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.

A225760 Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT).

Original entry on oeis.org

2287674594, 983574906769, 16155706018465, 24267609913869, 72461523834219, 367110963344658, 473161567692022, 8504240238563547, 9271267603660839, 13796686490781630, 28200194168137420, 68964192934317607, 121927568913483970, 125247439852891719, 280877330289234924, 288885660249168850
Offset: 1

Views

Author

Frank M Jackson, May 15 2013

Keywords

Comments

A PPT can be drawn as a closed lattice polygon with the hypotenuse intersecting no lattice points other than at its start and end. Consequently the PPT is subject to Pick's theorem.

Examples

			a(1) = 2287674594 as it is the first count of internal lattice points within more than one PPT. It has (a, b) = (18108, 252685) and (28077, 162964).

Links

Crossrefs

Cf. A024407, A225414.

Programs

  • Mathematica
    getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getlist[j_] := (newlist=getpairs[j]; Table[(newlist[[m]][[1]]^2-newlist[[m]][[2]]^2-1) (2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, {m, 1, Length[newlist]}]); maxterms=4000; table=Sort[Flatten[Table[getlist[2p+1], {p, 1, 2maxterms}]]]; n=1; table1={}; While[n
  • PARI
    is(n)=my(b,s,N=2*n);fordiv(n>>valuation(n,2),a,if(gcd(b=N/a+1, a+1)==1 && issquare(b^2+(a+1)^2) && s++>1, return(1)));0 \\ Charles R Greathouse IV, May 15 2013

Formula

If integers a < b are the perpendicular sides of a PPT, then Pick's theorem gives the count of internal lattice points, I = (a-1)*(b-1)/2 and is comparable to the area, A = a*b/2.

Extensions

a(8) and beyond from Frank A. Stevenson, Nov 29 2023

A226028 Array T(j,k) of counts of internal lattice points within all Pythagorean triangles (see comments for array order).

Original entry on oeis.org

3, 22, 17, 49, 103, 43, 69, 217, 244, 81, 156, 305, 505, 445, 131, 187, 671, 709, 913, 706, 193, 190, 793, 1546, 1281, 1441, 1027, 267, 295, 799, 1819, 2781, 2021, 2089, 1408, 353, 465, 1249, 1828, 3265, 4376, 2929, 2857, 1849, 451, 498, 1937, 2863, 3277, 5131, 6331, 4005, 3745, 2350, 561
Offset: 1

Views

Author

Frank M Jackson, May 23 2013

Keywords

Comments

The array of counts of internal lattice points within all Pythagorean triangles T(j,k) is arranged so that its first column is the ordered counts of internal lattice points within the k-th primitive Pythagorean triangle (PPT) A225414(k) and the j-th column is j multiples of these PPT side lengths.
Let the k-th PPT have integer perpendicular sides a, b then its j-th multiple has area A = j^2*a*b/2 and the count of lattice points intersected by its boundary is B = j*(a+b+1) by the application of Pick's theorem the count of internal lattice points within it is I = (j^2*a*b-j*(a+b+1)+2)/2.

Examples

			Array begins
    3,   17,  43,  81, 131, ...
   22,  103, 244, 445, ...
   49,  217, 505, ...
   69,  305, ...
  156, ...

Links

Crossrefs

Cf. A126587 (first row), A225414 (first column).

Programs

  • Mathematica
    getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getpptpairs[j_] := (newlist=getpairs[j]; Table[{(newlist[[m]][[1]]^2-newlist[[m]][[2]]^2-1)(2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, newlist[[m]][[1]]^2-newlist[[m]][[2]]^2, 2newlist[[m]][[1]]*newlist[[m]][[2]]}, {m, 1, Length[newlist]}]); lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@ IntegerPartitions[#1 +dim-1, {dim}], 1] &, maxHeight], 1]; array[{x_, y_}] := (pptpair=table[[y]]; (x^2*pptpair[[2]]*pptpair[[3]])/2-x(pptpair[[2]]+pptpair[[3]]+1)/2+1); maxterms=20; table=Sort[Flatten[Table[getpptpairs[2p+1], {p, 1, maxterms}], 1]][[1;;maxterms]]; pairs=lexicographicLattice[{2, maxterms}]; Table[array[pairs[[n]]], {n, 1, maxterms(maxterms+1)/2}]

A226575 Ordered excesses of internal lattice point counts of scaled up primitive Pythagorean triangles (PPT's) (see comments).

Original entry on oeis.org

4, 24, 48, 72, 160, 168, 180, 300, 448, 504, 520, 768, 784, 900, 1080, 1152, 1176, 1320, 1584, 1620, 1920, 2200, 2232, 2268, 2548, 2904, 3108, 3744, 3784, 3808, 3840, 4416, 4680, 4732, 5508, 5880, 5880, 5928, 6624, 6720, 6732, 7600, 8568, 8760, 9280, 9900
Offset: 1

Views

Author

Frank M Jackson, Jun 12 2013

Keywords

Comments

Every PPT with perpendicular legs a, b and hypotenuse c can be scaled up by the value of its hypotenuse to form a lattice triangle in two configurations. The first is where the scaled perpendicular legs a*c and b*c lie parallel to the coordinate axes. The second is where only the scaled hypotenuse c*c lies parallel to one coordinate axis. a(n) is the excess of internal lattice point counts of the second config. over the first and n is the ordered occurrence. There are multiple occurrences of this excess for different scaled PPT's. a(n) == 0 (mod 4).

Examples

			a(6) = 168 as the PPT (20,21,29) when scaled by 29 to (580,609,841) has a lattice point count of 176002 (config. 1) and 176170 (config. 2). Hence E = 168 and it is the 6th occurrence.

Links

Crossrefs

Cf. A225414, A226028.

Programs

  • Mathematica
    getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getlist[j_] := (newlist=getpairs[j]; Table[(newlist[[m]][[1]]^2+newlist[[m]][[2]]^2-1)(newlist[[m]][[1]]-newlist[[m]][[2]])(newlist[[m]][[2]]), {m, 1, Length[newlist]}]); maxterms=10; table=Sort@Flatten@Table[getlist[2p+1], {p, 1, maxterms}][[1;;maxterms]]

Formula

For config. 1 the internal lattice count I = (c^2*a*b-c*(a+b+1)+2)/2. For config. 2 the internal lattice count I = (c^2*a*b-(a+b+c^2)+2)/2. So the excess of config. 2 over 1 is E = (c-1)*(a+b-c)/2.
Showing 1-3 of 3 results.

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