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 1-3 of 3 results.

A218864 Numbers of the form 9*k^2 + 8*k, k an integer.

Original entry on oeis.org

0, 1, 17, 20, 52, 57, 105, 112, 176, 185, 265, 276, 372, 385, 497, 512, 640, 657, 801, 820, 980, 1001, 1177, 1200, 1392, 1417, 1625, 1652, 1876, 1905, 2145, 2176, 2432, 2465, 2737, 2772, 3060, 3097, 3401, 3440, 3760, 3801, 4137, 4180, 4532, 4577, 4945, 4992
Offset: 1

Views

Author

Jason Kimberley, Nov 08 2012

Keywords

Comments

Numbers m such that 9*m + 16 is a square. - Vincenzo Librandi, Apr 07 2013
Equivalently, integers of the form h*(h + 8)/9 (nonnegative values of h are listed in A090570). - Bruno Berselli, Jul 15 2016
Generalized 20-gonal (or icosagonal) numbers: r*(9*r - 8) with r = 0, +1, -1, +2, -2, +3, -3, ... - Omar E. Pol, Jun 06 2018
Partial sums of A317316. - Omar E. Pol, Jul 28 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(18*n-17))*(1 + x^(18*n-1))*(1 - x^(18*n)) = 1 + x + x^17 + x^20 + x^52 + .... - Peter Bala, Dec 10 2020

Crossrefs

Characteristic function is A205987.
Numbers of the form 9*m^2+k*m, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), A057780 (k=6), this sequence (k=8).
Cf. A074377 (numbers m such that 16*m+9 is a square).
Cf. A317316.
For similar sequences of numbers m such that 9*m+i is a square, see list in A266956.
Cf. sequences of the form m*(m+i)/(i+1) listed in A274978. [Bruno Berselli, Jul 25 2016]
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), this sequence (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Programs

  • Magma
    a:=func; [0]cat[a(n*m): m in [-1,1], n in [1..20]];
  • Mathematica
    Array[(18 # (# - 1) - 7 (-1)^#*(2 # - 1) - 7)/8 &, 48] (* or *)
    CoefficientList[Series[x (1 + 16 x + x^2)/((1 + x)^2*(1 - x)^3), {x, 0, 47}], x] (* Michael De Vlieger, Jun 06 2018 *)

Formula

a(n) = (18*n*(n - 1) - 7*(-1)^n*(2*n - 1) - 7)/8. - Bruno Berselli, Nov 13 2012
G.f.: x*(1 + 16*x + x^2)/((1 + x)^2*(1 - x)^3). - Bruno Berselli, Nov 14 2012
Sum_{n>=2} 1/a(n) = (9 + 8*Pi*cot(Pi/9))/64. - Amiram Eldar, Feb 28 2022

A195151 Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7
Offset: 0

Views

Author

Omar E. Pol, Sep 14 2011

Keywords

Comments

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.
All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

Examples

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...
		

Crossrefs

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

Programs

A317326 Multiples of 26 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).
Partial sums give the generalized 30-gonal numbers.
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
From Bruno Berselli, Jul 27 2018: (Start)
Also, this type of sequence is characterized by:
O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;
E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;
a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;
a(n) = (m/2)^((1 + (-1)^n)/2)*n;
a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

Crossrefs

Cf. A252994 and A005408 interleaved.
Column 26 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).
Cf. A316729.

Programs

  • Julia
    [13^div(1+(-1)^n,2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018
  • Mathematica
    Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

Formula

a(2*n) = 26*n, a(2*n+1) = 2*n + 1.
From Bruno Berselli, Jul 27 2018: (Start)
O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.
E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).
a(n) = -a(-n) = (7 + 6*(-1)^n)*n.
a(n) = 13^((1 + (-1)^n)/2)*n.
a(n) = 2*a(n-2) - a(n-4). (End)
Multiplicative with a(2^e) = 13*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(3-s)). - Amiram Eldar, Oct 26 2023
Showing 1-3 of 3 results.