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.

A051870 18-gonal (or octadecagonal) numbers: a(n) = n*(8*n-7).

Original entry on oeis.org

0, 1, 18, 51, 100, 165, 246, 343, 456, 585, 730, 891, 1068, 1261, 1470, 1695, 1936, 2193, 2466, 2755, 3060, 3381, 3718, 4071, 4440, 4825, 5226, 5643, 6076, 6525, 6990, 7471, 7968, 8481, 9010, 9555, 10116, 10693, 11286, 11895, 12520
Offset: 0

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Author

N. J. A. Sloane, Dec 15 1999

Keywords

Comments

Also, sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Apr 26 2008
This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - T. D. Noe, Apr 13 2011
Also sequence found by reading the line from 0, in the direction 0, 18, ... and the parallel line from 1, in the direction 1, 51, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, Jul 18 2012
Partial sums of 16n + 1 (with offset 0), compare A005570. - Jeremy Gardiner, Aug 04 2012
All x values for Diophantine equation 32*x + 49 = y^2 are given by this sequence and A139278. - Bruno Berselli, Nov 11 2014
This is also a star enneagonal number: a(n) = A001106(n) + 9*A000217(n-1). - Luciano Ancora, Mar 30 2015

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
  • Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing, 2012, page 6.

Links

Crossrefs

Cf. A014634, A014635, A033585, A033586, A033587, A035008, A069129, A085250, A129271-A129278, A139278.

Programs

Formula

G.f.: x*(1+15*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(n) = 16*n + a(n-1) - 15, with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 06 2010
a(16*a(n)+121*n+1) = a(16*a(n)+121*n) + a(16*n+1). - Vladimir Shevelev, Jan 24 2014
E.g.f.: (8*x^2 + x)*exp(x). - G. C. Greubel, Jul 18 2017
Sum_{n>=1} 1/a(n) = ((1+sqrt(2))*Pi + 2*sqrt(2)*arccoth(sqrt(2)) + 8*log(2))/14. - Amiram Eldar, Oct 20 2020
Product_{n>=2} (1 - 1/a(n)) = 8/9. - Amiram Eldar, Jan 22 2021

A129276 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk-k) in the squared q-factorial of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 8, 1, 1, 42, 106, 42, 1, 1, 241, 1558, 1558, 241, 1, 1, 1444, 23589, 53612, 23589, 1444, 1, 1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1, 1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2007

Keywords

Comments

Dual triangle is A129274.
Central terms form a bisection of A127728.

Examples

			Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 4 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, (42), 71, 96, (106), 96, 71, (42), 19, 6, (1)],
where the terms enclosed in parenthesis form row 4.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 8, 8, 1;
1, 42, 106, 42, 1;
1, 241, 1558, 1558, 241, 1;
1, 1444, 23589, 53612, 23589, 1444, 1;
1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1;
1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1; ...

Links

Crossrefs

Cf. A129277 (column 1), A129278 (column 2); A127728 (central terms), related triangles: A129274, A128564, A008302 (Mahonian numbers).

Programs

  • Mathematica
    faq[n_, q_] := Product[(1-q^k)/(1-q), {k, 1, n}]; t[0, 0] = t[1, 0] = t[1, 1] = 1; t[n_, k_] := SeriesCoefficient[faq[n, q]^2, {q, 0, (n-1)*k}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 26 2013 *)
  • PARI
    T(n,k)=if(n==0,1,polcoeff(prod(i=1,n,(1-x^i)/(1-x))^2,(n-1)*k))

Formula

T(n,k) = [q^(nk-k)] Product_{i=1..n} { (1-q^i)/(1-q) }^2 for n>0, with T(0,0)=1.
Row sums = (n!)^2/(n-1) for n>=2.

A129277 Column 1 of triangle A129276; a(n) is the coefficient of q^n in the squared q-factorial of n+1.

Original entry on oeis.org

1, 2, 8, 42, 241, 1444, 8867, 55320, 349009, 2220242, 14215521, 91487834, 591285123, 3834960060, 24947236547, 162704291214, 1063516446543, 6965286759424, 45696734431169, 300262228345720, 1975679169075314
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2007

Keywords

Links

Crossrefs

Cf. A129276, A129278.

Programs

  • PARI
    a(n)=polcoeff(prod(i=1,n+1,(1-x^i)/(1-x))^2,n)

Formula

a(n) = [q^n] Product_{i=1..n+1} { (1-q^i)/(1-q) }^2.
Showing 1-3 of 3 results.

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

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