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 A274979 #86 Mar 11 2025 12:29:53
%S A274979 0,1,15,18,46,51,93,100,156,165,235,246,330,343,441,456,568,585,711,
%T A274979 730,870,891,1045,1068,1236,1261,1443,1470,1666,1695,1905,1936,2160,
%U A274979 2193,2431,2466,2718,2755,3021,3060,3340,3381,3675,3718,4026,4071,4393,4440,4776,4825
%N A274979 Integers of the form m*(m + 7)/8.
%C A274979 Nonnegative values of m are listed in A047393.
%C A274979 Also, numbers h such that 32*h + 49 is a square.
%C A274979 Equivalently, numbers of the form i*(8*i + 7) with i = 0, -1, 1, -2, 2, -3, 3, ...
%C A274979 Infinitely many squares belong to this sequence.
%C A274979 The first bisection is A139278, and 0 followed by the second bisection gives A051870.
%C A274979 Generalized 18-gonal (or octadecagonal) numbers (see the third comment). - _Omar E. Pol_, Jun 06 2018
%C A274979 Partial sums of A317314. - _Omar E. Pol_, Jul 28 2018
%C A274979 Exponents in expansion of Product_{n >= 1} (1 + x^(16*n-15))*(1 + x^(16*n-1))*(1 - x^(16*n)) = 1 + x + x^15 + x^18 + x^46 + .... - _Peter Bala_, Dec 10 2020
%C A274979 Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. They are also the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, k >= 5. In this case k = 18. - _Omar E. Pol_, Apr 25 2021
%H A274979 Bruno Berselli, <a href="/A274979/b274979.txt">Table of n, a(n) for n = 1..1000</a>
%H A274979 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A274979 O.g.f.: x^2*(1 + 14*x + x^2)/((1 + x)^2*(1 - x)^3).
%F A274979 E.g.f.: (3*(2*x + 1)*exp(-x) + (8*x^2 - 3)*exp(x))/4.
%F A274979 a(n) = (8*(n-1)*n - 3*(2*n-1)*(-1)^n - 3)/4.
%F A274979 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 6. - _Wesley Ivan Hurt_, Dec 18 2020
%F A274979 From _Amiram Eldar_, Feb 28 2022: (Start)
%F A274979 Sum_{n>=2} 1/a(n) = (8 + 7*(sqrt(2)+1)*Pi)/49.
%F A274979 Sum_{n>=2} (-1)^n/a(n) = 8*log(2)/7 + 2*sqrt(2)*log(sqrt(2)+1)/7 - 8/49. (End)
%F A274979 a(n) = (n-1)*(4*n+3)/2 if n is odd and a(n) = n*(4*n-7)/2 if n is even. - _Chai Wah Wu_, Mar 11 2025
%e A274979 100 is in the sequence because 100 = 25*(25+7)/8 or also 100 = 4*(8*4-7).
%e A274979 From _Omar E. Pol_, Apr 24 2021: (Start)
%e A274979 Illustration of initial terms as vertices of a rectangular spiral:
%e A274979 46_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _18
%e A274979 | |
%e A274979 | 0 |
%e A274979 | |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A274979 | 1 15
%e A274979 |
%e A274979 51
%e A274979 More generally, all generalized k-gonal numbers can be represented with this kind of spirals, k >= 5. In this case k = 18. (End)
%t A274979 Select[m = Range[0, 200]; m (m + 7)/8, IntegerQ] (* _Jean-François Alcover_, Jul 21 2016 *)
%t A274979 Select[Table[(m(m+7))/8,{m,0,200}],IntegerQ] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,1,15,18,46},50] (* _Harvey P. Dale_, May 07 2019 *)
%o A274979 (Sage)
%o A274979 def A274979_list(len):
%o A274979 h = lambda m: m*(m+7)/8
%o A274979 return [h(m) for m in (0..len) if h(m) in ZZ]
%o A274979 print(A274979_list(199)) # _Peter Luschny_, Jul 18 2016
%o A274979 (Magma) [t: m in [0..200] | IsIntegral(t) where t is m*(m+7)/8];
%o A274979 (Python)
%o A274979 def A274979(n): return (n>>1)*((n<<2)+(3 if n&1 else -7)) # _Chai Wah Wu_, Mar 11 2025
%Y A274979 Cf. sequences of the form m*(m+k)/(k+1) listed in A274978.
%Y A274979 Cf. similar sequences listed in A299645.
%Y A274979 Cf. A317314.
%Y A274979 Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), this sequence (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
%K A274979 nonn,easy
%O A274979 1,3
%A A274979 _Bruno Berselli_, Jul 15 2016