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 A278310 Numbers m such that T(m) + 3*T(m+1) is a square, where T = A000217. 7
 3, 143, 4899, 166463, 5654883, 192099599, 6525731523, 221682772223, 7530688524099, 255821727047183, 8690408031080163, 295218051329678399, 10028723337177985443, 340681375412721826703, 11573138040695364122499, 393146012008229658338303, 13355391270239113019379843 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Equivalently, both m+1 and 2*m+3 are squares for nonnegative m. Corresponding triangular numbers T(m): 6, 10296, 12002550, 13855048416, 15988853699286, 18451128064030200, 21292585958400815526, ... Square roots of T(m) + 3*T(m+1) are listed by A082405 (after 0). Negative values of m for which T(m) + 3*T(m+1) is a square: -1, -2, -26, -842, -28562, -970226, -32959082, ... LINKS Colin Barker, Table of n, a(n) for n = 1..650 Index entries for linear recurrences with constant coefficients, signature (35,-35,1). FORMULA O.g.f.: x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)). E.g.f.: (exp((1-sqrt(2))^4*x) + exp((1+sqrt(2))^4*x) - 10*exp(x))/8 + 1. a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3. a(n) = 34*a(n-1) - a(n-2) + 40 for n>2. a(n) = a(-n) = ((1 + sqrt(2))^(4*n) + (1 - sqrt(2))^(4*n))/8 - 5/4. a(n) = 4*A001109(n)^2 - 1. a(n) = -A029546(n) + 38*A029546(n-1) + 3*A029546(n-2) for n>1. Lim_{n -> infinity} a(n)/a(n-1) = A156164. Floor(sqrt(a(n))) = A182189(n) - 2. a(n) - a(n-1) = 4*A046176(n) for n>1. EXAMPLE 3 is in the sequence because T(3) + 3*T(4) = 6 + 3*10 = 6^2. For n=5 is a(5) = 5654883, therefore floor(sqrt(5654883)) = 2377 = A182189(5) - 2 = 2379 - 2. MAPLE P:=proc(q) local n; for n from 3 to q do if type(sqrt(2*n^2+5*n+3), integer) then print(n); fi; od; end: P(10^9); # Paolo P. Lava, Nov 18 2016 MATHEMATICA Table[((1 + Sqrt[2])^(4 n) + (1 - Sqrt[2])^(4 n))/8 - 5/4, {n, 1, 20}] RecurrenceTable[{a[1] == 3, a[2] == 143, a[n] == 34 a[n - 1] - a[n - 2] + 40}, a, {n, 1, 20}] LinearRecurrence[{35, -35, 1}, {3, 143, 4899}, 50] (* G. C. Greubel, Nov 20 2016 *) PROG (Sage) def A278310():     a, b = 3, 143     yield a     while True:         yield b         a, b = b, 34*b - a + 40 a = A278310(); print([next(a) for _ in range(18)]) # Peter Luschny, Nov 18 2016 (PARI) Vec(x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)) + O(x^50)) \\ G. C. Greubel, Nov 20 2016 (MAGMA) Iv:=[3, 143]; [n le 2 select Iv[n] else 34*Self(n-1)-Self(n-2)+40: n in [1..20]]; CROSSREFS Subsequence of A000466. Cf. A000217, A001109, A029546, A046176, A082405, A156164, A182189. Cf. A278438: numbers m such that T(m) + 2*T(m+1) is a square. Cf. A078522: numbers m such that 3*T(m) + T(m+1) is a square. Cf. similar sequences with closed form ((1 + sqrt(2))^(4*r) + (1 - sqrt(2))^(4*r))/8 + k/4: A084703 (k=-1), A076218 (k=3), this sequence (k=-5). Sequence in context: A195937 A162363 A102965 * A195936 A323998 A172145 Adjacent sequences:  A278307 A278308 A278309 * A278311 A278312 A278313 KEYWORD nonn,easy,changed AUTHOR Bruno Berselli, Nov 17 2016 STATUS approved

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Last modified February 28 23:19 EST 2020. Contains 332353 sequences. (Running on oeis4.)