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A220755 Numbers n such that n^2 + n(n+1)/2 is an oblong number (A002378).

%I A220755 #43 Jun 13 2015 00:54:37
%S A220755 0,1,28,117,2760,11481,270468,1125037,26503120,110242161,2597035308,
%T A220755 10802606757,254482957080,1058545220041,24936732758548,
%U A220755 103726628957277,2443545327380640,10164151092593121,239442505350544188,995983080445168597,23462921979025949800
%N A220755 Numbers n such that n^2 + n(n+1)/2 is an oblong number (A002378).
%C A220755 Numbers n such that 6*n^2 + 2*n + 1 is a square. - _Joerg Arndt_, Apr 14 2013
%C A220755 a(n+4) - a(n) is divisible by 40. (a(n+2) - a(n)) mod 10 = period 4: repeat 8, 6, 2, 4. See A000689. - _Paul Curtz_, Apr 15 2013
%C A220755 For this 5 consecutive terms recurrence,the main (or principal) sequence is: CRR(n)= 0, 0, 0, 0, 1, 1, 99, 99, 9702, 9702,... . - _Paul Curtz_, Apr 16 2013
%C A220755 Also numbers n such that the sum of the octagonal numbers N(n) and N(n+1) is equal to the sum of two consecutive triangular numbers. - _Colin Barker_, Dec 09 2014
%H A220755 Vincenzo Librandi, <a href="/A220755/b220755.txt">Table of n, a(n) for n = 1..1000</a>
%H A220755 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,98,-98,-1,1).
%F A220755 G.f.: x^2 * (1+27*x-9*x^2-3*x^3) / ( (1-x)*(1-10*x+x^2)*(1+10*x+x^2) ). - _Giovanni Resta_, Apr 14 2013, adapted by _Vincenzo Librandi_ Aug 13 2014
%F A220755 a(n) = ((-(-1)^n+sqrt(6))*(5+2*sqrt(6))^(n-1)-((-1)^n+sqrt(6))*(5-2*sqrt(6))^(n-1)-2)/12. - _Bruno Berselli_, Apr 14 2013
%F A220755 a(n) = a(n-1) + 98*a(n-2) - 98*a(n-3) - a(n-4) + a(n-5).
%t A220755 LinearRecurrence[{1, 98, -98, -1, 1}, {0, 1, 28, 117, 2760}, 30] (* _Giovanni Resta_, Apr 14 2013 *)
%t A220755 CoefficientList[Series[x (1 + 27 x - 9 x^2 - 3 x^3)/((1 - x) (1 - 10 x + x^2) (1 + 10 x + x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 13 2014 *)
%o A220755 (C)
%o A220755 #include <stdio.h>
%o A220755 typedef unsigned long long U64;
%o A220755 U64 rootPronic(U64 a) {
%o A220755     U64 sr = 1L<<31, s, b;
%o A220755     while (a < sr*(sr+1))  sr>>=1;
%o A220755     for (b = sr>>1; b; b>>=1) {
%o A220755             s = sr+b;
%o A220755             if (a >= s*(s+1))  sr = s;
%o A220755     }
%o A220755     return sr;
%o A220755 }
%o A220755 int main() {
%o A220755   U64 a, n, r, t;
%o A220755   for (n=0; n < 3L<<30; n++) {
%o A220755     a = n*(n+1)/2 + n*n;
%o A220755     t = rootPronic(a);
%o A220755     if (a == t*(t+1)) {
%o A220755         printf("%llu\n", n);
%o A220755     }
%o A220755   }
%o A220755 }
%o A220755 (PARI) concat([0], Vec( x * (1+27*x-9*x^2-3*x^3) / ( (1-x)*(1-10*x+x^2)*(1+10*x+x^2) ) + O(x^66) ) )  /* _Joerg Arndt_, Apr 14 2013 */
%o A220755 (Maxima) makelist(expand(((-(-1)^n+sqrt(6))*(5+2*sqrt(6))^(n-1)-((-1)^n+sqrt(6))*(5-2*sqrt(6))^(n-1)-2)/12), n, 1, 25); /* _Bruno Berselli_, Apr 14 2013 */
%Y A220755 Cf. A000217, A005449 (n^2 + n(n+1)/2).
%Y A220755 Cf. A011916 (numbers n>=0 such that n^2 + n(n+1)/2 is a triangular number).
%Y A220755 Cf. A220186 (numbers n>=0 such that n^2 + n(n+1)/2 is a square).
%Y A220755 Cf. A220185 (numbers n>=0 such that n^2 + n(n+1) is an oblong number).
%Y A220755 (Example of a family of main sequences: A131577, A024495, A000749, A139761. )
%Y A220755 Cf. A251793.
%K A220755 nonn,easy
%O A220755 1,3
%A A220755 _Alex Ratushnyak_, Apr 13 2013
%E A220755 a(11)-a(21) from _Giovanni Resta_, Apr 14 2013