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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]].

%I A141575 #2 Mar 30 2012 17:34:26
%S A141575 1,2,2,13,17,21,185,245,305,425,7361,12833,18817,32321,47873,215171,
%T A141575 271051,328691,449251,576851,853171,12334505,21164697,31341961,
%U A141575 55836009,86013257,164203785,212610281,532365557,659940697,793109789,1076412613
%N 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]].
%C A141575 General Lucas-like Binet sequences
%C A141575 where Prime[m]starts at 1:
%C A141575 a(n)=((Prime[n]+gap[n]*Sqrt[Prime[m])^n+(Prime[n]-gap[n]*Sqrt[Prime[m])^n)/2.
%C A141575 Row sums are:
%C A141575 {1, 4, 51, 1160, 119205, 2694186, 583504495, 12222749556, 4868938911913,
%C A141575 3621654266405174, 21636046625243691}
%F A141575 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]].
%e A141575 {1},
%e A141575 {2, 2},
%e A141575 {13, 17, 21},
%e A141575 {185, 245, 305, 425},
%e A141575 {7361, 12833, 18817, 32321, 47873},
%e A141575 {215171, 271051, 328691, 449251, 576851, 853171},
%e A141575 {12334505, 21164697, 31341961, 55836009, 86013257, 164203785, 212610281},
%e A141575 {532365557, 659940697, 793109789, 1076412613, 1382639597, 2065328317, 2442521189, 3270431797},
%e A141575 {40436937953, 68810349217, 102354570337, 185966400481, 293310073697, 587469359713, 778486092257, 1259085279457, 1553019848801},
%e A141575 {7312866926183, 15217609281335, 25813998655559, 56317915837223,
%e A141575 101380456546055, 246072307427783, 351480840333479, 643872497781095,
%e A141575 837435900955463, 1336749872660999}, {512759709537725, 608866569299409,
%e A141575 709085196658213, 922088454409101, 1152233212894709, 1665820807145925,
%e A141575 1950209769575213, 2576571400365309, 2919512658836837, 3667365684348213,
%e A141575 4951533162173037}
%t A141575 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[%]
%Y A141575 Cf. A011943, A081336, A034478.
%K A141575 nonn,tabl,uned
%O A141575 1,2
%A A141575 _Roger L. Bagula_, Aug 18 2008