 2015

Keywords

Comments

Intersection of A000566 and A074378 (even triangular numbers divided by 2). - Michel Marcus, Jun 20 2015

Examples

			18 is in the sequence because 18 is the 3rd heptagonal number, and 2*18 is the 8th triangular number.

Links

Crossrefs

Cf. A000217, A000566, A074378, A259156-A259162, A259164-A259167.

Programs

  • Mathematica
    LinearRecurrence[{1, 2079362, -2079362, -1, 1}, {18, 189, 37727235, 393298308, 78448579122960}, 20] (* Vincenzo Librandi, Jun 20 2015 *)
  • PARI
    Vec(-9*x*(2*x^4+19*x^3+33170*x^2+19*x+2)/((x-1)*(x^2-1442*x+1)*(x^2+1442*x+1)) + O(x^20))

Formula

G.f.: -9*x*(2*x^4+19*x^3+33170*x^2+19*x+2) / ((x-1)*(x^2-1442*x+1)*(x^2+1442*x+1)).

A333292 Triangle read by rows: T(m,n) = Sum_{ 1 <= i <= m, 1 <= j <= n, gcd(i,j)=1 } i*j, for 1 <= n <= m.

Original entry on oeis.org

1, 3, 5, 6, 14, 23, 10, 18, 39, 55, 15, 33, 69, 105, 155, 21, 39, 75, 111, 191, 227, 28, 60, 117, 181, 296, 374, 521, 36, 68, 149, 213, 368, 446, 649, 777, 45, 95, 176, 276, 476, 554, 820, 1020, 1263, 55, 105, 216, 316, 516, 594, 930, 1130, 1463, 1663, 66, 138, 282, 426, 681, 825, 1238, 1526, 1958, 2268, 2873
Offset: 1

Views

Author

N. J. A. Sloane, Mar 23 2020

Keywords

Comments

The last two diagonals are A333293, Sum_{k=1..n-1} k^2*phi(k) + n^2*phi(n)/2, and A319087, Sum_{k=1..n} k^2*phi(k), where phi = A000010. Is there a similar formula for the general term?

Examples

			Triangle begins:
1,
3, 5,
6, 14, 23,
10, 18, 39, 55,
15, 33, 69, 105, 155,
21, 39, 75, 111, 191, 227,
28, 60, 117, 181, 296, 374, 521,
36, 68, 149, 213, 368, 446, 649, 777,
45, 95, 176, 276, 476, 554, 820, 1020, 1263,
55, 105, 216, 316, 516, 594, 930, 1130, 1463, 1663,
...

Links

Crossrefs

First two columns are A000217 and A074378, rightmost two diagonals are A333293 and A319087.
Main diagonal is A319087.
Cf. A320541.

Programs

  • Maple
    T:= (m, n)-> add(add(`if`(igcd(i, j)=1, i*j, 0), j=1..n), i=1..m):
seq(seq(T(m, n), n=1..m), m=1..12);  # Alois P. Heinz, Mar 23 2020

A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(4*k + 1)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 1, 3, 3, 2, 6, 3, 4, 8, 4, 9, 9, 6, 15, 10, 12, 20, 12, 22, 23, 18, 35, 26, 30, 46, 32, 51, 54, 45, 76, 62, 71, 99, 76, 111, 117, 104, 160, 136, 154, 205, 167, 230, 244, 223, 319, 286, 319, 406, 349, 456, 484, 458, 619, 570, 632, 779, 695
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Comments

Number of partitions of n into parts congruent to {0, 3, 5} mod 8.
Convolution inverse of A244465.

Examples

			For n=23 the a(23)=6 solutions are 3+3+3+3+3+3+5, 3+3+3+3+3+8, 3+3+3+3+11, 3+5+5+5+5, 5+5+5+8, and 5+5+13.

Links

Crossrefs

Cf. A000009, A000041, A006950, A007742, A047622, A074378, A195850, A244465, A308400.

Programs

  • Mathematica
    nmax = 68; CoefficientList[Series[1/Sum[(-x)^(k (4 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Product[1/((1 - x^(8 k - 5)) (1 - x^(8 k - 3)) (1 - x^(8 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Sum[PartitionsP[k] (-x)^k, {k, 0, nmax}]/Sum[PartitionsQ[2 k] (-x)^k, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 / Sum_{k>=0} (-x)^A074378(k).
G.f.: Product_{k>=1} 1 / ((1 - x^(8*k - 5)) * (1 - x^(8*k - 3)) * (1 - x^(8*k))).
G.f.: ( Sum_{k>=0} A000041(k)*(-x)^k ) / ( Sum_{k>=0} A000009(2*k)*(-x)^k ).
a(n) ~ sqrt(sqrt(2) - 1) * exp(sqrt(n)*Pi/2) / (2^(9/4)*n). - Vaclav Kotesovec, May 25 2019
a(n) = a(n-3) + a(n-5) - a(n-14) - a(n-18) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 5, 14, 18, ... is the sequence A074378. - Ludovic Schwob, Aug 04 2021

A154565 One-half of averages of twin prime pairs of A001318.

