%I A033954
%S A033954 0,7,22,45,76,115,162,217,280,351,430,517,612,715,826,945,1072,1207,
%T A033954 1350,1501,1660,1827,2002,2185,2376,2575,2782,2997,3220,3451,3690,3937,
%U A033954 4192,4455,4726,5005,5292,5587,5890,6201,6520,6847,7182,7525,7876,8235
%N A033954 n*(4*n+3). Also, second 10-gonal (or decagonal) numbers.
%C A033954 Write 0,1,2,... in clockwise spiral; sequence gives numbers on positive 
               x axis.
%C A033954 The equations 1 + 2 = 3 and 3^2 + 4^2 = 5^2 set the stage for considering 
               whether it is also true that 5^3 + 6^3 = 7^3 and 7^4 + 8^4 = 9^4. 
               Reflecting on Fermat's Last Theorem or resorting to a calculator 
               dispels any hope that either of the two equations could be correct. 
               However, it is true that 5^3 + 6^3 + 2 = 7^3 and 7^4 + 8^4 + 64 = 
               9^4. More significantly, each of these equations is the first of 
               an infinite sequence of equations featuring consecutive integers 
               that conform to the spirit of the equations mentioned in A000384. 
               For n>0, a(n)^4+(a(n)+1)^4 +...+(a(n)+n)^4 +(4*A000217(n))^3 = (a(n)+n+1)^4+...+(a(n)+2n)^4; 
               e.g., 7^4+8^4+(4*1)^3=9^4; 22^4+23^4+24^4+(4*3)^3=25^4+26^4; see 
               also 045944 - Charlie Marion (charliemath(AT)optonline.net), Dec 
               8 2007
%D A033954 S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 
               1998) 188; 30 (#4, 1999-2000), 246-250.
%D A033954 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 2nd ed., 1994, p. 99.
%H A033954 Emilio Apricena, <a href="a035608.png">A version of the Ulam spiral</
               a>
%F A033954 G.f.: x(7+x)/(1-x)^3. - Michael Somos, Mar 03 2003
%F A033954 a(n)=8*n+a(n-1)-9 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 12 2009]
%e A033954 16 17 18 19 ...
%e A033954 15 4 5 6 ...
%e A033954 14 3 0 7 ...
%e A033954 13 2 1 8 ...
%e A033954 For n=2, a(2)=8*2+0-9=7; n=3, a(3=8*3+7-9=22; n=4, a(4)=8*4+22-9=45 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
%t A033954 s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 8}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 16 2008]
%o A033954 (PARI) a(n)=4*n^2+3*n
%Y A033954 Same as A033951 except start at 0. Cf. A002943.
%Y A033954 a(n)=A001107(-n)=A074377(2n).
%Y A033954 Sequences from spirals: A001107, A002939, A007742, A033951, A033952, 
               A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y A033954 Cf. A002620.
%Y A033954 Sequence in context: A031053 A063130 A171441 this_sequence A159227 A081274 
               A038764
%Y A033954 Adjacent sequences: A033951 A033952 A033953 this_sequence A033955 A033956 
               A033957
%K A033954 nonn,easy
%O A033954 0,2
%A A033954 N. J. A. Sloane (njas(AT)research.att.com).