A303295 -id:A303295 - cp's OEIS frontend

A217367 a(n) = ((n+7) / gcd(n+7,4)) * (n / gcd(n,4)).

0, 2, 9, 15, 11, 15, 39, 49, 30, 36, 85, 99, 57, 65, 147, 165, 92, 102, 225, 247, 135, 147, 319, 345, 186, 200, 429, 459, 245, 261, 555, 589, 312, 330, 697, 735, 387, 407, 855, 897, 470, 492, 1029, 1075, 561, 585, 1219, 1269, 660, 686, 1425, 1479, 767, 795, 1647

Offset: 0

Jean-François Alcover, Oct 01 2012

The 7th sequence (p=7) of the family A060819(n)*A060819(n+p).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Quasi-polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A060819, A181318.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(2+3*x+6*x^4-6*x^5+4*x^6-3*x^7)/(1-x+x^2-x^3)^3)); // G. C. Greubel, Sep 20 2018

a[n_] := n*(n+7)/(2*Mod[1 + Floor[n/2], 2] + 2); Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 01 2012 *)
CoefficientList[Series[x (2 + 3 x + 6 x^4 - 6 x^5 + 4 x^6 - 3 x^7) / (1 - x + x^2 - x^3)^3, {x, 0, 33}], x] (* Vincenzo Librandi, Jul 17 2013 *)

my(x='x+O('x^50)); Vec(x*(2+3*x+6*x^4-6*x^5+4*x^6-3*x^7)/(1-x+x^2-x^3)^3) \\ G. C. Greubel, Sep 20 2018

a(n) = n*(n+7)/(2*mod(1 + floor(n/2), 2) + 2).
G.f.: x*(2 + 3*x + 6*x^4 - 6*x^5 + 4*x^6 - 3*x^7)/(1 - x + x^2 - x^3)^3.
From Peter Bala, Aug 07 2022: (Start)
a(n) = numerator of n*(n+7)/4.
a(n) is quasi-polynomial in n: if p(n) = n*(n+7)/4 then a(4*n) = p(4*n), a(4*n+1) = p(4*n+1), a(4*n+2) = 2*p(4*n+2) and a(4*n+3) = 2*p(4*n+3) = A303295(n+1) for n >= 1. (End)
Sum_{n>=1} 1/a(n) = 697/735 + Pi/14. - Amiram Eldar, Aug 16 2022

A342300 Least nonnegative number greater than the previous number which is simultaneously an n-gonal and (n+1)-gonal number.

Original entry on oeis.org

0, 1, 3, 36, 9801, 40755, 121771, 297045, 631125, 1212751, 2158695, 3617601, 5773825, 8851275, 13117251, 18886285, 26523981, 36450855, 49146175, 65151801, 85076025, 109597411, 139468635, 175520325, 218664901, 269900415, 330314391, 401087665, 483498225, 578925051, 688851955, 814871421

Offset: 0

Robert G. Wilson v, Jun 04 2021

Also the least nontrivial number simultaneously an n and (n+1)-gonal number for n greater than one.
0 and 1 are always terms of any sequence of polygonal numbers of a particular rank beginning with index 0.
Since the formula for the k-th n-gonal number P(n,k) is k*(4+k*(n-2)-n)/2, one can extrapolate for the non-geometrical terms 0, 1 and 2.
Indices of the n and (n+1)-gonal numbers by pairs: {0, 0} {1, 1}, {3, 2}, {8, 6}, {99, 81}, {165, 143}, {247, 221}, {345, 315}, {459, 425}, {589, 551}, {735, 693}, {897, 851} ..., .
{x, y} of the above are {8n^2 + 10n - 3, 8n^2 - 10n - 7} for n>3 (A303295).
In the first 1000 terms, 1 is congruent to 0 (mod 6), 333 are congruent to 1 (mod 6), and 666 are congruent to 3 (mod 6).

			a(3) is the least triangular and square number > 3, which is 36: A001110(2).
a(4) is the least square and pentagonal number > 36, which is 9801: A036353(2).

Index entries for linear recurrences with constant coefficients, signature (6,-15, 20,-15,6,-1).

Cf. A001110, A036353, A046180, A048903, A048906, A048924, A203627, A189216.

a[n_] := Intersection[ Table[ PolygonalNumber[n, i], {i, 2, 10000}], Table[ PolygonalNumber[n + 1, i], {i, 2, 10000}]][[1]]; a[0] = 0; a[1] = 1; Array[a, 30, 0] (* Or *)
a[n_] := a[n] = 6a[n - 1] -15a[n - 2] +20a[n - 3] -15a[n - 4] +6a[n - 5] -a[n - 6]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 36; a[4] = 9801; a[5] = 40755; a[6] = 121771; a[7] = 297045; a[8] = 631125; a[9] = 1212751; Array[a, 30, 0]

a(n) = 32n^5 - 112n^4 + 70n^3 + 93n^2 - 57n - 35 for n > 3; a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 36.

G.f.: x*(1 - 3*x + 33*x^2 + 9610*x^3 - 17556*x^4 + 23575*x^5 - 17753*x^6 + 7122*x^7 - 1189*x^8)/(1 - x)^6. - Stefano Spezia, Jun 08 2021

cp's OEIS Frontend

A217367 a(n) = ((n+7) / gcd(n+7,4)) * (n / gcd(n,4)).

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