Original entry on oeis.org

2, 15, 51, 210, 330, 651, 1365, 1650, 1926, 3480, 5430, 5985, 6501, 11310, 16485, 16590, 21660, 25026, 27270, 28635, 35190, 38001, 39285, 46905, 48690, 58905, 64170, 90651, 109485, 143376, 148995, 151845, 190995, 311676, 316251, 332526
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 12 2009

Keywords

Links

Crossrefs

Cf. A154563, A074378.

Programs

  • Mathematica
    q=3;lst={};s=0;Do[s+=n/q;If[Floor[s]==s,If[PrimeQ[2*s-1]&&PrimeQ[2*s+1],AppendTo[lst,s]]],{n,0,8!}];lst

A193470 Square array A(n,k) (n>=1, k>=0) read by antidiagonals: A(n,0) = 0 and A(n,k) is the least integer > A(n,k-1) that can be expressed as a triangular number divided by n.

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 0, 1, 5, 6, 0, 7, 2, 14, 10, 0, 2, 9, 5, 18, 15, 0, 1, 3, 30, 7, 33, 21, 0, 3, 6, 9, 34, 12, 39, 28, 0, 15, 4, 11, 11, 69, 15, 60, 36, 0, 4, 17, 13, 13, 21, 75, 22, 68, 45, 0, 1, 5, 62, 15, 20, 24, 124, 26, 95, 55, 0, 5, 12, 17, 66, 30, 35, 38, 132, 35, 105, 66
Offset: 1

Views

Author

Peter Luschny, Jul 27 2011

Keywords

Examples

			n\k  0   1   2    3    4     5     6     7
------------------------------------------
1 |  0   1   3    6   10    15    21    28    A000217
2 |  0   3   5   14   18    33    39    60    A074378
3 |  0   1   2    5    7    12    15    22    A001318
4 |  0   7   9   30   34    69    75   124    A154260
5 |  0   2   3    9   11    21    24    38    A057569
6 |  0   1   6   11   13    20    35    46    A154293
7 |  0   3   4   13   15    30    33    54    A057570
8 |  0  15  17   62   66   141   147   252    A157716

Crossrefs

Cf. A061782, A011772.

Programs

  • Maple
    A193470_rect := proc(n,k) local j,i,L; L := NULL; j := 0; while nops([L]) < k do add(i/n, i=1..j); if type(%,integer) then L := L,% fi; j := j+1 od; L end:
seq(print(A193470_rect(n, 12)),n = 1..8);
  • Mathematica
    a[, 0] = 0; a[n, k_] := a[n, k] = For[j = a[n, k-1]+1, True, j++, If[Reduce[m > 0 && j == m(m+1)/(2n), m, Integers] =!= False, Return[j]]]; Table[a[n-k, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 07 2016 *)

A267654 Irregular triangle of palindromic subsequences. Every row has 2*n+1 terms. From the second row, there are only two alternated numbers: 2*n+4 and 2*n+2.

Original entry on oeis.org

2, 4, 2, 4, 6, 4, 6, 4, 6, 8, 6, 8, 6, 8, 6, 8, 10, 8, 10, 8, 10, 8, 10, 8, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16
Offset: 0

Views

Author

Paul Curtz, Jan 19 2016

Keywords

Comments

Row sums = 2, 10, 26, 50, ... = A069894(n).
Starting from A053186(n) =
0, for b(n)
0, 1, 2, for c(n)
0, 1, 2, 3, 4, for d(n)
0, 1, 2, 3, 4, 5, 6,
etc,
a(n) is used for
1) b(n+1) = b(n) + (a(0)=2) i.e. 0, 2, 4, 6, ... = A005843(n).
2) c(n+3) = c(n) + (period 3:repeat 4, 2, 4) i.e. 0, 1, 2, 4, 3, 6, 8, ... = A265667(n).
3) d(n+5) = d(n) + (period 5:repeat 6, 4, 6, 4, 6) i.e. 0, 1, 2, 3, 4, 6, 5, 8, 7, 10, ... = A265734(n).
Etc.
a(n) has a companion with the same terms,differently distributed,yielding permutations of the nonnegative numbers. See A265672.
a(n) other writing (by pairs):
2, 4, 2, 4,
6, 4, 6, 4,
6, 8, 6, 8, 6, 8, 6, 8,
10 8, 10, 8, 10, 8, 10, 8,
10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12,
14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12,
etc.
First column: A168276(n+2). Second column: A168273(n+2).
Row sums: 12, 20, 56, 72, ... = 4*A074378(n+1).
The last term of the successive rows is the number of their terms.
Main diagonal: A005843(n+1).

Examples

			The triangle is
2,
4, 2, 4,
6, 4, 6, 4, 6,
8, 6, 8, 6, 8, 6, 8,
etc.

Crossrefs

Cf. A007395, A005843, A008586, A016825, A053186, A069894, A074378, A086520, A168273, A168276, A265667, A265672, A265734.

Programs

  • Mathematica
    Table[2 (n - 1) + 2 (Boole@ OddQ@ k + 1), {n, 0, 7}, {k, 2 n + 1}] // Flatten (* Michael De Vlieger, Jan 19 2016 *)

Formula

a(n) = 2 * A086520(n+2).
a(2n) = 4*n + 2 times 4*n + 2 = 2, 2, 6, 6, 6, 6, 6, 6, 10,....
a(2n+1) = 4*(n+1) times 4*(n+1) = 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 12, ....
Previous Showing 21-30 of 32 results. Next

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