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.
%I A093722 #45 Dec 30 2024 10:51:00
%S A093722 0,1,3,7,8,14,20,29,31,42,52,66,69,85,99,118,122,143,161,185,190,216,
%T A093722 238,267,273,304,330,364,371,407,437,476,484,525,559,603,612,658,696,
%U A093722 745,755,806,848,902,913,969,1015,1074,1086,1147,1197,1261,1274,1340
%N A093722 Integers of the form (k^2 - 1) / 120.
%C A093722 This is "one-fifteenth of triangular numbers (integers only)". - _Vladimir Joseph Stephan Orlovsky_, Mar 04 2009
%C A093722 The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^n)/( (1 - q^(10*n-2))*(1 - q^(10*n-8)) ) = 1 - q - q^3 + q^7 + q^8 - q^14 - q^20 + + - - ... . - _Peter Bala_, Dec 26 2024
%H A093722 Harry J. Smith, <a href="/A093722/b093722.txt">Table of n, a(n) for n = 1..20000</a>
%H A093722 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F A093722 |A113430(n-1)| is the characteristic function of the numbers in A093722.
%F A093722 a(-1 - n) = a(n). a(n) = (A057538(n) * 2 - 1) / 120.
%F A093722 G.f.: -x^2*(1+2*x+4*x^2+x^3+4*x^4+x^6+2*x^5) / ( (1+x)^2*(x^2+1)^2*(x-1)^3 ). - _R. J. Mathar_, Jun 09 2013
%F A093722 From _Peter Bala_, Dec 26 2024: (Start)
%F A093722 a(n) is quasi-polynomial in n
%F A093722 a(4*n) = n*(15*n + 1)/2; a(4*n+1) = (3*n + 1)*(5*n + 2)/2;
%F A093722 a(4*n+2) = (3*n + 2)*(5*n + 3)/2; a(4*n+3) = (n + 1)*(15*n + 14)/2.
%F A093722 For 0 <= k <= 3, a(4*n+k) = (N_k(n)^2 - 1)/120, where N_0(n) = 30*n + 1, N_1(n) = 30*n + 11, N_2(n) = 30*n + 19 and N_3(n) = 30*n + 29. (End)
%p A093722 A093722 := proc(q) local n;
%p A093722 for n from 0 to q do
%p A093722 if type(sqrt(120*n+1), integer) then print(n);
%p A093722 fi; od; end:
%p A093722 A093722(1500); # _Peter Bala_, Dec 26 2024
%t A093722 Select[Table[(n^2-1)/120,{n,0,700}],IntegerQ] (* _Harvey P. Dale_, Nov 26 2010 *)
%o A093722 (PARI) {a(n) = (((n\4 * 3 + n%4) * 10 + (-1)^(n\2))^2 - 1) / 120 } /* _Michael Somos_, Oct 17 2006 */
%Y A093722 Cf. A002381, A057538, A113430, A214429, A379210.
%K A093722 nonn,easy
%O A093722 1,3
%A A093722 _Michael Somos_, Apr 13 2004
%E A093722 More terms from _Harvey P. Dale_, Nov 26 2010
%E A093722 Offset corrected to 1 by _Ray Chandler_, Jul 29 2019