A048922
Indices of 9-gonal numbers which are also octagonal.
Original entry on oeis.org
1, 425, 286209, 192904201, 130017145025, 87631362842409, 59063408538638401, 39808649723679439625, 26830970850351403668609, 18084034544487122393202601, 12188612452013470141614884225
Offset: 1
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I:=[1, 425, 286209]; [n le 3 select I[n] else 675*Self(n-1)-675*Self(n-2)+1*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Dec 23 2011
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LinearRecurrence[{675,-675,1},{1,425,286209},30] (* Vincenzo Librandi, Dec 23 2011 *)
Join[{1},Transpose[NestList[{Last[#],674Last[#]-First[#]-240}&, {1,425}, 10]][[2]]] (* Harvey P. Dale, Feb 05 2012 *)
A048923
Indices of octagonal numbers which are also 9-gonal.
Original entry on oeis.org
1, 459, 309141, 208360351, 140434567209, 94652689938291, 63795772583840701, 42998256068818693959, 28980760794611215887441, 19532989777311890689441051, 13165206129147419713467380709
Offset: 1
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I:=[1, 459, 309141]; [n le 3 select I[n] else 675*Self(n-1)-675*Self(n-2)+Self(n-3): n in [1..15]]; // Vincenzo Librandi, Dec 24 2011
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LinearRecurrence[{675,-675,1},{1,459,309141},30] (* Vincenzo Librandi, Dec 24 2011 *)
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
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).
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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]
A378245
Numbers that are both k-gonal and (k+1)-gonal for some k >= 3.
Original entry on oeis.org
1, 36, 1225, 9801, 40755, 41616, 121771, 297045, 631125, 1212751, 1413721, 2158695, 3617601, 5773825, 8851275, 13117251, 18886285, 26523981, 36450855, 48024900, 49146175, 65151801, 85076025, 94109401, 109597411, 139468635, 175520325, 218664901, 269900415, 330314391
Offset: 1
a(2) = 36 is both the 8th triangular and the 6th square number.
a(3) = 1225 is both the 49th triangular and the 35th square number.
a(5) = 40755 is both the 165th pentagonal number and the 143th hexagonal number.
The subdiagonal of
A189216 is also a subsequence.
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upto(limit) = my(terms=List(1)); for(k=3, oo, my(found=0); for(n=2, oo, my(a = (2*n - 1)^2, b = (4*n*(3*n - 5) + 6), c = (8*(n-1)^2 + 1), s = (a*k^2 - b*k + c), v = n * (n*k - k - 2*n + 4) / 2); if(issquare(s), my(t = sqrtint(s) + k - 3); if(t % (2*(k-1)) == 0, listput(terms, v); found += 1)); if(v >= limit, break)); if(found == 0, break)); Vec(vecsort(terms)); \\ Daniel Suteu, Dec 08 2024
Showing 1-4 of 4 results.
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