A141575 A gap prime-type triangular sequence of coefficients: gap(n)=Prime[n+1]-Prime[n]; t(n,m)=If[n == m == 0, 1, If[m == 0, ((Prime[n] + gap[n])^ n + (Prime[n] - gap[n])^n)/2, ((Prime[n] + gap[n]*Sqrt[Prime[m]])^n + (Prime[n] - gap[n]*Sqrt[Prime[m]])^n)/2]].

1, 2, 2, 13, 17, 21, 185, 245, 305, 425, 7361, 12833, 18817, 32321, 47873, 215171, 271051, 328691, 449251, 576851, 853171, 12334505, 21164697, 31341961, 55836009, 86013257, 164203785, 212610281, 532365557, 659940697, 793109789, 1076412613

Offset: 1

Author

Roger L. Bagula, Aug 18 2008

Keywords

• nonn
• tabl
• uned

Comments

General Lucas-like Binet sequences

where Prime[m]starts at 1:

a(n)=((Prime[n]+gap[n]*Sqrt[Prime[m])^n+(Prime[n]-gap[n]*Sqrt[Prime[m])^n)/2.

Row sums are:

{1, 4, 51, 1160, 119205, 2694186, 583504495, 12222749556, 4868938911913,

3621654266405174, 21636046625243691}

Examples

			{1},
{2, 2},
{13, 17, 21},
{185, 245, 305, 425},
{7361, 12833, 18817, 32321, 47873},
{215171, 271051, 328691, 449251, 576851, 853171},
{12334505, 21164697, 31341961, 55836009, 86013257, 164203785, 212610281},
{532365557, 659940697, 793109789, 1076412613, 1382639597, 2065328317, 2442521189, 3270431797},
{40436937953, 68810349217, 102354570337, 185966400481, 293310073697, 587469359713, 778486092257, 1259085279457, 1553019848801},
{7312866926183, 15217609281335, 25813998655559, 56317915837223,
101380456546055, 246072307427783, 351480840333479, 643872497781095,
837435900955463, 1336749872660999}, {512759709537725, 608866569299409,
709085196658213, 922088454409101, 1152233212894709, 1665820807145925,
1950209769575213, 2576571400365309, 2919512658836837, 3667365684348213,
4951533162173037}
		

Crossrefs

Cf. A011943, A081336, A034478.

Programs

Mathematica

gap[n_] := Prime[n + 1] - Prime[n]; t[n_, m_] := If[n == m == 0, 1, If[m == 0, ((Prime[n] + gap[n])^n + (Prime[n] - gap[n])^n)/2, ((Prime[n] + gap[n]*Sqrt[Prime[m]])^n + (Prime[n] - gap[n]*Sqrt[Prime[m]])^n)/2]]; Table[Table[FullSimplify[t[n, m]], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

gap(n)=Prime[n+1]-Prime[n]; t(n,m)=If[n == m == 0, 1, If[m == 0, ((Prime[n] + gap[n])^ n + (Prime[n] - gap[n])^n)/2, ((Prime[n] + gap[n]*Sqrt[Prime[m]])^n + (Prime[n] - gap[n]*Sqrt[Prime[m]])^n)/2]].